Transport properties in multichannel systems · Transport properties in multichannel systems Scuola...

Transport properties in multichannel systems

Scuola di Dottorato in Scienze Astronomiche,

Chimiche, Fisiche e Matematiche “Vito Volterra”

Dottorato di Ricerca in Fisica – XXV Ciclo

Candidate

Laura Fanfarillo

ID number 697501

Thesis Advisor

Dr. Lara Benfatto,

Prof. Claudio Castellani

A thesis submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Physics

October 2012

Thesis not yet defended

Laura Fanfarillo. Transport properties in multichannel systems.

Ph.D. thesis. Sapienza – University of Rome© 2012

version: December 7, 2012

email: [email protected]

mailto:[email protected]

To my parents

Abstract

High Temperature Superconductors (HTSC), cuprates and pnictides, can be seen asprototypes of multichannel systems. The rich phase diagram reveals the coexistence(and/or competition) of several orders near the superconducting one. Many are theanomalies characterizing the so-called normal phase of these compounds. In thiswork we show that, by taking properly into account the interplay between severaldegrees of freedom, it is possible to explain some of the anomalous properties of thenormal state of HTSC.

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Contents

Introduction ix

1 The puzzling world of High Tc Superconductors 11.1 HTSC: Prototype of Multichannel Systems . . . . . . . . . . . . . . 11.2 Introduction to HTSC: Experimental Overview . . . . . . . . . . . . 2

1.2.1 General Properties and the phase diagram . . . . . . . . . . . 21.2.2 Phenomenology of the Underdoped Cuprates . . . . . . . . . 81.2.3 Anomalous Properties of the Normal State of Pnictides . . . 14

1.3 Open Theoretical Issues in the Physics of HTSC . . . . . . . . . . . 191.3.1 p-h and p-p Degrees of Freedom in Underdoped Cuprates . . 201.3.2 Hole and Electron Interacting Bands in Pnictides . . . . . . . 22

2 Quantum Field Theory Approach to Transport Problems 27

2.1 Transport by Linear Response Theory . . . . . . . . . . . . . . . . . 272.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.2 Kubo Formula for the conductivity . . . . . . . . . . . . . . . 30

2.2 ScF contribution to transport . . . . . . . . . . . . . . . . . . . . . . 342.2.1 Standard Theory of ScF . . . . . . . . . . . . . . . . . . . . . 342.2.2 Functional Integral Approach to ScF in multichannel systems 41

2.3 Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.3.1 Hall Transport in Kubo formalism . . . . . . . . . . . . . . . 462.3.2 Sign of RH and Character of the Carriers . . . . . . . . . . . 53

2.4 Vertex Corrections and Boltzmann Theory . . . . . . . . . . . . . . . 55

3 Multichannel Interaction Effects on ScF in Cuprates 59

3.1 Preliminary on Slave Boson Approach to the t-J model . . . . . . . . 593.2 Functional Integral Analysis . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.1 p-p and p-h Channel Decomposition of the Interaction . . . . 633.2.2 Mean-Field Results . . . . . . . . . . . . . . . . . . . . . . . . 643.2.3 Beyond Mean-Field . . . . . . . . . . . . . . . . . . . . . . . 66

4 Unconventional Hall Effect in Pnictides 774.1 A precedent: Unconventional Hall Effect in Cuprates . . . . . . . . . 774.2 Hall Transport in multiband systems . . . . . . . . . . . . . . . . . . 81

4.2.1 Generalization to the Multiband Systems: More is different . 814.3 Hall Effect in Pnictides . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3.2 Vertex corrections: Currents and Conductivities . . . . . . . 85

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4.3.3 Comparison with the experiments . . . . . . . . . . . . . . . 91

Conclusions 93

List of Acronyms 97

Bibliography 99

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Introduction

The two main families of High Temperature Superconductors (HTSC), cupratesand pnictides systems, show a common denominator: several degrees of freedom(spin, charge, orbital) are at play. From this point of view we can refer to theseas multichannel systems. The rich phase diagram represents the first signature ofthe presence of several kind of correlations, ranging from the Antiferromagnetic(AF) interaction in the undoped systems to the pairing one responsible for theSuperconducting (SC) behavior. Thus, while dealing with HTSC, it is crucial totake into account the presence of interactions in several channels and their inter-play, whose understanding still represents a theoretical challenge. Moreover, in thecase of pnictide systems the multichannel feature is related also to the their elec-tronic structure, characterized by several sheets of the Fermi Surface (FS). Thismakes the multiband character of superconductivity an unavoidable ingredient ofpnictides, that must be accounted for by any theoretical approach. During thePh.D. program my activity has been deeply oriented toward issues connected to thephysics of cuprates and pnictides HTSC seen as multichannel systems. In particularI focused on theoretical aspect of fluctuation contributions and transport propertiesof the "normal" state of these SC compounds. In this analysis, taking into accountthe presence of several degrees of freedom represents the key point in order to clarifymany of the anomalies experimentally observed in HTSC.

In the first part of this work I devoted my attention to the physics of cupratesystems. The discovery in 1986 of high-temperature superconductivity in cuprates[1] has been a turning point in the physics of the correlated electron systems. Inthese systems d-wave superconductivity appears when the parent compounds, thatare Mott insulators, are doped. While the searching of the pairing glue remainsa very important question, it appeared from the very beginning that an equallyimportant issue is the understanding of their complex phase diagram, originatingfrom the presence of interactions in several channels, whose relevance in differentphenomena is still debated. A typical example is provided by the depression in theDensity of States (DOS) observed in the normal phase of the underdoped cupratesand often referred to as the Pseudogap (PG). Nowadays no agreement has beenreached about its origin. The different scenarios which attempt to account for thisphenomenon may be separated into two classes [2]. A first class of models attributesthis feature to a precursor of pairing i.e. it considers the PG as originated bythe particle-particle (p-p) channel correlations. A second class of models insteadattributes the PG to correlations in the particle-hole (p-h) channel, involving bothcharge/spin fluctuations and current-current correlations.

The multichannel character of the interactions is also relevant within the context

ix

of the Superconducting Fluctuations (ScF) in cuprates. In particular in underdopedcuprates ScF should be relevant for transport and thermodynamics because of theirlow superfluid densities and short correlation lengths. Actually, such ScF havebeen widely highlighted in the PG region by several experimental measurements,ranging from diamagnetism [3, 4, 5, 6] and Nernst effect [4, 7] to paraconductivity[8, 9, 10]. In particular, the survival of a large Nernst signal up to temperaturesmuch larger than the SC transition temperature Tc in the underdoped region, hasbeen interpreted as the evidence for vortex-like phase fluctuations with a Kosterlitz-Thouless (KT) character due to the quasi-two-dimensional (2D) nature of the system[3, 4]. At the same time, several authors [8, 9] claimed that paraconductivity inunderdoped cuprates simply follows the T dependence expected for the quasi-2DAslamazov-Larkin (AL) regime of Gaussian Ginzburg-Landau (GL) ScF close toTc [11]. This outcome motivated various investigations in the attempt to explainthe experimental data on Nernst signal and diamagnetism within a GL framework[12, 13]. Interestingly, only recently [14] it has been noticed that, regardless of thespecific character (KT vs GL) of the ScF, there is an overall discrepancy in thestrength of the ScF contribution to the conductivity and diamagnetism, which isnot expected in conventional superconductors.

This quantitative disagreement between the ScF contribution to diamagnetismand conductivity could be related to the influence of p-h interactions on the ScFin a doped Mott insulator. This topic has been the subject of the first part of thePh.D. work. The functional integral method turns out to be a powerful tool inthis kind of analysis since it allows to take easily into account the interplay be-tween several degrees of freedom. We derived explicitly the ScF contribution todiamagnetism and conductivity above Tc and we demonstrated that their differentmagnitude is a direct outcome of the proximity to the Mott-insulator phase in thepresence of current-current interactions [15]. Our result gives a possible explanationof the suppression of the fluctuating contribution to the conductivity with respectto diamagnetism as experimentally observed in underdoped cuprates.

In the second part of the Ph.D. program I moved my attention to pnictidesystems. The discovery of superconductivity in iron-based superconductors [16]renewed the interest on the properties of interacting multiband systems. Pnictidesshare many similarities with cuprate superconductors, e.g., the layered structure,the proximity to a magnetic phase [17], the relatively large ratio between the SC gapand the critical temperature Tc [18, 19, 20], and the small superfluid density [17].Like in cuprates, the proximity of very different phases induces one to consider moreinteracting channels, moreover in pnictides the FS topology forces the analysis ofmultiband models. Indeed, all the families of pnictides are semimetals, with severalsmall hole and electron pockets at the Fermi level, originating from almost emptyhole and electron bands. Such a topology has been predicted by Local DensityApproximation (LDA) calculations and confirmed by several experiments sensitiveto the FS structure, as Angle-Resolved PhotoEmission Spectroscopy (ARPES) [19,20, 21, 22, 23] and de Haas-van Alphen (dHvA) magnetization measurements, [24,25, 26]. In addition, since the calculated electron-phonon coupling cannot accountfor the high values of Tc [27, 28], it has been suggested that the pairing glue couldbe provided by Spin Fluctuations (SpF) exchanged between electrons in differentbands [28, 29]. Thus, pnictides are expected to be somehow different from other

x

multiband superconductors (like MgB2) where the main coupling mechanism isintraband [30, 31].

This scenario raises interesting questions with regard to the appropriate de-scription of the physical properties of a multiband system dominated by interbandinteractions, both in the normal and SC state. A first issue emerges by the compari-son between the band structure predicted by LDA calculations and the one observedexperimentally. In fact, although an overall qualitative agreement is found, inter-esting discrepancies still remain, such as a substantial shrinking of the experimentalFermi areas determined by dHvA experiments with respect to the ones predictedby LDA. Such discrepancy can be accounted for assuming a shift of the real bandswith respect to LDA predictions. Notably, such shifts have systematically differentsigns in hole-like and electron-like bands and lead always to Fermi areas smallerthan expected in LDA. Recently it has been proposed that such a shrinking of theFS areas could be explained as an effect of the interband coupling between electronand hole bands mediated by low-energy SpF [32]. Indeed, by taking into accountwithin an Eliashberg analysis the strong p-h asymmetry of pnictides, one can showthat the self-energy corrections in each band acquire a finite real part, which givesrise to a shift of the bands. In particular, when interactions have a predominantinterband character, the sign of the shift agrees with the experimental observation,leading to a shrinking of the FS.

The FS shrinking is thus an example of the unconventional physics emergingfrom the interband coupling between bands having opposite (hole vs electron) char-acter. Another example of anomalous hole and electron mixing is the Hall Effect.Indeed the experiments on Hall transport in pnictides highlighted several anomaliesboth concerning the order of magnitude of the Hall coefficient RH and its temper-ature/doping dependence (see for example [33, 34]). Indeed, within a semiclassicalBoltzmann-like approach, RH is expected to be almost zero in slightly doped com-pounds, since in compensated metals the contribution of holes and electrons shouldcompensate each other, while from experiments it is found to have a large absolutevalue with a strong temperature dependence. These experimental outcomes makepresently the scenario of the transport properties of pnictides extremely anomalous,and prompt the need of a theoretical investigation in the framework of multibandapproach.

The analysis of the unconventional transport properties of the normal phase ofpnictides has represented the central topic of the second part of this Ph.D. work.Motivated by previous works on unconventional Hall effect in cuprates [35, 36],we analyzed the Hall transport by including the effects of AF vertex corrections.This can be done by generalizing the Eliashberg approach introduced in [32, 37], inwhich the SpF propagator is consider momentum-independent, to the momentum-dependent case. As we shall see the effect of AF vertex corrections is importantsince also a very small amounts of dopants lead to a strong renormalization of thebare velocities, with the dressed currents acquiring the character of the majoritycarrier. This effect is more pronounced at low temperature and low doping wherethe AF fluctuation are stronger. As a consequence an Hall coefficient with a largeabsolute value and a strong temperature dependence is found in agreement with theexperiments. Our description represents a convincing unified theoretical picture ableto explain, at the same time, the unusual features of the Hall transport in the wholephase diagram (temperature and doping dependence) concerning the normal phase

xi

of pnictides [38].Finally the multiband character and the interband nature of the interaction are

also crucial in the analysis of ScF in pnictides. This topic has been addresses in[39] at the very beginning of the iron-age of superconductivity. The more basicinformation that can be extracted, once the theoretical background is established,concerns the dimensionality of the system under consideration. For weak-coupledlayered materials the standard theory of ScF predicts the existence of a dimen-sional crossover near TC in the behavior of the ScF contribution to several physicalquantities (e.g. to the conductivity). Nonetheless the crossover temperature is de-termined by the degree of anisotropy of the system, so that for very anisotropic(quasi 2D) compounds the three-dimensional (3D) regime is never seen (too nearto Tc for being experimentally accessible). Recently a renaissance of interest occursin the analysis since several experimental probes seem to give opposite outcomesconcerning the degree of anisotropy (i.e. the dimensionality) of some pnictide sys-tems, such as LiFeAs. In the very last part of the Ph.D program we focused onthis topic. Preliminary results reveal that the estimate of the crossover temperaturein a multiband anisotropic system hides several subtleties that one needs to takeinto account. We will anticipate some preliminary results [40] in Chapter 2 in thecontext of the discussion of the ScF theory.

The plan of the Thesis is the following:

In Chapter 1, after an experimental overview of HTSC about some selectedtopics, we will point out some theoretical open issues that will be the subject of thefollowing Chapters.

In Chapter 2 we will introduce the theoretical tools used in this work. TheKubo formula for Direct Current (DC) and Hall conductivity will be derived. Aftera review of the main results of the standard theory of GL ScF, the functional integralapproach to the analysis of ScF will be discuss, also with connection to the analysisof multichannel systems.

In Chapter 3 the interplay between the p-h and the p-p degree of freedoms incuprates will be analyze in the framework of a slave boson approach of the t-Jmodel. The inclusion of current-current interactions, overlooked until now in theliterature, does not change the outcomes of a Mean-Field (MF) analysis, whilegiving relevant contributions at the Gaussian level in the analysis of fluctuations.The ScF contributions to conductivity and diamagnetism are explicitly computedalso in connection to recent experiments in underdoped cuprates.

In Chapter 4 we will analyze the unconventional transport properties found inthe normal phase of pnictides. The analysis of the unconventional Hall effect incuprates is presented as an interesting precedent in which vertex corrections areneeded in order to recover the experimental findings. The generalization of thegauge-invariant procedure in the calculus of DC and Hall transport to a multibandmodel is then carried on. Finally, the results in the analysis of a minimal two-bandmodel appropriate for pnictides are discussed.

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Chapter 1

The puzzling world of High Tc

Superconductors

In the first chapter we will describe the main features of Fe- and Cu-based HighTemperature Superconductors (HTSC) emphasizing similarities and differences be-tween these compounds. After reviewed the general properties of the HTSC systems,we will present the outcomes of some (selected) experiment on transport in the nor-mal phase of these compounds. From this complex experimental scenario emergemany theoretical open issues. We will present in details some aspects of these thatwill be the subject of the following Chapters.

1.1 HTSC: Prototype of Multichannel Systems

In many cases the physical properties of electrons in crystalline solids can be an-alyze in a good approximation by means of independent-particle models (nearly-free electron approximation). Within this approximation we forget about inter-actions between carriers so that the motion of electrons can be seen as that ofa single non-interacting electron in a periodic lattice of ions. The cooper oxidessystems are a remarkable case where interactions between electrons cannot be ne-glected. At composition for which conventional single-electron theories would pre-dict a good metal, these materials present instead an insulating behavior charac-terized by an Antiferromagnetic (AF) order. With a small chemical substitutionshigh-temperature superconductivity arises beside a plethora of new and anomalousphenomena.

At the beginning of the 2008 the family of the HTSC gained a new member, theiron-based class of superconductors [16]. The discovery of superconductivity in thisnew class of materials represented a turning point in the field of superconductivitysince, although the cuprates demonstrated that high temperature superconductivitywas possible, the iron-based materials prove that this phenomenon is not limited to asingle class of compounds. At the very beginning the analogies between iron-basedand cuprate superconductors suggested that a similar route to high-temperaturepairing could be at play in these two classes of materials. However, a large exper-imental evidence has been accumulated so far that significant differences betweenpnictides and cuprates are also important, starting from the very basic fact thatpnictides have a multiband structure.

1

2 1. The puzzling world of High Tc Superconductors

Even a cursory look at the phase diagram (Fig. 1.1) and the general proper-ties of these systems reveal an intricate interplay between several relevant degreesof freedom. In particular,while magnetism and superconductivity are antitheticalstates in elemental superconductors, after the discovery of pnictides, it seems notby chance to find this two phases one beside the other in the same phase diagram.This suggests that magnetism could be more a friend than a foe of superconduc-tivity in exotic superconductors. The anomalous properties of the normal stateare another common denominator of these systems. This is no surprising since theinterplay between several interacting channels leads unavoidably to the emergenceof several anomalies with respect to the metallic normal phase of a conventionalsuperconductor i.e. deviations from a Fermi Liquid (FL) behavior.

Figure 1.1. From [41]: Comparison between phase diagram of cuprates and pnictides.

The presence of several degrees of freedoms and the occurrence, and possibly thecompetition, of so many states of the matter, makes the comprehension of HTSCsystems a very hard task. The analysis of the interplay between the several interac-tion channels and the understanding of their influence on the physical observablesallow one to potentially isolate the essential degrees of freedom that must be takeninto account in the analysis of these compounds and could shed new light on thecomprehension of the high-temperature superconductivity mechanism.

1.2 Introduction to HTSC: Experimental Overview

1.2.1 General Properties and the phase diagram

Despite the wide variety of HTSC compounds, they all share a layered crystallo-graphic structure. Cuprate systems are made up of one or more copper-oxygenplanes separated by layers of other atoms (typically rare-earth) that act as chargereservoir (see Fig. 1.2). In Fe-bases superconductors we have a planar layer of ironatoms joined by tetrahedrally coordinated pnictogen (P, As) or chalcogen (S, Se, Te)anions arranged in a stacked sequence separated by alkali, or rare-earth and oxy-gen/fluorine blocking layers. In particular, the 1111 systems have alternating FeAsand RE(O,F) sheets (RE= Rare Earth= La,Nd,Ce,etc.), the 122 systems have twoFeAs layers sandwiched between A (=Ba,Sr,K) layers, while the 11 system (LiFeAsand FeSe) present a very elementary structure without filler species (see Fig. 1.4)

1.2 Introduction to HTSC: Experimental Overview 3

Figure 1.2. From [42]: Crystal structure of La2−xSrxCuO4. The basic building block isa perovskite structure consisting of the CuO6 octahedron seen in the center of the unitcell, surrounded by eight La ions. The octahedra are elongated, giving rise to a layeredstructure with quasi-2D CuO2 sheets, which are the common block of all HTSC cuprates.In La2−xSrxCuO4, Sr substitution for La, or the addition of interstitial oxygen, changesthe carrier concentration on the CuO2 planes.

Looking at the electronic structure cuprates have 2D lattices of 3d transitionmetal ions as the building blocks. The multiband electronic structure can be reduceto a low-energy effective one-band model since the in-plane subset of Cu d-orbitalseg (d3z2−r2 and dx2−y2) are both present near the Fermi level but the planar dx2−y2

is quite dominant. On the other hand in Fe-based materials the out-of-plane pnic-togen hybridize well with the t2g Fe d-orbitals and all three of them have weightat the Fermi Surface (FS). Moreover the overlap between the d-orbitals is impor-tant. As a consequence the minimal model for Fe-based materials is essentiallymultiband. This is the first striking difference between these to classes of materialsFig.1.3. The multiband character of the electronic structure of Fe-based materials,first predicted by Density Functional Theory (DFT) calculus in the Local DensityApproximation (LDA) [44, 45], was soon confirmed by direct Angle-Resolved Pho-toEmission Spectroscopy (ARPES) [19, 20, 21, 22, 23]. We will further discuss thispoint in Section 1.2.3.

The second difference between Cu- and Fe-based materials concerns the stateof the parent (undoped) compound. In cuprates it is found to be an insulator,and should be classified as a Mott insulator. Indeed, in the copper-oxygen layerthere is an odd number of electrons per unit cell. According to band theory, theband is half-filled so that a metallic behavior is expected. Nevertheless due to the


Figure 1.3. Schematics of two-dimensional cross-sections of FSs for cuprates and pnictides.For weakly doped cuprates, the FS has a single sheet, and filled states (the ones closerto Γ point) occupy about a half of Brillouin zone area. In pnictides, the FS consists ofmultiple sheets - 2 hole pockets centered at Γ, two elliptical electron pockets centeredat (0, π) and (π, 0).

strong Coulomb repulsion the system prefers to avoid double occupancy (i.e. toput two electrons/holes on the same ion), so that the ground state is an insulator.The undoped Mott insulator exhibits AF order of the spin localized on copperatoms. The Neél temperature TN ranges from 250 and 400 K, depending on thematerial. The origin of the AF order can be easily understood since in the AFconfiguration neighboring spins are oppositely aligned and one can gain energy byvirtual hopping between neighboring Cu-sites. Concerning the Fe-based materialsthe parent compound is a (bad) metal characterized by a AF order below TN ∼ 140K, composed of iron spins aligned in a stripe-like structure (see Fig. 1.4). Theundoped phase thus appears like an AF Spin Density Wave (SDW) metal where theelectrons appear to be more delocalized with respect to the cuprates systems.

The metallic behavior of Fe-based systems reveals a moderate degree of theelectronic correlation in pnictides and suggests that the Mott physics of a half-filledHubbard model is not a good starting point for pnictides. While the AF order in aMott insulator arises because the spins can lower their energy if they are antiparallelto their neighbors, the SDW emerges from an instability of the paramagnetic FS.This picture is corroborated by the experimental observation of a "nesting condition"between the bands [19, 20, 21, 22, 23]. Indeed, several hole and electron pockets arelocated in different parts of the Brillouin Zone (BZ), "nested" by the characteristicwave-vector of magnetic order Q = (π, π). This means that distinct sections of FScan be made to match via translation by the momentum vector that identifies theperiodicity of the magnetically-ordered state (See Fig. 1.5 for a schematic picture).As we will comment later, within this picture, magnetism emerges also as the naturalcandidate for the superconducting glue: as doping increases, long-range magneticinstability is suppressed in favor of a residual exchange of Spin Fluctuations (SpF)between hole and electron pockets that leads to pairing.

In both cases by doping the systems one can manipulate the magnetism to im-prove electrical conduction and achieve superconductivity. The doping operationcan be performed in different ways. In cuprates it is accomplished by replacing theelements out of the CuO layers with others having different valence or adding extra


Figure 1.4. From [43]: Crystallographic and magnetic structures of the iron-based super-conductors. a) The five tetragonal structures of Fe-based superconductors. b) Sketchof the active planar iron layer common to all superconducting compounds (iron in red,pnictogen/chalcogen in gold). The dashed line indicates the size of the unit cell of theFeAs-type that includes two iron atoms due to the staggered anion positions.

G

M

Q

Q

Figure 1.5. Simplified scheme of the FS of pnictides with hole pockets around the Γ pointof the BZ and electron ones around the M point. The ordering vector of the AF orderQ = (π, π) connects the two pockets having different character.

out-of-plane oxygen, e.g. La2−xSrxCuO2 (hole doping), Nd2−xCexCuO2 (electrondoping), and YBa2Cu3O6+δ (extra oxygen). The additional electron or hole is thenassumed to dope the plane in an itinerant state. The electronic couplings in theinter-layer direction are very weak: it is indeed strongly believed that supercon-ductivity is related to processes occuring in the copper-oxigen planes. In Fe-basedmaterials similar phase diagrams are obtained by different routes, for example byreplacing the spacer ion as in LaFeAsO1−xFx and Sr1−xKxFe2As2, or by in-planesubstitution of Fe with Co or Ni as in Ba(Fe1−xCox)2As2 and Ba(Fe1−xNix)2As2,or by replacing Ba with K, Ba1−xKxFe2As2. These heterovalent substitutions dopethe FeAs or FeP plane in analogy to the case of cuprates. Another possibility is


doping via isovalent substitutions as for example in BaFe2(As1−xRux)2. In thesecases "dopants" can act as potential scatterers and change the electronic structurebecause of differences in ionic sizes or simply by diluting the magnetic ions withnonmagnetic ones. However the phase diagrams also in these cases are quite simi-lar, challenging workers in the field to seek a systematic structural observable whichcorrelates with the variation of Tc.

Looking at Fig. 1.1 one sees that while the phase diagram of Fe-based materialsappears more symmetric between r.h.s. and the l.h.s. side (electrons-, hole- dopingrespectively), that of cuprates is asymmetric since the character of the doping influ-ences the way in which antiferromagnetism is suppressed. On the hole-doped sidethe AF order is already shattered at doping of the order of x ∼ 0.03 or even less,on the electron-doped side instead the AF state persists to much higher doping,leaving a relatively narrow window of superconductivity. In what follows we willfocus on the properties of hole-doped cuprate systems.

Looking at the evolution from magnetism to superconductivity in cuprates thelong-range ordered Néel phase vanishes before superconductivity appears, while inFe-based materials the competition between these orders can take several forms.At a critical doping value LaFeAs shows an abrupt and first-order like evolution,whereas in the 122 systems the superconducting phase coexists with magnetismover a finite range and then persists to higher doping Fig. 1.6.

Figure 1.6. Left: Phase diagram of fluorine doped LaO1−xFxFeAs (from [46]), showing afirst-order transition. Right: Coexistence of AF and SC state in the phase diagram ofBa1−xKxFe2As2(from [47]).

Summarizing, it is a common feature of both classes the suppression of the AFlong-range order of the undoped compounds, the emergence of a SC state at somefinite doping and its suppression at higher doping, such that Tc forms a "dome" inthe phase diagram. The maximal Tc (the top of the dome) fixes the value xopt of theoptimal doping. Under- and overdoped regimes are distinguished by doping levelsx < xopt, x > xopt respectively.

The SC state of HTSC is highly unconventional and cannot be understood withinthe usual BCS picture of phonon mediated superconductivity. Also in the case of


pnictides the hypothesis of conventional mechanism of exchange of lattice vibrationsas pairing glue is ruled out by theoretical estimates that show that the electron-phonon coupling is not strong enough to produce the observed critical temperatures[27]. The proximity to an AF phase emerges as common denominator of unconven-tional superconductivity. The possibility arises that the superconductivity couldbe driven by magnetic interaction between carriers. Such a theoretical perspec-tive had been already discussed many years ago in the context of cuprates, andonly recently it has been revisited in connections to iron-based superconductors. Inthe context of cuprates no consensus has been reached yet and SpF remains onlyone of the possible candidates for the pairing glue [2]. On the contrary in pnic-tides there is a general agreement about the relevance of SDW fluctuations as thepairing mechanism. This picture has been made more precise in the last years byseveral theoretical calculations based on realistic five-orbitals models where pair-ing is studied within an Random Phase Approximation (RPA) [48, 49] or on therenormalization-group approach to simplified two or four band models [29, 50]. Ithas been shown that the problem can be efficiently mapped into effective low-energymodels with only an interband repulsion, that can be accommodated by a supercon-ducting order parameter that changes sign between hole and electron pockets, theso-called s± symmetry. However, despite the general agreement on the possibilityof having this overall sign change between hole and electron sheets, no theoreticalor experimental consensus has been reached yet about the structure. Indeed, thegap structure seems to be strongly material dependent, in particular is not clearif the SC order parameter is fully isotropic on each pocket or instead it has nodes(zeros) on some part of the FSs [51]. On the other hand in the context of cupratesit is now well established that the SC state is universally d-wave with cos kx−cos ky

structure [2, 42, 52]. Another difference in the SC phase of cuprates and pnictidescan be finally detected in the behavior of the superfluid density ρs. In cupratesdue to the proximity to the Mott insulator ρs is small and vanishes with decreasingdoping (leading to the insulating phase at x → 0). The anomalous scaling of ρs

with doping can only be accounted for by including correlation effects [53]. On thecontrary in pnictides the experimental findings on ρs [46, 54, 55] give evidence of aconventional scaling with the number of carriers once the almost empty characterof the bands has been taken into account [56].

Although unconventional, the SC state appears nonetheless as the less anoma-lous one of the phase diagram of HTSC. The normal state above Tc representsindeed the more complex and anomalous phase. In cuprate systems the non-SCphase around the optimal doping above Tc, often referred to as the "strange metal"phase, presents numerous properties at odds with the FL theory, as for example un-usual power law of the transport properties. In particular the electrical resistivityis linear in a wide range of temperatures rather than quadratic as expected for aFL system [57], on the other hand also the Hall coefficient RH shows a temperaturedependence not expected for a FL system, where RH is constant. The scenario iseven more complex since signs of the magnetism seems to persist far into this regimeshowing up as local order on short length scales observed in muon spin relaxationmeasurements [58]. Decreasing the doping level the non-FL behavior is made evenmore pronounced. In the underdoped regime a large number of experiments haveprobed the existence of a depression in the Density of States (DOS), often referredto as the Pseudogap (PG) phase [2]. The PG appears below a crossover temper-


ature T ∗. This crossover temperature, of the order of 400 K at low doping level,decreases by increasing doping, however if T ∗ should intersect or merge with Tc isa very moot point since the two possibilities have been associated with radicallydifferent physical explanations of the PG phenomena (see Fig. 1.10) [2]. We willfurther discuss the anomalies of the underdoped phase of cuprates in Section 1.2.2.Although the normal phase of Fe-based material does not manifest a robust PGbehavior, it is far from being conventional due to the multiband character of thesystem. Indeed, the persistence of local AF correlations far from the SDW phase,highlighted in [59], opens new question about the role of the AF degrees of freedomand their interplay with superconductivity. Moreover the presence of several (inter-acting) hole and electrons bands could be the source of the odd transport propertiesof these materials. For example both Direct Current (DC) and Hall conductivitymeasurements has been often interpreted as due to the coexistence of carriers withdifferent scattering times: large and temperature independent for one type, smalland temperature dependent for the other one. For example the direct comparisonbetween DC and Hall conductivity in 122 compounds performed by the group ofF. Rullier-Albenque [33] suggests that the hole contribution to the transport canbe neglected at low T in most of the phase diagram. The situation is still unclearsince although the presence of several conducting channels is not surprising due tothe multiband structure of pnictides, however no theoretical proposal emerged sofar to explain why one channel should be much less conducting than the other. Wewill further discuss this topic in what follows since this is one of the focuses of thepresent work.

For both cuprates and pnictides the standard FL behavior of the normal stateis recovered only in the overdoped region of the phase diagram.

1.2.2 Phenomenology of the Underdoped Cuprates

The Pseudogap

Historically the anomalous behavior of the electronic properties of underdopedcuprates was discovered from a drop of the spin susceptibility χs measured byNuclear Magnetic Resonance (NMR) shifts [60] below a temperature T ∗. Ther-modynamical probes (e.g. specific-heat measurements) soon allowed to establishthe strongly reduction of the DOS. However, nowadays, the ARPES technique isthe most powerful tool to investigate the doping and temperature dependence ofthe PG, since it gives a k-resolved information on the spectral properties of thesingle-particle excitation of the system.

In ARPES measurements the opening of the SC gap below Tc is reflected inthe shift, with respect to the Fermi level, of the leading edge of the photoelec-tron energy distribution curve. The doping and temperature evolution of the PG inBi2Sr2CaCu2O8 was systematically investigated in [61] At low temperature ARPESdata are consistent with a gap having d-wave symmetry. The gap is maximal nearthe M-point (0, π) and vanishes along the line connecting (0, 0) and (π, π), wherethe excitation is often referred to as nodal quasiparticle. When temperature israised above Tc one would expect to recover a gapless FS. This is what happensin the overdoped sample, however in underdope regime the situation appears morecomplex. Above Tc the gapless region gradually expands to cover a finite region


near the nodal point, beyond which the PG gradually opens as one moves towards(0, π). The leading edge shift survives up to a temperature T ∗ at the M -points.This unusual behavior is sometimes referred to as the Fermi arc (see Fig.s 1.7,1.8).Note that the PG lineshape at the antinodal spot, (0, π), is extremely broad and

50 0Binding energy (meV)

14

70

95

120

150

180

T(K)

(a) (b) (c)

Γ

Y

T (K)0 50 100 150

Mid

poin

t (m

eV)

abccba

M

50 0 50 0

0

-10

20

10

(d) (e)

Figure 1.7. From [61] (a-c) ARPES spectra (solid curves) for underdoped Bi2Sr2CaCu2O8

(Tc = 85K) taken at different k points along the FS shown in (d). The dotted lines de-fines the chemical potential as determined by the reference spectra from polycrystallinePt (red lines) in electrical contact with the sample. Note the pullback of the spectrumfrom the FS for T > Tc. Note the closing of the spectral gap at different temperaturesfor different k. This feature is also apparent in the plot (e) of the midpoint of theleading edge of the spectra as a function of T .

completely incoherent, while the onset of superconductivity is marked by the ap-pearance of a small coherent peak at this gap edge (Fig. 1.7). On the other handthe lineshape is relatively sharp along the nodal direction even above Tc. A narrowlineshape in the nodal direction has also been observed in other cuprates so thatthe notion of relatively well defined nodal excitations in the normal state is mostlikely a universal feature.

The continuous evolution of the PG into the superconducting gap below Tc

and its d-wave-like wave-vector dependence has driven the interpretation of the PGphenomena as a precursor effect of the SC phase. According to this preformed-pairscenario the crossover temperature T ∗ acquires the meaning of the pairing tempera-ture at which fluctuating pairs start forming, but without coherence owing to largefluctuations of the superconducting order parameter. Lowering the temperature,


Γ

M

M Γ Γ

Y Y Y

MM

MM

Figure 1.8. From [61]: Schematic illustration of the temperature evolution of the FS inunderdoped cuprates. The d-wave node below Tc (left panel) becomes a gapless arcabove Tc (middle panel) which expands with increasing T to form the full FS at T ∗

(right panel).

the coherence is eventually established as soon as T < Tc and superconductivityappears. The occurrence of large superconducting pair fluctuations in underdopedcuprates is expected due to the low dimensionality induced by the layered structure(the quasi-2D systems), as well as to the short coherence length ξ0 of the super-conducting pairs, of the order of few lattice spacing (typically, ξ0 ∼ 10 − 20Å) andit has been widely experimentally observed as we will discuss later in this Section.Within this scheme, the phase diagram of cuprates can be interpreted in termsof a crossover from Bose-Einstein Condensation (BEC) of preformed pairs to BCSsuperconductivity, as the doping is varied [62].

However the situation is far from being clear. Actually, as opposite to the inter-pretation of the PG phenomena as precursor paring (i.e. originated by interactionsin the particle-particle (p-p) channel), some experimental evidences suggest a majorrole of other kind of interactions lying in the particle-hole (p-h) channel. For exam-ple a recent analysis of ARPES data by Kaminski and coworkers [63] reveals differentsignatures associated to the PG and the SC state that seem to support a competing-orders scenario as the origin of the PG states. This observation however leaves openthe question about what should be the interaction in the p-h channel driving theopening of the PG (e.g. spin-fluctuation, d-density wave, charge/spin instability re-lated to the presence of a Quantum Critical Point, microscopical current-interactionand so on. For a review see [2]).

Analysis of Superconducting Fluctuations (ScF)

During this work we focused on some anomalies in the ScF behavior detected in theunderdoped phase of cuprates. In order to clarify the experimental outcomes thatmotivated our interest in this topic, in this section some experimental evidencesand their interpretation about the PG phase are presented in more details, and thepuzzling outcomes of the analysis of ScF in this context is reviewed.

The analysis of ScF contribution to transport seems to support a conventionalregime of the ScF. For example the analysis of the fluctuation contribution to theDC conductivity above Tc (the so-called paraconductivity) in underdoped cupratesis found in agreement with the prediction of a standard Ginzburg-Landau (GL)


analysis of the ScF [8, 9], showing a T dependence of the data as expected for thequasi-2D Aslamazov-Larkin (AL) regime [11] (we will review the standard theoryof ScF in Section 2.2.1). In particular in [8] Leridon and coworkers carried on asystematic analysis of ScF in the underdoped regime both within a GL approach(i.e. conventional ScF of modulus and phase) and a Kosterlitz-Thouless (KT) one(pre-formed pair scenario with the pair amplitude already defined above Tc andfluctuations of the phase). The outcomes of their observation turned out to beconsistent with an GL picture of the ScF, suggesting that the SC pair amplitude isnot already defined above Tc.Another interesting analysis of ScF transport has been also carried out by Alloul andcoworkers [9]. They suppress the superconductivity by means of a magnetic fieldand study the transverse magnetoresistence of the sample. At high temperature inthe normal state, the variation of resistivity δρ = ρ(H, T )−ρ(0, T ) increases with H2

as usually observed for the classical transverse magnetoresistance in low magneticfield. Deviations from the quadratic behavior are observed at low temperatures(below Tonset) for magnetic fields weaker than a certain value H ′

c, due to the onsetof ScF. One can thus measure Tonset as the last temperature for which a quadraticbehavior is observed for any H, i.e. the temperature for which H ′

c → 0. Thisprocedure also allows to extrapolate the zero field resistivity ρn(T, 0) in the absenceof ScF (see Fig. 1.9). The measurement of the resistivity at zero field ρ(T ) allowsto fix at the same time Tonset and T ∗. T ∗ is defined as the temperature below whichthe resistivity ρ(T ) is not linear in T , while Tonset is the temperature below whichρ(T ) and ρn(T ) do not coincide anymore.

Figure 1.9. From H. Alloul et al. Europhys. Lett. 91, 37005 (2010): Left: Fieldvariation of δρ/ρ0 plotted versus H2 for decreasing temperatures down to T ∼ Tc in aundoped YBa2Cu3O6+xsample. For T > 130K the magnetoresistance follows the H2

dependence, but at lower T the contribution of ScF increases the low field conductivity.This occurs below an onset field H ′

c(T ) (arrow) which is plotted in the inset togetherwith ∆σSCF (T, 0). Both quantities become negligible at a temperature T ′

c = 130K,which defines the onset of ScF. Right: T variation of the resistivity in the same sample.The full square data for ρn(T, 0) are deduced from data showed in the left panel. Theenlargement in inset (a) better exhibits the ScF contribution (hatched area). In inset(b)the resistivity decrease with respect to the high T linear extrapolation (dashed line inthe main panel) due to the opening of the PG is displayed together with that due toScF.


In [9] they found for the undoped samples T ∗ always above Tonset (see Fig. 1.9),however, in contrast with previous estimate [3, 7, 4], near optimal doping theyobserve a crossing of the PG and ScF lines with T ∗ < Tonset (see Fig. 1.10). Ifconfirmed by further experiment, this result could be an unambiguous evidencethat the PG cannot be simply due to preformed pairs.

Figure 1.10. From H. Alloul et al. Europhys. Lett. 91, 37005 (2010). Possible phasediagrams for the cuprates. The phase-transition lines delimiting the AF and the SCstates are well established. The two crossover lines lying above Tc signal the openingof the PG, T ∗ and the onset of ScF, Tν . Their relative position depends whether apreformed-pair scenario (b) or a competing-order scenario (a) applies.

The analysis of Nernst effect and diamagnetism above Tc is instead stronglydisputed. The explanation within a standard GL framework of the large Nernstsignal and the strong diamagnetism detected in the underdoped cuprates, seemsnot reliable, being too strong the fluctuating effects with respect to those expectedin a Gaussian regime of ScF described by the GL approach [12]. For this reason thedetection of a large Nernst effect and of diamagnetism above Tc [3, 7, 4] are ofteninvoked in favor of a preformed pair scenario related to a vortex-like phase fluctua-tions with a KT character. However it has been suggested by several authors [5, 9]a possible role of the disorder in the enhancement of both the range of temperaturein which the Nernst effect is observed and the intensity of the signal. Moreover alsofrom the theoretical point of view various investigations have been performed inthe attempt to explain the experimental data on Nernst signal and diamagnetismwithin a GL framework [12, 13]. In particular in [13] it has been shown that theinclusion of vertex corrections to the Gaussian theory resulting from the PG yieldsto a Nernst signal in good agreement with the data of underdoped samples.

Regardless of the KT or GL character of fluctuations, there is an outcome ofthe experiments that is not expected for conventional superconductors, i.e. an over-all discrepancy between the magnitude of the ScF contributions to diamagnetismand paraconductivity. However, from the experimental point of view, performinga systematic analysis of the ScF on the same sample is a difficult task since ev-ery experiment requires a sample preparation that not always allows to perform


other testes. Thus only recently the possibility to perform paraconductivity anddiamagnetism measurements on similar samples, namely a thin layer of underdopedLa2−xSrxCuO4, has uncovered this unconventional features of the ScF in under-doped cuprates [14]. The discrepancy between the two ScF contribution is foundto be surprising strong with the ScF contribution to the conductivity about twoorders of magnitude smaller than the fluctuating diamagnetism in the same systemFig.1.11.

Figure 1.11. From [14] : A comparison of fluctuation conductivity with diamagnetism insimilarly doped La2−xSrxCuO4crystal.

The analysis of Bilbro et al. in [14] is carried out in the context of a vortexunbinding picture for 2D superconductors, using the Halperin-Nelson (HN) analysis[64] of the ScF based on the Bardeen-Stephen (BS) theory [65] of the vortex motion.Motivated by the assumption that very close to Tc vortices are the principal degreesof freedom in quasi-2D materials, they fit the 2D susceptibility and conductancedata using the usual relations used in the vortex unbinding picture (discussed in[64])

δχd = −kBT

Φ20d

ξ2(T ) δGs =1

Φ0nf µ(1.1)

where ξ is the correlation length, Φ0 is the flux quantum, µ the vortex mobility, andnf ∼ ξ−2 is the areal density of thermally excited vortices. The ratio between theseto quantities can be used to give a determination of the vortex diffusion constantD(T ) ≃ δχd/δGs using only experimental data. Since conventional vortex dynamicswould predict a much larger fluctuation conductivity given the size of diamagnetism,Bilbro and coworkers comment the experimental outcome as the observation ofa regime of fast vortices. The definition of such anomalous D coefficient leadsto two possible conclusions. If the quantities detected are indeed completely dueto ordinary KT-like vortex fluctuation then the vortex properties appear highlyunusual and explaining these anomalies represents a well posed theoretical challenge.On the other hand, it is possible that the diamagnetic response could be enhanced


by another no-SC contribution coming from another interacting channel. This isprecisely the point of view that we will explore in Chapter 3.

1.2.3 Anomalous Properties of the Normal State of Pnictides

During this work we focused on some anomalies in the transport properties of pnic-tides trying to draw a general picture able to account for both DC and Hall transportin the normal phase of pnictides. In this Section we will first highlight some relevanteffects of the interband interaction and electronic correlation as come out from thecomparison of LDA prediction with ARPES or quantum oscillation measurements.Then we will point out the anomalous transport properties of pnictides.

The discovery of superconductivity in Fe-based materials has represented agolden age of DFT/LDA calculations. In fact this kind of analysis contributedto the emerging understanding of the physics of the Fe-based much more thanin the case of cuprate superconductors, where strong correlations do not allow areliable Mean-Field (MF) level analysis. The right topology of the FS has beenpredicted [45] before ARPES measurements could establish the electronic structureof Fe-based materials, as well as the correct magnetic ordering in the normal state[28]. The most successful proposal regarding the pairing symmetry so far has beenmade based on band structure calculations [28].

The FS topology highlighted by LDA calculation consists generally in four mainbands at the Fermi level with 2D character: two hole-like pockets around the Γpoint and two electron-like pockets around the M point [27, 28, 44, 45]. A fifththree-dimensional (3D) band crossing the Fermi level at the Γ point is sometimespredicted by LDA calculations [44, 45], even if its position with respect to the chem-ical potential is strongly affected by the interlayer distance. ARPES experimentsboth in 1111 [21] and 122 systems [19, 20, 22, 23] confirm the theoretical LDApredictions both in the multiband topology of the FSs and in the fulfillment of thenesting condition Fig. 1.12An alternative technique probing the FS topology, which is not momentum resolvedbut has the advantage of being a bulk probe, is based on de Haas-van Alphen (dHvA)magnetization measurements, which allow one to estimate the size of the Fermi ar-eas and the effective mass m∗ for each Fermi sheet [24, 25]. Also in this case anoverall qualitative agreement is found between LDA calculations and experiments.

However interesting discrepancies emerge. In general, we can distinguish be-tween high energy discrepancies and low energy discrepancies. ARPES data inBa0.6K0.4Fe2AS2 [19, 20] and LaFePO [21] for instance find overall bandwidths afactor of 2 smaller than the LDA calculations, possibly related to the static elec-tronic correlation. On the other hand, the ARPES observation of Fermi velocitiesa factor 4 larger than the LDA calculated low-energy ones [20] points out a rele-vant role of dynamical quantum renormalization effects associated with a low-energyboson-mediated interaction. Besides the bandwidth renormalization, a second strik-ing result emerging from dHvA is a substantial shift of the bands with respect tothe Fermi level when compared to LDA calculations. Even though a certain degreeof inaccuracy could be present in DFT calculations, the persistent observations ofsuch shifts suggests a robust feature in these materials whose origin is unclear. Theauthors of [32] ascribed this effect to the interband character of the interaction in


Figure 1.12. From [19]: ARPES data of Ba0.6K0.4Fe2As2. Three-dimensional plot of thesuperconducting-gap size ∆ measured at 15 K on the three observed FS sheets (shownat the bottom as an intensity plot) and their temperature evolutions (inset). Notice thesame |∆| on the α, γ bands connected by the nesting vector. This feature support theexchange of SpF between nested bands as the pairing glue.

a multiband picture with almost empty bands (i.e. in a condition of strong p-hasymmetry). Motivated also by the anomalous properties of transport in pnictidesa further analysis of this mechanism highlights some important consequences of theshrinking on the number of carriers involved in transport [37, 66]. We will discussthe outcomes of this analysis in the next Section.

From the experimental point of view the great excitement following the discoveryof pnictides has triggered a very intense experimental activity. A complete overviewabout transport measurements in pnictides is beyond the possibilities of this Ph.D.Thesis. We will focus in what follows in DC and Hall conductivity measurementsin the normal state of pnictides. As illustrative example we will discuss in detailsthe results of few experimental papers in order to highlight the general outcomes ofDC transport measurements in pnictides.

At the very beginning experimentalists manly focused on the analysis of 122system due to the availability of good quality large single crystals. This technicalaspect is particularly important in the analysis of transport since in polycrystals theanalysis might be biased by the presence of different doped patches in the samplesand by the mixing of different directions. Almost simultaneously in 2009 two inter-esting papers appeared on transport in 122 compounds [33, 34]. They performeda systematic analysis on Ba(Fe1−xCox)2As2 over a wide range of temperature anddoping. In order to get more insight into the doping-evolution of transport prop-erties they not only measured the resistivity ρ(T ) of various doped sample, but


also performed Hall effect measurement. In a semiclassical picture the Hall coef-ficient gives a direct estimate of the character and of the density of the carries,RH = ±1/ne (+ for holes, − for electrons), thus its analysis in a multiband systemin the presence of both hole and electrons is crucial in the understanding of thetransport mechanism. Hall measurements on the same sample up to a magneticfield H = 8 T, confirmed the linear dependence in the magnetic field of the Halleffect as expected in the weak field limit (ωcτ ≪ 1 where ωc = eH/m is the cy-clotronic frequency and τ the scattering rate). The experimental data are presentedin Fig.s 1.13,1.14

Figure 1.13. From [33]: Left panel: Temperature dependence of the in-plane resistivityρ(T ) of Ba(Fe1−xCox)2 As2 single crystals. For the sake of clarity, the data for theundoped parent and low doped non SC samples, which evolve differently than for higherCo content, are plotted with dashed lines. Notice the singularities associated with theSDW transitions.Right panel: T variation of the Hall number |nH | for a set of samples.Here, nH (e/Fe) = 0.32/RHx(10−9m3/C). For the undoped compound, the drop belowTN occurs in two steps, the value on the plateau indicated by an arrow being 0.074e/Fe. Inset: Raw data for the magnetic samples showing the drops of RH at TN . Thedata for the most overdoped sample have been kept for comparison.

A cursory look at the data allows to draw out some general outcomes:

• the onset of the AF order reflects in a general step-like behavior of the con-ductivities.

• the resistivity data are quite conventional. In particular in the paramagneticphase the linear T -dependence of ρ(T ) is the same for almost any doping levelx. No evidences of multiband transport are visible at first sight.

In particular concerning the Hall transport:

• no deviations from the linear H-dependence are observed for any doping level

• there is on overall strong T -dependence of the Hall data, with a softening athigh temperature.

• in slightly doped material, i.e. an almost compensated metal, an almost van-ishing RH is expected on the basis of a semiclassical picture (see Eq. (1.2)below). Experimental data at low doping show in contrast a large value of|RH |, with RH ∼ 10−9m3/C at room temperature.


Both experimental group analyzed their data in the framework of a semiclassicaltwo-band model:

σ(= ρ−1) = σe + σh, RH =1e

(nhµ2h − neµ2

e)(nhµh + neµe)2

, (1.2)

where σα, (α = e, h), is the electron/hole conductivity, nα the carrier density, andµα = e2τα/mα the corresponding mobility, with τα being the transport scatteringtime and mα the effective carrier mass in each band. As above mentioned, one isthe central outcome of this analysis: fitting transport data within a semiclassicalapproach requires to assume a large disparity of the mobility of holes and electronin order to achieve reasonable value of RH . Indeed RH is found large and negativeat low doping, where nh ∼ ne, so that the only possibility to obtain an "electronic"(i.e. negative) RH from Eq. (1.2) is to consider µh/µe → 0.

Under this assumption the agreement is quite good at high temperature andhigh doping level (see Fig. 1.15); however the theoretical calculation misses thetemperature dependence when T is low enough, not only in the SDW regime butalso in the normal phase (Tc(x = 0.05) ∼ 70K), and the accuracy gets worse andworse as the doping decreases. The authors of [34] do not explain the origin of thisdiscrepancy and only comment about the possible relevance of "somewhat" differenttemperature dependence for µh, µe.Along the same path also Rullier-Albenque and coworkers [33] arrive at the conclu-sion that the only way to explain their Hall data at low temperatures and low dopingwithin a semiclassical picture is to assume a drastically suppressed hole mobilities.

The discussion above holds also for hole-doped compounds. In these cases in-deed the experimental (temperature- and doping-) behavior of ρ and RH is the sameof those presented here, but with the opposite sign of RH Fig. 1.16.

Concerning the 1111 compounds, a systematic and exhaustive survey has not yetbeen performed, also due to the limited availability of clean single crystal. Howeverthe experimental findings are quite similar also for 1111 compounds suggesting thatthe source of these anomalies is not a peculiarity of a given structure configuration.As an illustrative example we report in Fig. 1.17 some measurements of in-planeresistivity and Hall coefficient on LaFePO samples [67, 68]. It is worth noting herethe strong temperature dependence of the resistivity that is not observed insteadin 122 samples: in LaFePo indeed one finds ρ(100K)/ρ(10K) ∼ 10 (see inset in Fig.1.17).

The ρxy(T, H) for all samples has the same linear H-dependence in the wholetemperature range, indicating no anomalous Hall effect in this system. The Hallcoefficients for all the samples are negative and have a large magnitude. Near roomtemperature, all three compounds have similar values, RH ∼ −4 10−9m3/C. If asingle band model is assumed, using the semiclassical results RH = −1/nec we canestimate n ∼ 0.1 electrons per unit volume, i.e. a very low carrier density. Boththe order of magnitude and the temperature variation is not explained in [67, 68],they only suggest the possible relevance of several conducting channels. In the 1111compounds the requirement to fit within a two-band semiclassical picture the Halldata, as well as the resistivity ones, has even more dramatic consequence, since oneneeds to assume more that one order of magnitude of disparity between the two


Figure 1.14. From [34]: Upper panel: Temperature dependence of the Hall coefficient RH

at 9 T. The red dashed line at x . 0.07 follows TN , below which RH rises sharply, indi-cating a dramatic change of the carrier concentration and scattering rate, as discussedin the text. The blue dotted line outlines the superconducting region. The lower panel(from left to right) represents the FSs calculated for nonmagnetic Ba(Fe1−xCox)2As2

for x =0, 0.2 and 0.3.

Figure 1.15. Electron concentration extracted from the Hall coefficient presented in themain text, as compared with the calculated Hall concentrations (solid line), assumingan x-dependent ratio of the electron and hole mobilities, µh/µe = 1.3x. The insetshows the calculated volumes of the hole and electron pockets as a function of electrondoping, in the rigid band approximation.

1.3 Open Theoretical Issues in the Physics of HTSC 19

Figure 1.16. From [34]: Temperature dependence of RH for the parent phase, 4% and 25%K-doped Ba1−xKxFe2As2 crystals, 2% Co-doped and 4% Co-doped Ba(Fe2−xCox)2As2

crystals. Note that a very small amount of K-doping leads to a sudden sign change ofRH in the AF state. The inset shows RH near the AF transition for selected dopings.

band mobilities.

These experimental observations open new questions about the appropriate de-scription of the scattering mechanism in pnictides. First of all a theoretical ex-planation that could account for such a large disparity between the scattering rateof holes and electrons is still lacking. For example, the inclusion of spin [69] ororbital [70] fluctuations within realistic models can account at most for a factor of2 of anisotropy between the average quasiparticle lifetime on the electron and holepockets, not enough to explain neither the absolute value of RH nor its T depen-dence reported in [33, 34, 67, 68, 71, 72, 73, 74, 75]. Moreover so large anisotropyin the scattering rate of the hole and electron bands, should give strong anomaliesin many other transport measurements. Nowadays several authors claim evidenceof two very different scattering rates in pnictides from the analysis of optical datawhere the flat mid-infrared optical conductivity is sometimes attributed to a verybroad Drude-like intraband contribution [76, 77, 78]. However this interpretationhas been questioned by several authors on the basis of the presence in pnictides oflow-energy interband optical transitions [66, 77, 79, 80].

A convincing framework to explain the unconventional properties of the Halltransport in pnictides is thus still lacking and appears as a primary need in order tounderstand and hence elucidate the role of the scattering mechanisms in pnictides.

1.3 Open Theoretical Issues in the Physics of HTSC

In the previous section we reviewed few (selected) experimental findings in HTSC.A detailed overview of theoretical models is beyond the aim of this Ph.D. thesis.


Figure 1.17. Left panel: Electrical resistivity versus temperature T measured in the ab-plane. In the inset the full T -range measured [67]. Right panel: RH vs T. Inset showsthe magnetic field H dependence of Hall resistivity ρxy at T = 200K for the samplewith x = 0.05. ρxy is proportional to H up to 5 T [68].

However it seems useful to collect some general ideas about possible strategies inthe theoretical description of HTSC in order to address the questions arisen in theexperiments discussed in the previous section.

1.3.1 p-h and p-p Degrees of Freedom in Underdoped Cuprates

The coexistence, and possibly the competition, between many interaction channelsleads to unusual property as widely highlight in the context of underdoped cuprates.When so many are the relevant degrees of freedom it is difficult even to isolate the"key interaction" of the correct theory. The strong debate about the origin of thePG is a typical example of this complexity. The situation is even more involvedsince also assuming to be able to establish the more relevant interacting channel,the interplay with the others could lead to unexpected results.

In this framework the analysis of ScF appears a relevant topic. In particularunrevealing the origin of the huge anisotropy between the ScF to conductivity anddiamagnetism pointed out in [14] appears a tempting task which could shed newlight on the understanding of the interplay of the several interactions channels andtheir role in the physics of cuprates. In such analysis a microscopic theory that ac-count for ScF effects is needed. Only starting from a microscopic model Hamiltonianand including the several relevant degrees of freedom we can control how transportand thermodynamic properties are affected by the several interacting channels. Aswe will discuss in the next Chapter (Section 2.2.2) this can be easily done by meansof the functional integral approach in the description of Gaussian regime of GLfluctuations.

Within this framework the experimental observation of fast vortices must bereformulated by means of a more conventional ScF language that is valid bothwithin the GL and the KT theory. Actually, in 2D the contribution of ScF tothe conductivity δσ and diamagnetism δχd can be expressed in terms of the SC


correlation length by means of Eq.s (1.1) within the KT language, or as:

δχd = −e2T

3πdξ2(T ), δσ =

e2

16~d

ξ2(T )ξ2

. (1.3)

in a GL formalism (see Section 2.2.1). From Eq.s (1.3) one can express the ratioδσ/δχd, up to universal constants, as δσ/|δχd| ∝ ξ2

σ(T )/ξ2χd

(T ) where ξσ, ξχdare

the correlations length extracted from paraconductivity and diamagnetism mea-surements, respectively. Since one would expect the same length scale involved,one should get ξ2

σ/ξ2χd

= 1, i.e. δσ/δχd ∼ O(1), thus the experimental findings ofδσ/δχd ∼ 10−2 is totally unexpected also in the context of a GL analysis.

What changes between the GL and KT approach lies in the T dependence of theξ(T ) divergence at Tc (power-law within the GL approach and exponential withinthe KT theory), but not in the possibility of express the ScF contributions in termsof the correlation length ξ(T ), so that the discussion above holds in any case. Thediscrepancy pointed out by Bilbro et al can be always recast in both cases as adisparity of the correlation lengths extracted from conductivity and diamagnetism,i.e. ξ2

σ(T )/ξ2χd

(T ) ∼ 10−2.The occurrence of this effect in the framework of the underdoped cuprates sug-

gests a possible source of this anomaly in a non-trivial interplay between the SC p-pinteractions and other degrees of freedom belonging to the p-h channel. The crucialpoint is to account for any relevant channel. Since we are dealing with transportproperties, one could expect that a major role is played by current-current interac-tions. The possible relevance of this kind of interactions to the physics of cuprateshas been suggested within several contexts, ranging from the gauge-theory formu-lation for the t− J model [2] to the theoretical approaches emphasizing the role ofmicroscopic currents [81, 82].

Dealing with cuprates the analysis of a model containing the effects of strongelectronic correlation is required. From a theoretical point of view, a good candidateto study a strongly correlated material with a d-wave superconducting phase is thet-J model. The Hamiltonian acts in the Hilbert space with no double occupiedsites, as it is appropriate in the strong-coupling limit of the Hubbard model. Itis composed of a first term representing the electron hopping t and a second onedescribing the Heisenberg exchange interaction J between electron spins

H = −t∑

〈i,j〉 σ

(c†iσcjσ + H.c.) + J

∑

〈i,j〉

[

SiSj −14

ninj

]

− µ∑

i σ

c†iσciσ, (1.4)

c†iσ(ciσ) is the fermion creation (annihilation) operator and Si = Ψ†

iσ2 Ψi is the

spin operator, σ are the Pauli matrices and Ψi = (ci↑

ci↓). ni =

∑

σ c†iσciσ and 〈i, j〉

denotes nearest-neighbor pairs on a square lattice. A possible approach is to studythe t-J model in the slave boson formulation: indeed, strong electron correlationcan be handled by introducing slave particles together with the constrain that thesum of the electron and slave-particle density is equal to one on every lattice site.The slave boson MF theory of the model produces a phase diagram which agreesqualitatively with the phases observed in the cuprate superconductors [2]. Howeverthis approach disagrees with experiments in same aspects. One example is thebehavior of the superfluid density at low temperature. In HTSC cuprates ρs(T ) isexperimentally found to scale as ρs(T ) = ρs(0) + αT , where ρs(0) is proportional to


the hole concentration x and α is a weakly doping-dependent constant. As abovementioned the scaling of ρs(0) with doping can be accounted for by taking intoaccount the correlation effects. In the framework of the t-J model in the slaveboson formulation indeed the x dependence of ρs at T = 0 is correctly recovered.However the temperature dependence is found to be strongly suppressed by a factorx2. Such a result has triggered a great investigation in this issue. Recently Ng et al.[83] analyzed the problem from the point of view of the FL theory by calculating therenormalization of the superfluid density below Tc due to the presence of current-current interactions. Even though this approach did not solve the puzzle of thesuperfluid stiffness ρs, it opened new questions concerning the role of current-currentinteractions also above Tc.

Moreover notice that the presence of the density-density interactions, by intro-ducing Hartree-Fock (HF) corrections to the quasiparticle dispersion, leads to adifference between the quasiparticle current and velocity in the usual Landau FLlanguage [84]. The effect of this dichotomy on the GL functional for ScF can beaccounted for within a general field-theory for the t-J model [2, 83], which includesfluctuations both in the p-p and p-h channel.

The analysis of the interplay between the p-h interactions (in particular thecurrent-current ones) and the ScF will be the topic of Chapter 3 where we dis-cuss the recent results obtained in [15]. Here by computing the AL contributionto diamagnetism and conductivity it has been shown that the role of Landau FLcorrections differs in the static or dynamic limit. As a consequence the ScF contri-bution to diamagnetism and conductivity scales with different prefactors, leading tothe suppression of paraconductivity with respect to fluctuation diamagnetism whenthe Mott insulator is approached, in analogy with experiments in cuprates.

1.3.2 Hole and Electron Interacting Bands in Pnictides

In this section we want to give the theoretical framework in which the analysisof pnictides should be carried on and to point out some open issue that has beenaddressed during the Ph.D. work. The multiband character and the dominance ofinterband interaction emerge as unavoidable ingredients of any theoretical modeling.

Assuming the exchange of SpF as the main pairing mechanism in pnictides wecan model these compounds as multiband systems where carriers of several bandsinteract via the exchange of SpF. Within this picture the Eliashberg theory appearsas appropriate in the description of the superconductivity of pnictides. Historicallythe Eliashberg theory had been developed in order to describe intermediate andstrong coupling superconductors. In these systems indeed the BCS theory turnsout to be less accurate. The reason can be traced back to the fact that the in-teraction in BCS Hamiltonian is instantaneous, while in reality it should be re-tarded. Within the Eliashberg theory the electron-electron interaction mediated bya bosonic mode is indeed local in space and retarded in time, (reflecting the delayin the development of the screening effects). Within this approach we find a pair ofcoupled integral equations which relate a complex energy gap function and a com-plex renormalization parameter for the superconducting state to the electron-bosonand the electron-electron interactions in the normal state. The BCS gap equationthus can be obtained as the limit of weak coupling of the Eliashberg gap equations.


This approach can be used in the context of pnictides since within a spin mediatedparing picture the spin mode represents the bosonic mode that mediates the directelectron-electron interactions in the Eliashberg theory. It has been shown also thateven though the analysis of several thermodynamic quantities in terms of a two-band BCS approach seems to fit quite well the experimental data, others features,as for example the magnitude of the gaps, can be accounted only using more sophis-ticated four-band models and going beyond a BCS-like approach [85]. A generaloverview about the Eliashberg theory can be found in literature [86]. The use ofEliashberg approach to the physics of pnictides and its consequence are discussedin [32, 37, 66, 85].

Let us review here few of the very recent results obtained using Eliashbergtheory in the description of pnictides [32, 37, 66, 85]. Within this approach someanomalies has been interpreted as evidence of interband dominant interactions.This is the case of the FS-shrinking effect. As mentioned above, the comparisonbetween the band structure predicted by LDA calculations and the one observedexperimentally reveals several discrepancies. A striking result emerging from dHvAis a substantial shift of the bands with respect to the Fermi level when compared toLDA calculations. Indeed, the accurate determination of the FS areas provided bydHvA gives values smaller then those expected by LDA. Such discrepancy can beaccounted for assuming a shift of the LDA bands. Notably, such shifts have differentsigns in hole-like and electron-like bands, being downward for the hole-bands andupward for the electron-like ones [24]. This effect seems to persist to higher energies,as one can infer from ARPES, which gives a systematical reduction of the energydifference between the bottom of the electron bands and the top of the hole bandswith respect to the values predicted by LDA [22, 23].

The shrinking of the FS in pnictides has been attributed in [32] to the presenceof interband interactions in a multiband system with almost empty bands, where thep-h asymmetry forces to develop a finite-band Eliashberg approach (to be contrastedto the standard infinite-band approximation). Indeed, by modeling the interbandSpF as a bosonic mode with local propagator D(ωl) the self-energy of each electronicband in the Matsubara space can be written as:

Σα(iωn) = −T∑

m,β

Vα,βD(ωn − ωm)Gβ(iωm), (1.5)

where α, β are band indexes, and Vα,β (Vα,β = Vβ,α) is the coupling of the retardedboson-mediated multiband interaction and Gα(z) is the local Green’s function forthe α band, namely:

Gα(z) =∫ Emax,α

Emin,α

dǫNα(ǫ)1

z − ǫ− Σα(z) + µ. (1.6)

Here Nα(ǫ) is the non-interacting electronic DOS of the α band, with upper andlower band edges Emax,α and Emin,α, respectively, and we drop the spin index sinceit does not play any role in the following.

In the conventional Eliashberg analysis, one usually assumes that the distanceof the chemical potential from the band edges is much larger than the typical bosonenergy scale, so that one can approximate the DOS with its value at the Fermi


level, Nα(ǫ) ≈ Nα(µ), and one can extend the integration limits over ǫ in Eq. (1.6)to ±∞. In this way one is enforcing the p-h symmetry and the Matsubara self-energy Σα(iωn) results purely imaginary. Enforcing implicitly the p-h symmetrycan be however quite dangerous in systems as the iron-pnictides where each band isstrongly away from the half-filling. As a byproduct of taking into account the p-hasymmetry, the self-energy acquires a finite real part χα(iωn) ≡ ReΣα(iωn) 6= 0,whose low energy limit χα = χα(iωn=0) gives rise to a band shift that can be ingeneral different in each band. While this effect is usually disregarded in single-bandsystems, because it can be absorbed in a redefinition of the chemical potential, inthe multiband case it can lead to observable relative shifts of the various bandswith respect to the Fermi level. In particular, when the interaction has a dominantinterband character it can be shown [32] that the hole bands are shifted downwardsand the electron ones are shifted upwards, leading to the shrinking of the (non-interacting) LDA bands observed in the experiments (see Fig 1.18).

Figure 1.18. From [32] Intensity map of the spectral function for the interacting holebands (left panel) and the two degenerate electron bands (right panel), obtained byusing LDA parameters renormalized by a factor 2 and interband potential V = 0.46eV. The dashed lines represent the non-interacting parabolic bands, and the horizontalsolid line is the chemical potential. The ticks on the top mark the value of the Fermivectors in the presence (solid yellow ticks) and in the absence (white dashed ticks) ofinteraction. Note the shrinking of the FSs due to the coupling to the retarded bosonicmode.

Apart from the FS shrinking the coupling to an interband low-energy mode hasseveral consequences on the transport properties. Indeed, one can see that twodifferent quantities must be introduced to characterize the carrier concentration ineach band: the total carrier density n, as given by the integration of the DOS at allenergies up to the Fermi level, and the coherent carrier density n, that is associatedto the Fermi area as n = k2

F /2π. While in a standard single-band system theydo coincide even in the presence of interactions (as guaranteed by the Luttingertheorem [87]), here n is almost unchanged when the interaction is switched on,


while n is strongly affected, as showed by the FS shrinking. More interestingly, itturns out [37] that the transport properties at low energy, as the DC conductivity,depend on n:

σ =ne2τ

m∗(1.7)

Here the interaction effects enter in three places: the scattering time τ , the effectivemass m∗ (see Section 2.3.1) and the coherent carrier density n. While the formertwo effects are present also in a single-band case, the latter one is a peculiarity ofpnictides that has been often overlooked in the literature in the analysis of experi-mental data.

Let us stress that the reviewed results has been derived assuming that the bosonpropagator is momentum independent. Within this approximation we do not havevertex corrections. However the inclusion of these could be important in order toexplain the unconventional features of transport highlighted by the experiments.

The case of cuprates set a precedent. Although the role of AF degrees of freedomin the context of cuprates is a very debated topic, it is worthy to note that severalanomalies of the normal phase could be explained within a Nearly AntiferromagneticFermi Liquid (NAFFL) picture, not least the presence of the PG in the underdopedphase [2, 88]. Within this framework Kontani and coworkers [35, 36] demonstratedthat only by taking into account vertex corrections, the correct behavior of the Hallcoefficient RH(T, x) can be derived. Apart the validity of the NAFFL approach tocuprates, the outcomes of this kind of analysis open new questions about the roleof AF fluctuations in connection to the physics of pnictide systems (where actuallythe use of a NAFFL picture seems less questionable).

In Chapter 4 we directly compute the DC and Hall conductivities for a modelmultiband system in the presence of interband-dominating interactions, treatedwithin a finite-band Eliashberg approach, but, with the momentum dependenceof the boson propagator i.e. including the effect of vertex corrections. We willdiscuss, within this framework, how the unconventional Hall effect in the normalphase of pnictides, usually interpreted as related to a strong anisotropy between thescattering rate of holes and electrons, could be ascribed to the large contribution ofAF vertex corrections as recently highlighted in [38].

Chapter 2

Quantum Field Theory

Approach to Transport

Problems

The quantum-field theory will be the main tool used in the present project. In thischapter, after some basic definitions, we will review the main results of the LinearResponse Theory concerning the transport properties [89, 90]. First of all we willrecover the DC expression for the conductivity on the basis of the Kubo formula. Wewill also discuss the contribution of ScF to conductivity within the framework of thestandard theory of ScF [11]. Moreover we will show how it is possible to recover thesame results within a functional integral approach. Such a formalism represents apowerful technique to systematically investigate the topic in multichannel systems,since the generalization of this method in the presence of many interacting channelsor in the case of a multiband system is straightforward. After the DC conductivity,also the Kubo formula for Hall transport in a nearly free-electron system is derivedfollowing [90]. This result opens new questions about the analysis of transport incuprates and pnictides systems, since some of the assumptions required in the stan-dard derivation are no more valid in the context of HTSC. Finally, we will commentat the end of this Chapter about the connection between the Kubo-approach deriva-tion of transport properties and Boltzmann picture since this turn out to be a crucialpoint in the physics of HTSC.

2.1 Transport by Linear Response Theory

Linear response theory is an extremely widely used concept in all branches ofphysics. It simply states that the response to a weak external perturbation isproportional to the perturbation, and therefore all one needs to understand is theproportionality constant.

〈δA(t)〉 =∫ ∞

−∞dt′χAB(t− t′)B(t′). (2.1)

The derivation of the general formula for the linear response of a quantum systemto a perturbation can be found in literature [89] and is well known as Kubo Formula

χRAB(t− t′) = θ(t− t′)〈[A(t), B(t′)]〉, (2.2)

27

28 2. Quantum Field Theory Approach to Transport Problems

which expresses the linear response of a quantity A(t) to a perturbation B(t′) Here〈...〉 is the equilibrium average with respect to the non-perturbed Hamiltonian. Thisis the main remarkable and useful result of the linear response theory, because theinherently non-equilibrium quantity A(t) has been expressed as a retarded corre-lation function of the system at the equilibrium. The explicit calculation of theretarded response function is usually carried out by means of the solution of theequivalent correlation function in the Matsubara formalism [89] that is defined as

χAB(τ) = −〈Tτ [A(τ), B(0)]〉

χAB(iωn) =∫ β

0dτeiωnτ χAB(τ), (2.3)

where Tτ is the time ordering operator in imaginary time, and ωn are the discreteMatsubara frequencies (even for boson, odd for fermions). The corresponding re-tarded function at real frequencies is obtained by analytic continuation

χAB(iωn → ω + iη) = χRAB(ω), (2.4)

where η is a positive infinitesimal. A lot of Kubo formulas can be computed, butsince we are interested in transport we will focus in what follows on the estimate ofthe conductivity tensor. After introducing some basic definitions in Section 2.1.2,we will discuss the Kubo formula for the optical conductivity. Since it has beenextensively studied so far [89] in Section 2.1.2 we will briefly review only the mainresults and we will discuss some implication concerning the importance of carryingon a conserving approximation in order to preserve the Gauge Invariance (GI) ofthe response function. For a detailed derivation we refer to [89].

2.1.1 Basic Definitions

In a continuum system the definition of the current simply follows from the deriva-tive of the Hamiltonian with respect to the gauge field A, which enters the Hamil-tonian through the so-called minimal substitution

p +e

cA⇒ −i~∇+

e

cA (2.5)

(where e > 0 since we used explicitly the fact that q = −e for the electron). Indeed,the minimal coupling is the one which guarantees that in the Hamiltonian formalismthe equations of motions of a particle in the electromagnetic (em) field are exactlythe Maxwell equations. As a consequence, in the presence of an em field only thekinetic part of the Hamiltonian is corrected:

K =1

2m

∫

drc+(r)[

−i~∇+e

cA(r)

]2

c(r), (2.6)

where c†(r)/c(r). are the creation/annihilation operators of a fermion in r. Thecurrent operator is then defined as:

J(r) = −∂H

∂A= − e

mc+(r)

(

−i~←→∇ + eA(r)

)

c(r) = −eJP (r)− e2

mn(r)A(r). (2.7)

The first term JP correspond to the current-density operator in the absence of fieldand is proportional to the electron velocity. This is the so called paramagnetic

2.1 Transport by Linear Response Theory 29

current operator. The second term is the diamagnetic term, proportional to theelectron density. This must be retained since when we do linear-response theoryone needs all the terms proportional to the perturbing field i.e. proportional to A.

Notice that thanks to Eq. (2.7) one can express in general the Hamiltonian as

H(A) = H(0) +∫

dr

[

eA(r) · JP (r) +e2

2A2(r)n(r)

]

. (2.8)

Since we are interested in describing lattice systems, let us consider the generalcase of an electronic system described by the Hamiltonian:

H = H0 + Hint, H0 = −∑

ij

∑

σ

tijc†iσcjσ − µ

∑

i

∑

σ

c†iσciσ, (2.9)

where the field operator c†iσ creates an electron of spin σ at the i-th site, tij is the

hopping parameter, µ is the chemical potential and Hint is the interaction term.For the kinetic part H0 a proper description for example of the cuprates is obtainedby restricting the sum to next-neighboring sites on a squared lattice that results ina band dispersion

εk = −2t(cos kxa + cos kya) + 4t′ cos kxa cos kya, (2.10)

where a is the lattice constant. However, to avoid unnecessary complications wewill also refer below to the simplest case t′ = 0, using units ~ = kB = c = 1. In alattice system one would like some extension of the minimal-coupling substitutionwhich guarantees an expansion of the Hamiltonian equivalent to Eq. (2.8) above, atleast up to terms of O(A2), so that one can perform calculations in linear-response.Such property is satisfied by the so-called Peierls Ansatz [91], which corresponds toinserting the gauge field A in Eq. (2.9) by means of the substitution (here q = −eis used)

ci → cieie∫ ri A·dr. (2.11)

In this case, it is clear that when the interaction term of the Hamiltonian is a density-density interaction the kinetic hopping term is modified, while the interaction termis gauge invariant. As a consequence, the kinetic term acquires a phase factor

K = −t∑

iδ

∑

σ

c†iσci+δσ → t

∑

iδ

∑

σ

c†iσci+δσeieA(ri)·δ, (2.12)

where δ = x, y is the versor of the displacement. This implies that the Hamiltoniancan be expanded to leading orders in A [91]:

H(Ai) ≈ H(0) +∑

j

[

eAi(rj)JPi (rj) +

e2

2A2

i (rj)τii(rj)

]

, (2.13)

so that we recover the expansion Eq. (2.8) for the lattice Hamiltonian Eq. (2.9).Thus the i-th component of the total electric current density operator is againexpressed as

Ji(r) = − ∂H

∂Ai= −eJP

i (r)− e2τii(r)Ai(r). (2.14)


where now the paramagnetic term and the diamagnetic one are found to be

JP (q) =1N

∑

k,σ

v(k)c†k−q/2σck+q/2σ , (2.15)

τii =1N

∑

k,σ

∂2εk

∂k2i

nk,σ, (2.16)

τii is the so-called diamagnetic tensor that generalizes to the lattice case the n/mterm of Eq. (2.7), while JP is the extension of the usual paramagnetic current tothe lattice. Here we used the standard definition for the velocity,

v(k) =∂εk

∂k. (2.17)

2.1.2 Kubo Formula for the conductivity

Consider a system of electrons subjected to an external em field E. This is givenby the electric potential, φext, and the vector potential, Aext:

E(r, t) = −∇rφext(r, t)− ∂tAext(r, t). (2.18)

To linear order in the external potential the coupling of the electrons to the em fieldis

Hext = −e

∫

dr n(r)φext(r, t) + e

∫

drJP (r) ·Aext(r, t), (2.19)

where JP is defined in Eq. (2.15). In order to obtain the Kubo formula one needsto evaluate the expectation value of the current induced by the external em field,〈J(r, ω)〉 at first order in the perturbation. One find

Jµ(q, ω) = σµν(q, ω) Eν(q, ω). (2.20)

Here we used the generalized quadrivector notation where the index µ = (i, 0)indicates spatial and time components respectively, Jµ = (Ji, J0) and J0 is theparticle density. σµν are the conductivity tensor components defined as

σµν(q, ω) = − ie2

V (ω + i0+)

[

ΠRJµJν

(q, ω)− 〈τµµ〉δµν(1− δν0)]

, (2.21)

where V is the unit-cell volume, ΠRJµJν

is the retarded current-current correlationfunction and τµµ is the diamagnetic tensor Eq. (2.16). The q → 0 limit of Eq.(2.21) defines the AC (optical) conductivity. It easy to verify that in this limit theoff-diagonal component of the conductivity tensor vanish by symmetry. So we areleft only with the diagonal components that are equivalent for an isotropic system.Moreover we can extract the real and imaginary part of the conductivity in order toanalyze the dissipative and the absorptive part. In what follows we will deal onlywith the former kind of processes, thus we will need to evaluate

Reσµµ(ω) =e2

V

ImΠJµJµ(q = 0, ω)ω

. (2.22)

The DC conductivity is obtained by taking the limit ω → 0 of Eq. (2.22). Let usnotice that the opposite limit (ω = 0, q → 0) describes a static electric field, which


is periodic in the space. In this case the charge will seek a new equilibrium afterwhich no current will flow. Thus is important to take the q → 0 limit first. Let usintroduce here also the definition of the em kernel Qµν to which we will often referin what follows

Jµ(q, ω) = Qµν(q, ω) Aν(q, ω), (2.23)

where

Qµν(q, ω) = e2[

ΠRJµJν

(q, ω) − 〈τµµ〉δµν(1− δν0)]

, (2.24)

σµν(q, ω) = −iQµν(q, ω)

V (ω + i0+). (2.25)

From Eq. (2.21) we have found that the conductivity of a given system requiredthe evaluation of the retarded current-current correlation function:

ΠRJµJν

(q, ω) =∫

dtθ(t− t′)eiω(t−t′)⟨[

JPµ (q, t)JP

ν (−q, t′)]⟩

, (2.26)

that can be evaluated as usual from the standard analytic continuation iΩm → ω+iηof the equivalent correlation function in the Matsubara formalism [89]

ΠJµJν (q, iΩm) =T

N

∫ β

0dτeiΩmτ 〈Tτ JP

µ (q, τ)JPν (−q, 0)〉, (2.27)

τ is the imaginary time, β = 1/T , Ωm = 2πmT is the bosonic Matsubara frequencyand N is the number of unit cells. According the diagrammatic technique this canbe decomposed in a series of diagrams with increasing numbers of single-particleGreen’s functions and interactions lines [89]. If we consider a non-interacting elec-tron system we have

Π0JµJν

(q, iΩm) = −2∑

k

[G0(k−q/2, iωn + iΩm)vµ(k)G0(k+q/2, iωn)vν(k)], (2.28)

where G−10 (k, iωn) = iωn − ξk is the bare Green’s functions, ωn = iπ(2n + 1) is the

fermion Matsubara frequency and the factor 2 is due to the spin summation.As soon as interactions are present in the system, we have to consider their

effect on ΠJµJν (q, iΩm). Always by means of the diagrammatic technique [89] it iseasy to demonstrate that one can perform a partial summation of diagrams to allorders by replacing in Eq. (2.28) each bare Green’s function G0 by the full Green’sfunction G defined by the Dyson equation:

G−1(k) = G−10 (k)− Σirr(k), (2.29)

where Σirr is the irreducible self-energy, we are using k = (k, iωn) notation (see Fig.2.1).

The current-current correlation function Π0JµJν

(q, iΩm) Eq. (2.28) computedwithin this approximation is the so-called bare-bubble. Within this approximationwe perform only a partial summation by dressing the bare Green’s functions. Actu-ally, the bare bubble does not contain the diagrams where interaction lines connectthe two electron Green’s functions.


G(k) = = +

Figure 2.1. The interactions dresses the single-particle bare Green’s function G0(k):G(k) = G0(k) + G0(k)Σirr(k)G0(k).

The sum of these kind of diagrams defines the so-called vertex corrections. Theinclusion of vertex corrections in the evaluation of the response function is veryimportant since is connected to the GI property of the derived response functionand to the charge conservation law. Indeed, since the vector potential can have anarbitrary gauge, we have to require the invariance of the theory under the gaugetransformation

Aν → Aν + ∂νf Aν(q)→ Aν(q) + iqνf (2.30)

where f is any function that depends on position and time, moreover the chargeconservation requires

∂tρ + ∂µJµ = 0 qµJµ(q) = 0. (2.31)

From Eq. (2.23), one can check that Eq.s (2.30), (2.31) are fulfilled as long as theem kernel satisfied the following relation

qµQµν(q, ω) = Qµν(q, ω)qν = 0. (2.32)

In general in order to guarantee the GI and the charge conservation, the bare bubbleapproximation is not enough and we need to include the contribution of the vertexcorrections. Let us estimate the contribution of these diagrams. We define theirreducible line-crossing diagram Λirr as the sum of all possible diagrams connectingboth the upper and the lower fermion line which cannot be cut into two pieces bycutting both the upper and the lower line just once, Fig. 2.2.

Λirr ≡ ≡ + + + . . .

Figure 2.2

Using Λirr we can now resum all diagrams in the current-current correlation functionas

ΠJµJν (q, iΩm) = −2∑

k

[G(k−)vν(k−, k+)G(k+)Jµ(k+, k−)], (2.33)

where k+ = (k+, iωn + iΩm, ), k− = (k−, iωn, ) with k± = k ± q/2. The velocityvµ(k) Eq. (2.17) is the unperturbed vertex, while the dressed vertex function isgiven by an integral equation (see Fig. 2.3):

Jµ(k+, k−) = vµ(k) +∑

k′

G(k′+)Jµ(k′

+, k′−)G(k′

−)Λirr(k − k′). (2.34)


= +Jµ(k+, k−) =

Figure 2.3

The full bubble defined by Eq. (2.33) gives now the correct response function andsatisfies Eq. (2.32).

Although Eq.s (2.29), (2.34) are exact, we always need to evaluate them un-der some approximations. It is possible to demonstrate that in order to fulfill Eq.(2.32), the approximation used must be conserving i.e. once the self-energy is eval-uated within some approximations, a consistent approximation for the vertex partis required. Moreover one can prove that a conserving approximation is guaran-teed as soon as the vertex part and the self-energy satisfy a relation known as theGeneralized Ward Identity (GWI). For a continuous system the GWI reads [92]

qµJµ(k+, k−) = G−1(k−)− G−1(k+). (2.35)

Let us check this result. Explicitly we want to verify that if a conserving approx-imation is performed i.e. the em response function Qµν , Eq. (2.23), is evaluatedwith the vertex part and the self-energy satisfying the GWI, Eq. (2.35), then theEq. (2.32) is automatically fulfilled and the GI is preserved.

From Eq. (2.33), using the GWI Eq. (2.35) we have

ΠJµJν (q, iΩm)qν = −2∑

k

vν(k)[G(k+)− G(k−)], (2.36)

now it is easy to check that, by integration per part, the r.h.s. of Eq. (2.36) reducesto the diamagnetic tensor definition

ΠJµJν (q, iΩm)qν = τννqν , (2.37)

so that from Eq. (2.24) we find

Qµν(q, ω)qν = ΠJµJν (q, iΩm)qν − τµµδµνqn = 0, (2.38)

that is the GI condition Eq. (2.32).From the above discussion it follows that for a non-interacting system the GWI

(2.35) is satisfied for the free electron Green’s function taken together with thebare vertex. In general for an interacting system the bare bubble approximation isinstead not enough. If one considers the lower order approximation, the so-calledHF approximation the self-energy Σ(k) is given by

Σ(k) =∑

k′

G(k′)V (k − k′). (2.39)

where V (k − k′) is a static potential. In this case the bare vertex does not satisfythe GWI condition Eq. (2.35). A conserving approximation requires to evaluatethe integral equation for the vertex function as

Jµ(k+, k−) = vµ(k) +∑

k′

G(k′+)Jµ(k′

+, k′−)G(k′

−)V (k− k′). (2.40)


since at the HF level the Λirr is simply given by V (k − k′). Now the evaluation ofthe current-current response function Eq. (2.33) using the dressed Green’s functionEq.s (2.29), (2.39) and the dressed vertex Eq. (2.40) leads to a gauge invariantresponse function that satisfies Eq. (2.32).

2.2 ScF contribution to transport

As discussed in the first chapter, in the context of HTSC the analysis of ScF emergesas a relevant issue because, due to the presence of fluctuating Cooper pairs aboveTc, a lot of precursor (and observable) effects of the superconducting state takeplace. For example the contribution of the ScF to the DC conductivity leads toan enhancement of the conductivity as Tc is approached, on the other hand alsothe magnetic behavior is affected by the presence of the fluctuating Cooper pairsshowing a precursor diamagnetism above Tc. The nature of ScF depends on whetherthe system is weakly or strongly coupled, on whether preformed pairs are presentor not and a wealth of physical information can be obtained from the analysis ofthe ScF contribution to the physical quantities, provided a theoretical backgroundis established to extract them. A phenomenological approach based on the GLfunctional allows one to describe ScF near the transition and to account for theircontribution to different thermodynamical and transport properties. However amicroscopic description of the fluctuation phenomena is more convenient since thisapproach allows the microscopic estimate of the phenomenological parameters ofthe GL theory. Moreover, besides the direct Cooper pairs contributions, othersindirect effects can take place even changing the single-particle properties. Theanalysis of these effects requires a microscopic treatment. In the next section wewill summarize the main result of the standard theory of ScF and discuss the ScFcontribution to conductivity and diamagnetism (for a complete review see [11]).Then we will derived the same results by means of the functional integral approachemphasizing the main advantage of this technique in the context of multichannelsystems.

2.2.1 Standard Theory of ScF

The Cooper Channel of the electron-electron Interaction:the ScF Propagator

Let us start the microscopic description of fluctuation phenomena in a supercon-ductor from the electron Hamiltonian. We will choose it in the simple BCS form1

H = H0 + HI =∑

k

∑

σ

ξkc†kσckσ − g

∑

k,k′,q

∑

σσ′

c†k+qσc†

−k−σc −k′−σ′c k′+qσ′ . (2.41)

Here g > 0 is the coupling constant of the electron-electron attraction which issupposed to be momentum independent and different from zero in a narrow domain

1Fluctuations in the framework of more realistic Eliashberg theory were studied in [93] where ithas been demonstrated that the strong coupling does not change drastically the results of the weakcoupling approximation. In particular in HTSC the BCS approach in the analysis of ScF turns outto be appropriate for our purpose.

2.2 ScF contribution to transport 35

around the FS. ξk is the dispersion measured from the Fermi level, notice that thechoise of an anisotropic band dispersion

ξk =k2

‖

2m− tz cos(kzd)− µ (2.42)

is suitable since both cuprates and pnictides are layered SC. Here tz is the hoppingbetween the layers, d is the interlayer distance and a periodic notation for thez-component of the momentum kz is used.

As usual we make use of the diagrammatic technique in the Matsubara formal-ism. The single-particle properties are given by the quasiparticle Green’s function.G0(k) = (iεn − ξk)−1. The effective electron-electron attractive interaction, drivingthe superconducting instability of the system at Tc, formally leads to the appearanceof a pole in the two-particle correlation function

L(k, k′, q) = 〈Tτ [c†k+qσc†

−k−σc −k′−σ′c k′+qσ′ ]〉, (2.43)

Tτ is the usual time ordering operator and a 4-vector notation is used. We con-sider only the Cooper channel (p-p channel) of the electron-electron interaction inthe two-particle correlation function. This defines the propagator of the fluctuatingCooper pairs, L(q), which will be call below the ScF propagator. By means of thediagrammatic technique we can draw down the complete series of diagrams for theScF propagator. In the ladder approximation (see Fig. 2.1) the Dyson equation forL(q) reads

L−1(q) = g−1 −Π(q), (2.44)

where the Π(q) is the p-p bubble defined by a loop of two single-particle Green’sfunction in the p-p channel

Π(q) = − T

N

∑

k

G(k + q/2)G(−k + q/2). (2.45)

Let us expand at the leading order the propagator in order to find a GL-likeexpression for the ScF

L−1(q, ωm) = g−1 −Π(q, ωm) = m + c‖q2‖ + czq2

z + γ|ωm|, (2.46)

where the mass of the propagator is defined a m = g−1 −Π0, (Π0 ≡ Π(0, 0)), whilehigher order of the expansion are given by the stiffness along the i-th directionci = −(∂2

qiΠ)qi=0/2, and the damping coefficient γ = −(∂ωΠ)ω=0. Let us notice that,

having in mind a layered system here we expand the p-p bubble taking into accounta possible anisotropy of the system, so that the coefficients of the q expansion aredifferent for the in-plane c‖ and the out-of-plane cz components.

The transition temperature is determined as the highest temperature at whichthe pole of L(q, ωm) occurs. Since higher orders of the q = (q, ωm) expansion movethe pole towards lower temperature, the equation for Tc is given by the condition

m|T =Tc = 0 ⇒ 1g

= Π0|T =Tc . (2.47)


Figure 2.4. The Dyson equation for the ScF propagator (wavy line) in the ladder approx-imation. The bold point correspond to the model electron-electron interaction g, solidlines represent electron Green’s functions, the loop of single-particle Green’s function isthe Π(q) operator.

After the integration over the Matsubara frequencies one finds

Π0 =1N

∑

k

f(ξk)− f(−ξk)2ξk

∼ NF

∫

dξtanh (βξk/2)

2ξ= NF ln(ω0/T ), (2.48)

where f(ξk) is the Fermi distribution, NF is the DOS at the Fermi level and ω0 isthe cut-off frequency given by the Debye energy for conventional BCS, i.e. phonon-mediated, superconductivity. The transition temperature thus is given by Tc ∼ω0 exp−(1/NF g). The equation for Tc Eq. (2.47) allows the cancellation of the cut-off dependence from the mass definition since, introducing the reduce temperatureε = ln T/Tc, we have

m =1g−NF ln(ω0/T ) = NF ε. (2.49)

The leading terms of the expansion of Π(q) define the in-plane and the out-of-planestiffness

c‖ = − 18N

∑

k

v2‖(k)

[

f ′(ξk)ξ2

k

− tanh (βξk/2)2ξ3

k

]

, (2.50)

cz = − 14N

∑

k

v2z(k)

[

f ′(ξk)ξ2

k

− tanh (βξk/2)2ξ3

k

]

, (2.51)

with v(k) defined in Eq. (2.17), and the damping coefficient that, in the BCS limit,is given by γ = πNF /8T .


In order to avoid confusion in what follows, let us introduce here also anotherformulation of the expansion Eq. (2.46). In fact, it can be often found in literaturewritten as

L−1(q, ωm) = N[

ǫ + η‖q2‖ + rz sin2(qzd/2) + (π/8T )|ωm|

]

, (2.52)

where d is the interlayer distance. Comparing Eq.s (2.46), (2.52) one finds thecorrespondence between the coefficients of the two formulation c‖ = Nη‖ and cz =Nd2rz/4.

By direct inspection of the ScF propagator Eq.s (2.46), (2.52) one can also definethe scale of length over which fluctuations decade, this is the so-called correlationlength

ξ2‖(T ) =

c‖

m=

η‖

ε=

ξ2‖

ε, ξ2

z (T ) =cz

m=

d2rz

4ε=

ξ2z

ε, (2.53)

where we define the constants ξ‖ = c‖/N = η‖, ξz = cz/N = d2rz/4 that canbe interpreted as the low temperature correlation lengts. Obviously in a layeredsystem the in-plane correlation length ξ‖ will be larger than the interlayer one ξz.However it is not the anisotropy between the correlation lengths that defines theanisotropy degree of a layered structure but rather the ratio between the out-of-plane correlation length ξ2

z (T ) and the interlayer distance d. This could appearreasonable in principle since when ξz(T ) ≫ d the layers are coupled so one couldexpect a more isotropic (i.e 3D) behavior of the ScF, while in the opposite limitξz(T )≪ d one could see the system like a quasi-2D structure. Since the correlationlength diverges as the transition is approached, one would expect a 2D-3D crossoverof the ScF in the vicinity of Tc. Later on we will discuss again this topic within theanalysis of paraconductivity.

The ScF Contribution to the em Response Operator

Once defined the ScF propagator L−1(q), in order to derive the contribution of ScFto transport and thermodynamic properties, we need to evaluate the contributionof the fluctuating Cooper pairs to the em kernel. This can be done by means ofthe standard perturbation theory. The explicit derivation of this quantity and theestimate of the contributions of ScF to the several physical quantities above Tc canbe found in literature [11]. Here we only review the results and discuss about theirsuitability for the analysis of HTSC.

Let us start from the Qµν(q.ω) kernel definition Eq. (2.24). In Fig. 2.5 we rep-resent the appropriate diagrams corresponding to the leading order of perturbationtheory in the ScF that can be drawn according to the rules of the diagrammatictechnique [11].

The first diagram represents the so-called AL contribution and turns out to bethe direct contribution due to the presence of the fluctuating Cooper pairs aboveTc. The contribution to the conductivity that comes out from this diagram iscalled paraconductivity. As we mentioned in the introduction to this section themicroscopic approach allows one to derive also indirect contributions connected toquantum effects or to changes in the single-particle properties. They are indeedrepresented by the others diagrams in Fig 2.5.

The first class of diagrams represent the so-called MT contribution. It is apurely quantum effect generated by the coherent scattering of the electrons coupled


(a) (b) (c)

Figure 2.5. Feyman diagrams for the leading-order contributions to the fluctution con-ductivity. Wavy lines are fluctuating propagators, thin solid lines are single-particleGreen’s functions. The three kind of diagrams represent: (a) AL, (b) MT, (c) DOScontributions respectively.

in a Cooper pair on the same elastic impurities. This contribution appears only intransport coefficients and have a temperature singularity that is similar to that ofthe paraconductivity, although, being extremely sensitive to electron phase-breakingprocesses and to the orbital pairing, it is typically negligible [11]. In particular theexperiments on HTSC have shown that the excess of conductivity above Tc canusually be explained in term of the paraconductivity alone. In cuprates this featurehas been interpreted as a consequence of the d-wave symmetry of pairing which killsthe MT processes [94].

Another indirect contribution comes from the others diagrams in Fig 2.5. Theyaccount for the suppression of the single-particle DOS at the Fermi level, indeedif some electrons are involved in (fluctuating) pairing they cannot simultaneouslyparticipate in charge transfer as single-particle excitations. The decrease of the sin-gle particle DOS at the Fermi level leads to a reduction of the Drude conductivity.This indirect correction appears side-by-side with the paraconductivity (being aconsequence of the fluctuating Cooper-pairs formation) has opposite sign and turnsout to be much less singular in comparison to the AL contribution, thus it can beneglected in the estimate of ScF effect near Tc. In what follows we will discuss thecontributions of the AL diagram only.

The AL contribution to the em kernel can be written as

δQALµν (q, ωm) = 4e2T

∑

Ωk

∫d3k

(2π)3Bµ(k+, Ωk, ωm)L(k, Ωk)

× Bν(k+, Ωk, ωm)L(k+, Ωk + ωm), (2.54)

where

Bµ(k, Ωk, ωm) = T∑

εn

∫d3k′

(2π)3vµ(k′)G(k′, εn+

)G(k′, εn)G(k − k′,−εn−), (2.55)

we used k+ = k + q, εn+= εn + ωm and εn−

= εn−Ωk. Close to the transition themost singular behavior of the integrals in Eq.s (2.54), (2.55) comes from the ScFpropagators, thus we will approximate Bµ(kΩk, ωm) ∼ B(q, 0, 0) and evaluate it atsmall q.


Paraconductivity and Diamagnetism

Starting from Eq. (2.54) the paraconductivity, derived for the first time by Asla-mazov e Larkin in [95], simply follows from the dynamic limit (q = 0, ω → 0) afteranalytical continuation of the Matsubara frequency ωm to the real frequency ω (seeSection 2.1.2). It is possible to evaluate both the in-plane and the out-of-plane con-tribution, but in the analysis of HTSC the in-plane paraconductivity δσxx turns outto be usually more useful, thus hereafter we will focus on this, µ = ν = x. Takinginto account that the integrals in Eq.s (2.54), (2.55) are dominated in the vicinityof Tc by the singular structure of the propagators several approximations can bedone and one finds that the leading term for the paraconductivity reads

δσAL =e2

πT

∫d3k

(2π)3B2

x(k)∫

dz

sinh2(z/2T )

(

ImLR(k,−iz))2

. (2.56)

Eq. (2.55) can be evaluate at small k as Bx(k) = −2Nη‖kx = −2c‖kx [11] andwe can insert the explicit form of the propagator Eq. (2.52) in Eq. (2.56). Theresulting expression for the paraconductivity is

δσAL =e2

16~d√

ε(ε + rz), (2.57)

where we restored the ~ unit. Notice that it has been demonstrated that Eq.(2.57) is general within a hydrodynamic description of the collective modes, anddoes not rely on any particular assumption about the pairing strength [96]. Bydirect inspection of Eq. (2.57), it is easy to verify that an intrinsic crossover takesplace. Indeed, if ε ≫ rz one can neglect the rz term in the denominator so thatδσAL ∼ 1/ε, while if we are in the opposite limit ε ≪ rz one can neglect the ε2

term and find δσAL ∼ 1/√

ε. This crossover is related to the spectrum anisotropy,indeed by definition rz = 4ξ2

z /d2. As already mentioned the degree of anisotropyof a system is intimately connected to the ratio between the correlation length ξz

and the interlayer distance d. The conditions ε ≫ rz (i.e. d ≫ ξz(T )), ε ≪ rz

(i.e. d ≪ ξz(T )) essentially define a crossover between a 2D character of the ScFand a 3D one, where the crossover temperature will be nearer to TC as the moreanisotropic will be the system

δσAL2D =

e2

16~d

1ǫ

ε≫ rz, (2.58)

δσAL3D =

e2

32~ξz

1√ǫ

ε≪ rz. (2.59)

Another consequence of the presence of fluctuating Cooper pairs above Tc con-cerns the behavior of the magnetic susceptibility of a superconductor above Tc. Thefluctuation-induced magnetic susceptibility is the precursor effect to the Meissnerdiamagnetism. It can be measured and turns out to be a not-negligible effect inHTSC as discussed in the first Chapter. The fluctuations contribution to the dia-magnetism can be derived starting from Eq. (2.24). It is connected to the staticlimit (q → 0, ω = 0) of the transverse component of the current-current responsefunction

δχd = −[

δQALt (q, ω = 0)/q2

]

q→0. (2.60)


The resulting expression for the ScF diamagnetism is

δχd = −e2T

3πd

ξ2‖

√

ε(ε + rz), (2.61)

where again one can observe the intrinsic dimensional crossover already discussedfor the paraconductivity.

From the experimental point of view the dimensional crossover of the ScF con-tribution near Tc is clearly visible in paraconductivity and diamagnetism data ofoptimally and overdoped cuprates (where there is a general agreement about thevalidity of the GL approach in the analysis of the ScF ). However, the interlayercoupling has a different relevance in the various families of cuprates, thus we canfind a substantial 3D behavior Eq. (2.59) in YBa2Cu3O6+x samples, while moreanisotropic Bi2Sr2CaCu2O8 or La2−xSrxCuO4 compounds show 2D fluctuations Eq.(2.58) also when Tc is approached, being the 2D-3D crossover too close to Tc to beclearly observed (see Fig.2.6)

Figure 2.6. From [97]: Paraconductivity for samples of YBCO-123 (triangles), BSSCO-2212(squares) and BSSCO-2223 (circles) plotted against ε = lnT/Tc on a ln-ln plot.The dotted and solid lines are the AL theory in 3D and 2D respectively. The dashedline is an extension of the ScF theory at higher temperature that is irrelevant for ouranalysis close to the transition.

Concerning the dimensional behavior of the ScF in pnictides, the experimentalscenario is still incomplete also due to the to the only recent availability of largeclean single crystal. However some anomalies has been already detected. For exam-ple a general disagreement of different experimental probes in the estimate of theanisotropy degree of several compounds is found. The case of LiFeAs is a remark-able example. A small anisotropy of the band dispersion is predicted by DFT [98]


in this material. Moreover the 3D character of the SC states is well established byseveral experimental technique ranging from upper critical field measurements [99],critical current anisotropy experiments [100] to directional heat transport measure-ments [101]. On the contrary NMR studies, by analyzing the spin lattice relaxationrate 1/T1T , reveal 2D AF SpF in the normal state [102] The presence of a 2D-3Dcrossover just above Tc [103] is supported by he experimental results for the ScFin this material that reveale a marked 2D fluctuation regime which extends up totemperatures very near to Tc [103, 104] (at least up to ε = ln T/Tc = 0.02). Thisobservation is strange since on the basis of the essential isotropic character of theband dispersion one could expect a pronounced 3D character of the ScF with acrossover temperature located not so close the criticality.

Recently we address this topic in [40]. Here we show that, while in a single-band system the only relevant parameter is the degree of anisotropy of the struc-ture defined by the strength of the interlayer coupling of the system, the crossovertemperature in a multiband system with dominant interband interaction is stronglyaffected by the anisotropy of the spin interactions themselves, possibly due to thechanging of nesting or orbital properties along the z-axis. Moreover, it has beenshown in [39], that the interband coupling between the bands leads to a differentcontribution of the various bands (depending on the band parameters) to the sin-gle collective mode which controls critical ScF. Thus the correctly estimate of the2D-3D crossover temperature of the ScF contribution could be strongly affected byseveral parameters, (i) the structural anisotropy of these compounds (d vs ξz), (ii)the intrinsic anisotropy of the pairing interaction, i.e. the kz dependence of theSC coupling possibly due to the changing of nesting or orbital properties along thez-axis, (iii) the interband nature of the pairing, which leads to a weighted contribu-tion of the various bands in the single collective mode which controls critical ScF.By taking into account these effects in a coherent scheme one can correctly estimatethe position of the crossover and reconcile the different experimental results.

2.2.2 Functional Integral Approach to ScF in multichannel systems

The GL functional obtained by means of the diagrammatic technique can be alsoderived making use of the method of functional integration. As we shall see, thefunctional integral approach allows one to treat on the same footing any relevant de-gree of freedom. For this reason it is a powerful tool for the analysis of multichannelsystem.

In this Section we aim to reformulate the standard procedure in the functionalintegral language. After that we will discuss the route to follow in order to generalizethe derivation in multichannel systems.

Functional Integral method

Let us start from the microscopic electron Hamiltonian Eq. (2.41), and let us con-sider also in this case only the Cooper channel of the electron-electron interaction.We can define the corresponding action as

S[ci(τ)] =∫ β

0dτc†i(τ)[∂τ − µ]ci(τ) + H[ci(τ)], (2.62)

where τ is the imaginary time and β = 1/T .


The complete description of the thermodynamic properties of the system can beobtained through the exact calculation of the partition function

Z = T r e−S[ci(τ)]. (2.63)

where the trace acts over all the degrees of freedom of the system and S is themicroscopic action defined by Eq. (2.62).

Near the SC transition, side by side with the fermionic electron states, fluctua-tion Cooper pairs of bosonic nature appear in the system. They can be describedby means of complex bosonic fields ∆ which can be treated as "Cooper pair wavefunctions". Thus in the order we need first to introduce the bosonic field and thento integrate out the electron degrees of freedom, such that we are left only withthe functional integration carried out over all possible configurations of the Cooperpairs wave functions:

Z =∫

D∆e−S[∆], (2.64)

S[∆] is the effective action of the system. Our task now consist in the explicitderivation of S[∆]. The standard procedure to derive the effective action can befind in literature [53], [105], and it can be summarized in four steps:

1. First of all we need to decouple the interaction term of the microscopic Hamil-tonian Eq. (2.41) in the p-p channel. This can be done by means of theHubbard-Stratonovich (HS) transformation that introduces the auxiliary com-plex bosonic field ∆ coupled to the operator Φq =

∑

k ck+q↑ck↓:

egΦ†Φ =∫

φe−|∆|2/g+(Φ†∆+h.c.), (2.65)

The HS field plays the role of the SC order parameter in the GL functional

2. Now our action is quadratic in the fermionic degrees of freedom. Thus we canintegrate out the fermions exactly. We are left with an effective action S[∆]only in terms of the HS field ∆

S =∑

q

|∆q|2g− Tr log Akk′ . (2.66)

where the trace acts over momenta, frequencies and spins. We used the no-tation k ≡ (k, iεn), q ≡ (q, iωm), while εn, ωm are the Matsubara fermionand boson frequencies, respectively. The Ak,k′ matrix is defined (in the usualNambu notation) as

− Akk′ =

(iεn − ξk)δkk′ ∆k−k′

∆∗k−k′ (iεn − ξk)δkk′

(2.67)

3. We can now fix the solution of the saddle-point equation at the MF value of theHS fields. Explicitly one has to decompose Eq. (2.67) as Ak,k′ = −G−1

0 δk,k′ +Σk−k′. Here G−1

0 δk,k′ contains the q = 0 saddle-point values ∆q=0 ≡ ∆0 of


the HS field obtained by minimization of the MF action while Σk−k′ containsthe fluctuating parts of the HS fields:

S =|∆q|2

g− Tr log

[

G−10 − Σk−k′

]

=|∆q|2

g− Tr log G−1

0

︸︷︷︸

SMF

− Tr log (1− G0Σk−k′)︸︷︷︸

SF L

, (2.68)

The solution of the MF equation ∂∆0SMF = 0

(1g− Π0

)

∆0 = 0 (2.69)

reduces to the standard BCS self-consistent equation for the superconductingorder parameter:

∆0 = 0 T > Tc,

g−1 =∫

dξN(ξ)tanh(βE/2)

2ET < Tc

(2.70)

where E =√

ξ2 + ∆20.

4. Beyond MF we need to consider SF L. It easy to verify that this term can bewritten as SF L =

∑

n1nTr[G0Σk−k′]n. This expression is exact since contains

all the terms of the expansion around ∆0. In order to analyze the ScF atGaussian level we consider the expansion up to the second order in HS fields.Moreover since we are interested in ScF above Tc we can put now ∆0 = 0 inAkk′ . The Gaussian effective action reads

SG = ∆†L−1∆, (2.71)

where L−1(q, ωm) = g−1 − Π(q, ωm) is the ScF propagator. We recover thesame expression obtained by means of the diagrammatic technique Eq. (2.46),thus expanding at the leading order the propagator one recover the usual GL-like expression for the ScF

L−1(q, ωm) = g−1 −Π(q, ωm) = m + c‖q2‖ + czq2

z + γ|ωm|, (2.72)

as we already discussed.

Once the expression for the propagator is obtained one can use the results of thestandard diagrammatic approach illustrated in the previous section for computingthe ScF contribution to the em kernel. However by means of the functional integralapproach we can go further and compute explicitly the em kernel Qµν and itsfluctuating part δQµν via functional derivation as

Qµν =δ2 ln Z[A]δAµ

q δAν−q

|Aµq =Aν

−q=0 (2.73)

Obviously in order to do this we need to introduce the gauge field A in our modelHamiltonian Eq. (2.41). This can be done via the minimal substitution for contin-uous system Eq. (2.5) or by means the Peierls ansatz (2.11) in lattice system. Then


we can follow the procedure illustrated above in order to define the effective actionS[A, ∆].

In order to compute the AL-like ScF contribution to the em kernel δQµν , weneed to compute the term order A∆2 in the effective action as:

S(A, φ) = −12

QMFµν Aµ

q Aν−q + Aµ

q F µ(∆) (2.74)

where Qµν is the MF part of the em kernel defined in Eq. (2.24), and F µ(∆) is afunction of the ∆ field which describes the connection of the em potential to the HSfield. The partition function is given by Z[A] =

∫D∆e−SGL(A,∆) so that the result

of the functional derivative Eq. (2.73) is

Qµν(q) =δ2 ln Z[A]

δAαq δAβ

−q

∣∣∣∣∣Aα

q =Aβ−q=0

= QMFµν (q) + δQµν(q) (2.75)

where QMFµν (q) simply follows from the quadratic term of Eq. (2.74), while δQµν(q)

is the contribution coming from the fluctuating mode coupled to A, and dependson the explicit form of the F(∆) function:

δQµν(q) = 〈F µ(∆)F ν(∆)〉. (2.76)

In this case we consider only the Cooper channel so that only the pairing ∆ fieldhas been introduced by means of the HS transformation. The relevant terms in theeffective action are given by

S = −12

QMFµν Aµ

q Aν−q − cqν2eAν

q′

(∆∗

q∆q−q′ + ∆∗q+q′∆q

), (2.77)

where summation over repeated indices is implicit. After some manipulation ofthe indices it easy to verify that in this case the em potential A is coupled withthe pairing field ∆ via a term ∼ c k ∆2

k, so that the fluctuation correction to thecurrent-current response function (2.76) is given by the usual AL contribution (seeEq. (2.54) with B(k) = −2 c k and Fig.2.7):

δQµν(q) = 4e2c2T∑

k,Ωk

(2k + q)µ(2k + q)νL(k, Ωk)L(k + q, Ωk + ωm), (2.78)

where L(k) = 〈∆∗k∆k〉 is the propagator of the ScF.

Multichannel Systems Analysis in the Functional Integral Approach

During this work we generalize the Functional Integral Method to the analysis oftransport properties of multichannel systems. Indeed, the HTSC materials un-der consideration are characterized by the presence of several interacting channels(Cooper, density, current, spin and so on). Moreover in pnictides the situation iseven more involved since the presence of several bands at the Fermi level allowsfor interband transitions, that seems to be the dominant interaction in these com-pounds.


Figure 2.7. Sketch of the AL contribution of the ScF to the current-current correlationfunction. The dashed lines represent the ScF propagator L(k, Ωk), while the red dotsrepresent the factor 2c. In comparison with the graphic illustration of Fig.2.5 the redsdots correspond the blocks B(k) = −2 c k.

The functional integral approach represents a very useful method, since thederivation of the effective action starting from the microscopic Hamiltonian is straight-forward and can be easily generalized to take into account many degrees of freedom.

In the presence of many channels of interaction one should decompose the in-teracting Hamiltonian in any relevant channel, not only the Cooper one. Once therelevant degrees of freedom are take under consideration, one has to introduce manyHS auxiliary fields as many the relevant channel of the interaction. This procedureleads to the definition of the effective action in terms of any of the relevant HS field.If one is interested in the transport properties in multichannel system the evaluationof the em kernel could appear in principle a hard task. However by means of theprocedure summarized above the estimate is straightforward:

• By means of the minimal substitution or of the Peierls Ansatz one introducesthe gauge field in the microscopic Hamiltonian.

• The interacting term of the Hamiltonian is written in terms of the relevantchannels. The HS transformation Eq. (2.65) has to be performed with theintroduction of the HS fields φi associated to the relevant interacting channel(with i channel index).

• After the integration of the fermions we are left with S[A, φi]. Now in orderto recover the AL-like ScF contribution to the em kernel we need to retain thecontributions of the fields which couple both to the gauge field A and to ∆2

since they will contribute to modify the F (∆) function in Eq. (2.76). Nowwe should solve the functional integrations over these relevant fields in thepartition function in order to define the effective action up to leading order inthe gauge field as

SGL(A, ∆) = −12

QMFµν Aα

q Aβ−q + Aµ

q F µ(∆), (2.79)

where F (∆) is the function that coupled the gauge field to the ∆ as obtainedafter the functional integration.

• The functional derivation Eq. (2.73) leads to the definition of the ScF contri-bution as

δQµν(q) = 〈F µ(∆)F ν(∆)〉 (2.80)


We will adopt this procedure in the analysis of the interplay between the p-h andp-p degrees of freedom in Chapter 3. Here we will discuss the subtleties connectedto the derivation of the effective action and of the current-current response func-tion by taking into account the presence of current-current interactions in a modelin which density-density interactions introduce HF correction to the bare dispersion.

On the other hand the functional approach allows also to handle with carethe multiband character of the system. In this case even by taken into accountthe interaction only in the Cooper channel we have to account for several pairingHS fields associated to several bands. The generalization of the derivation of theeffective action present some subtleties related to the dominance of the interbandinteractions since this mechanism leads to a mixture of attractive and repulsivecharacter of the interaction. This has been done in [39] in which the analysis of ScFfor a multiband system is discussed.

2.3 Hall Effect

Until now we discussed about DC conductivity, now we want to analyze the Halltransport. In particular in Section 2.1.2 we reviewed the Kubo formula for con-ductivity, here we want use the same approach to computed the gauge-invariantdescription of other transport phenomena, as for example the Hall effect.

While this approach to the analysis of conductivity is widely discussed in lit-erature [89], the derivation of a gauge-invariant response function by means of theKubo formula for the Hall conductivity, discussed for the first time by Fukuyamaet al. in [90], is less known. Thus we summarize the main steps of the derivationbefore discussing the results.

2.3.1 Hall Transport in Kubo formalism

In ordinary transport the em kernel defines the current J induced by the presenceof an em field A Eq. (2.23). On the other hand, in order to have an Hall current wehave to to consider the response to the perturbing gauge field A in the presence of auniform magnetic field along a perpendicular direction. Indeed, only in this case anelectromotive force along the perpendicular axis arises and the Hall coefficient RH

is finite. Explicitly, if we take the direction of a uniform current with density Jx asthe x-axis and apply a uniform magnetic field Hz along the z-axis, an electromotiveforce is induced along the y-axis and the Hall coefficient is then defined by RH =Ey/JxHz. When the magnetic field is weak enough, RH is given by

RH =σxy

σ2xxHz

. (2.81)

The standard result RH = −1/nec follows from the definition making use of theequilibrium condition E = −(v×H)/c and J = −ev. Hereafter we use c = 1 unit.

In what follows an expression for the current density is given by the linearresponse theory when the uniform electric field is weak enough. The results sum-marized here have been obtained so far by Fukuyama et al., for further details werefer to their paper [90].

2.3 Hall Effect 47

We consider a nearly free-electron system described by parabolic dispersion εp =p2/m. The perturbation is generated by the vector potential A(r) = Aqeiqr. Theconductivity tensor that describes the response of the system to this field can bewritten by the Kubo formula as

σµν(q, ω) =1iω

[

φµν(q, ω + iδ) − φµν(q, 0 + iδ)]

, (2.82)

with

φµν(q, ω + iδ) =T

V

∫ β

0dτ

∫ β

0dτ ′eiΩm(τ−τ ′)〈Tτ Jµ(q, τ)Jµ(q, τ ′)〉H , (2.83)

where the thermal average and the time evolution are determined by the Hamilto-nian

H = H(0)− J(−q)Aq. (2.84)

Since we are interested in the effect of a weak uniform magnetic field (generated bythe gauge field A), let us expand φµν up to linear order A

φµν ∼ Qµν + KαµνAα. (2.85)

The zero-th order kernel contribution Qµν describes the ordinary response Jµ ofthe system to a gauge field Aν Eq. (2.23) in the absence of magnetic fields. Inthe limit q → 0 only the diagonal components survive as discussed in the previousSection 2.1.2 and one recovers the standard definition for the condutivity σxx. Thelinear term Kµν defines instead the current Jµ induced by the gauge field Aν in thepresence of a uniform magnetic field generated by Aα. This is the em kernel thatwe need to evaluate in order to find the σxy conductivity needed for computing theHall coefficient Eq. (2.81)

σµν(q, ω) =1iω

Kαµν(q, ω)Aα

q . (2.86)

Let us evaluate explicitly the Kαµν kernel. At the first order in A we have

only two kind of relevant contributions from Eq. (2.83). In the first one we takeH = H(0) from Eq. (2.84) and the diamagnetic term for one J, so we have termslike

T re−(β−τ)H(0)eJPµ e−τH(0) e2

mn(−q)Aν

q.

The second kind of contributions contain instead the paramagnetic part for bothcurrent-density operators J and the linear A term from Eq. (2.84), so that in thelimit of weak field we can approximate the exponential as

e−H ∼ e−H(0)(1 + e JPα (−q)Aα

q).

These terms give rise to average of three JP operators. Hereafter we omit the Psuperscript of JP . We are left with

Kαµν(q, ω) =

e2

mδνα (Lµ(q, ω + iδ) −Lµ(q, 0 + iδ))

− e3 (Lαµν(q, ω + iδ) − Lα

µν(q, 0 + iδ)), (2.87)


Jµ

p+, ωnp−, ωn−

δνα Jν

p−, ωn− p+, ωn

p+, ωn−Jα Jν

Jµ(a) (b)

p−, ωn− p+, ωn

p−, ωn Jα

Jµ(c)

Figure 2.8. Contributions to the Kαµν kernel: (a) Lµ Eq. (2.88), (b-c) Lα

µν Eq. (2.89)

where

Lµ(q, iΩm) = −T

V

∫ β

0dτ

∫ β

0dτ ′eiΩm(τ−τ ′)〈Tτ Jµ(q, τ)n(−q, τ ′)〉, (2.88)

Lαµν(q, iΩm) = −T

V

∫ β

0dτ

∫ β

0dτ ′

∫ β

0dτ ′′eiΩm(τ−τ ′′)

〈Tτ Jµ(q, τ)Jα(−q, τ ′)Jν(0, τ ′′)〉. (2.89)

where the time evolution and the thermal average are defined by the H(0) Hamil-tonian. The contributions to Kα

µν of Eq. (2.87) are shown in Fig.2.8. Now we needto evaluate these correlations function under some approximations. The interactiondresses both the single-particle Green functions and the vertex as shown in Eq.s(2.29), (2.34) and we have already discussed the self-consistent approximation wemust use for the self-energy and the vertex part in order to ensure the GI of theresponse. To be specific let us discuss the case of scattering by a static potential 2

V (p− p′, ωn − ωn′) = V (p− p′)δn,n′ (2.90)

Let us consider the self-energy at the HF level Eq. (2.39), the vertex is given byEq. (2.40) that we repeat here for convenience

J(p+, ωn; p−, ωn−) = v(p) +

∑

p

G(p′+, ωn)J(p′

+, ωn; p′−, ωn−

G(p′−, ωn−

)V (p− k).

(2.91)Under this approximation the kernel Eq. (2.83) is given by (see Fig.):

Kαµν(q, Ωm) =

e2

mδνα

T

V

∑

p,n

G(p+, ωn)Jµ(p+, ωn; p−, ωn−)G(p−, ωn−

) +

− e3 T

V

∑

p,n


)Jα(p−, ωn−; p+, ωn)

× G(p+, ωn−)Jν(p+, ωn−

; p+, ωn)

− e3 T

V

∑

p,n


)Jν(p−, ωn−; p−, ωn)

× G(p−, ωn)Jα(p−, ωn; p+, ωn), (2.92)2The arguments that follow stay valid also in the case of dynamical interactions. In the case

in which the potential should be momentum-independent instead no vertex corrections rise in thecalculus and the GI follows simply by the calculus of the bare bubble.

2.3 Hall Effect 49

where p± = p± q/2 and ωn−= ωn − Ωm.

It can be proved that the GI is not broken for any finite q. However we areinterested in the weak-field limit effect thus we will retain only terms linear in themagnetic field i.e in q. To ensure the GI according to Eq. (2.86) the Kµνα kernel,Eq. (2.92), must satisfies

Kαµνqα = 0. (2.93)

We notice that if Kαµν has the following q-dependence

Kαµν ∼ qµδµν − qνδµα, (2.94)

then Eq. (2.93) is trivially satisfied. Moreover if Eq. (2.94) holds, then from Eq.(2.86) we find

σxy ∼ KαµνAα ∼ (qµδµν − qνδµα)Aα ∼ qxAy − qyAx ∼ (∇×A)z = Hz, (2.95)

so that we obtain directly a term linear in the magnetic field. Notice also that sincean imaginary unit comes out from ∇×A ∼ iq×Aq it cancels out in Eq. (2.86) sothat we will retain only the real part of the σxy conductivity.

To derive explicitly the structure of Eq. (2.94) one should expand Eq. (2.92)to linear order in q. The q-linear contributions can be found by expanding eitherthe arguments of the Green’s functions or the vertices. Let us discuss briefly thederivation:

• the first term in the r.h.s. of Eq. (2.92):

Here we have two contributions coming from the expansion of the argumentsof the single-particle Green’s functions so that for the vertex function we haveto consider only the q = 0 contribution of the vertex equation Eq. (2.91),i.e. Jµ(p, ωn, Ωm). The others contributions come from the derivative of thevertex coupled to the applied electric field, Jµ:

Jµ(p+, ωn; p−, ωn−) = Jµ(p, ωn, Ωm) + J ′

µ, (2.96)

which satisfies the integral equation derived by the expansion of Eq. (2.91):

J ′µ =

12

∑

ρ

qρ

∑

p′

V (p− p′)∂ρG(p′, ωn)Jµ(p′, ωn, Ωm)G(p′, ωn−)

−G(p′, ωn)Jµ(p′, ωn, Ωm)∂ρG(p′, ωn−).(2.97)

• the last two terms in the r.h.s. of Eq. (2.92):

For this triangular diagrams it easy to verify that the contributions obtainedby deriving the Green’s functions cancel out due to the isotropy of the system.Concerning the expansion of the others two vertexes let us observe that thevertex coupled to the magnetic field, Jα, does not carry any frequency, i.e.satisfies the vertex equation, Eq. (2.91) with Ωm = 0. As a consequence ithas the symmetry under the interchange between p− and p+ so that

Jα(p−, ωn; p+, ωn) ∼ Jα(p, ωn) +O(q2). (2.98)


Moreover in order to obtain a conserving approximation we can use the GWIEq. (2.35) from which follows

G2(p, ωn)Jα(p, ωn) = ∂αG(p, ωn). (2.99)

Finally for the third vertex Jν (the ones coupled to the induced electric field),the expansion gives

Jν(p±, ωn−; p±, ωn) = Jν(p, ωn, Ωm)± 1

2qρ∂ρJν(p, ωn, Ωm). (2.100)

By means of Eq.s (2.96), (2.97), (2.99), and (2.100) one can draw out the expansion.In the procedure several terms cancel out and one is left with four contributions:

Kαµν(q, iΩm) = −e3

2

[

qρJµ∂ρJν(G2Jα)G(−)− qρJµ∂αJν(G2Jα)G(−)

−qρJµ∂ρJν(G2(−)Jα)G + qρJµ∂αJν(G2(−)Jα)G]

.(2.101)

where

G = G(p, ωn), G(−) = G(p, ωn − Ωm), Jµ = Jµ(p, ωn, Ωm).

Now using the GWI Eq. (2.99) and the identity following from the symmetry of theisotropic system qρ · ∂ρ = δναqµ∂µ + δµαqν∂ν one finds

Kαµν(q, iΩm) = e3 (qµδµν − qνδµα)

T

V

∑

p,n

(Jµ∂νJν − Jν∂νJµ)G∂µG(−)− ∂µGG(−) (2.102)

where we added a factor two in order to take into account the spin multiplicity.Finally we consider the case in which J has the same symmetry of p, i.e. of v(p) =p/m

J(p, ωn, Ωm) = v(p) κ(p, iωn, iΩm), (2.103)

and use that Hα = i(qµδµν − qνδµα)Aα to get

σxy(ω) = ωce2

ω

T

V

∑

p, n

κ2(p, iωn, iΩm)vx(p)G∂µG(−)− ∂µGG(−)|iΩm→ω+iδ,

(2.104)where ωc = eHz/m is the cyclotron frequency.

The diagonal element of the conductivity tensor σxx derived in the previoussection Eq.s (2.22)-(2.33) is given by

σxx(ω) = −2e2

iω

T

V

∑

p, n

v2x(p)κ(p, iωn, iΩm)GG(−)|iΩm→ω+iδ, (2.105)

where the vertex part is evaluated within the same approximations used for theoff-diagonal element Eq. (2.104).

2.3 Hall Effect 51

After the n-summation over the Matsubara frequencies (for detailed derivationsee [89, 90]) the leading terms are given respectively by

σxx = − e2

V π

∑

p

v2x(p)

∫ ∞

−∞dz

∂f(z)∂z

κ(p, z)GR(p, z)GA(p, z), (2.106)

σxy = ωce2

V 2πi

∑

p

vx(p)∫ ∞

−∞dz

∂f(z)∂z

κ2(p, z) ×

GR(p, z)∂xGA(p, z) − ∂xGR(p, z)GA(p, z) (2.107)

where κ(p, z) = κ(p, z − iδ, z + iδ), GR/A(p, z) = G(p, iωn → z ± iδ) is the re-tarded/advanced Green’s function gives by

GR(p, z) = z − εp + µ− Σ′(p, z) − iΣ′′(p, z)−1, (2.108)

where Σ′ and Σ′′ are the real and the imaginary part of the self energy. We canexpand the self-energy in terms of z and εp. Assuming Σ′ weakly dependent on εp

and Σ′′ almost constant in z and εp we have

Σ(p, z) = Σ′(p, z) + iΣ′′(p, z) ≃ −λz − ξp − iΓ, (2.109)

where λ = −∂zΣ′|z=0, and Γ = −Σ′′(p, 0)|p=pF> 0. Thus

GR(p, z) = z(1 + λ)− ξp + iΓ−1. (2.110)

The quantity λ is an energy renormalization factors reflecting the energy dependentinteractions present in the system. For example, the electron-boson interactionsenhance the electron mass by a factor m∗ = (1 + λ) m [86].

At low temperature one can approximate ∂zf → δ(z). Moreover if the p-dependence of κ(p) is weak enough we can replace the momentum summation withan integral over the energy and approximate κ(p) ≃ κF .

For σxx we have

σxx =e2

π

2κF

Dm

∫ ∞

−∞dεN(ε)ε

1ξ2 + Γ2

≃ e2

π

2NF κF

Dm

∫ ∞

−∞dε(ξ + εF )

1ξ2 + Γ2

, (2.111)

where D is the dimension and comes out from∑

p p2x =

∑

p |p|2/D. Now the ∼ ξintegral is zero by symmetry and from the second integrand we obtain

σxx =ne2

m

κF

2Γ. (2.112)

Here n is the density of carriers and n = 2NF εF /D has been used. Notice thatas usual in the absence of vertex corrections (κF = 1) we would obtain that thetransport scattering rate Γ0

tr = 2Γ is twice the quasiparticle scattering rate. This is a


well known consequence of the fact that optical conductivity probes p-h excitations.In the general case one should however further account for vertex corrections. Thiscan be done usually by defining a renormalized scattering rate Γtr = 2Γ/κF so that

σxx =ne2

mτtr. (2.113)

τtr = 1/Γtr is the so-called transport scattering time. Thus we have found thestandard Drude-like form for the conductivity σxx with a scattering time definednot only by the imaginary part of the self-energy but also by the vertex correctionsτtr = 2κF τ i.e. a renormalization of the response function that cannot be reduced tosingle-particle processes. Indeed vertex corrections stem for diagrams that are nottaken into account by the renormalization of the self-energy since it is not possiblea factorization of these in two dressed Green’s function (see discussion below Eq.(2.29)).

Let us evaluate now σxy. We have

∂xGA = ∂x(−ξp − iΓ)−1 =1

(ξp + iΓ)2vx(p)

∂xGR = ∂x(−ξp + iΓ)−1 =1

(ξp − iΓ)2vx(p) (2.114)

so we can express σxy as

σxy = −2ωce2

πDm

∑

p

εpκ2(p)1

(ξ2p − iΓ2)2

. (2.115)

Again we can replace the momentum summation and evaluate the integral as in theprevious case. We obtain at the end

σxy = −ne2

mωc

(κF

2Γ

)2

= −ne2

mωcτ

2tr, (2.116)

using the same definition of Eq. (2.113) for τtr. Let us collect Eq.s (2.113)-(2.116)to obtain the final result. Since both in Eq.s (2.113), (2.116) the prefactor κF

stemmed from the vertex corrections can be always recast in a redefinition of atransport scattering time τtr, by definition it follows that

RH =σxy

σ2xxHz

= − 1ne

. (2.117)

Thus we recover the well known result for the Hall coefficient for a nearly free-electron system. The result Eq. (2.117) demonstrated that vertex correction for asingle parabolic band are irrelevant, i.e. it cancel out in the ratio σxy/σ2

xx. As wewill discuss later on this result shall be revised in the case of HTSC.

Let us comment finally about the sign of the Hall effect. We evaluated hereRH in the case of an electron system, but it is well known that in a semiclassicalpicture the sign of the Hall coefficient reveals the nature of the carriers. Thus thesame analysis for a system in which the carriers are holes should define a positiveRH . This can be easily check: for a nearly free-hole system we c an follow the

2.3 Hall Effect 53

above derivation, butthis time with v→ −v, and, as a consequence, J→ −J. Thuswe have that σxx ∝ Jx · vx = κF v2

x in Eq. (2.105) is exactly the same also foran hole band. On the other hand the σxy conductivity changes sign between holeand electrons carriers. Indeed from Eq. (2.104), although J∂xJ = κ2v∂xv remainspositive both for electrons and holes, the sign of ∂xG ∝ vx (see Eq. (2.114)) leadsto an opposite definition of σxy. Notice that the above discussion is valid as longas J ‖ v. More generally the sign of the Hall coefficient is not simply defined bythe nature of the carriers. We would like to clarify better this point since it will berelevant in the context of HTSC.

2.3.2 Sign of RH and Character of the Carriers

Let us start from the expression Eq.s (2.106), (2.107). In the clean limit (small Γ)we can use that

GRGA ≃ π

Γδ(z − ξ), GR∂µGA − ∂µGRGA ≃ 2iπ

Γ2δ(z − ξ)vµ. (2.118)

Moreover let us consider the general case in which J and v not necessarily areparallel. At low temperature, where −∂f/∂ξk → δ(ξk), the conductivities are givenby

σxx = e2∑

k

δ(ξk)vx(k)Jx(k)1Γ

, (2.119)

σxy

Hz= −e3

4

∑

k

δ(ξk)A(k)Γ2

, (2.120)

where A(k) is

A(k) = |v(k)|[

J(k)× (e‖ · ∇)J(k)]

· ez = |v(k)||J(k)|2(

dθJ(k)dk‖

)

. (2.121)

Here ez is the unit vector along the z axis, while

e‖ = (ez × v(k))/|v(k)|, (2.122)

is tangential to the FS at k since both ez and v(k) are perpendicular to the FS.θJ(k) is the angle between J(k) and the x-axis except for an arbitrary constant andd/dk‖ = e‖ · ∇ is the directional derivative along the k vector tangential to the FS.Thus by definition dθJ/dk‖ measures the variation of the angle θJ along the FS.The orientation of the FS, determined by the versor e‖ in Eq. (2.122), is oppositefor electrons and holes carriers since they have opposite velocities.

First of all let us consider again the simple case of a single parabolic band. Inthis case we verified that vertex corrections defines a renormalized current alwaysproportional to the bare velocity, that is J ‖ v. Either for electrons or holes we havedθJ/dk‖ = dθv/dk‖ where θv(k) is the angle between v(k) and the x-axis. Sincethe v vector is always perpendicular to the FS, the variation of θv is only due tothe orientation of the FS. Explicitly since for electrons the orientation of the FS isanticlockwise, θv is increasing moving Jk → Jk+δk along the FS, i.e. dθJ/dk‖ > 0.The opposite holds in the holes case (Fig. 2.9). As a consequence the character ofthe carriers fixes the sign of σxy as above mentioned.


Jk

Jk+δk

Jk

Jk+δk

(a) (b)

Figure 2.9

Let us consider now the more general case of a non-parabolic band. Now thevertex corrections appear relevant since they define a renormalized current not pro-portional to the bare one, so that J is no more parallel to v. Since θJ 6= θv, thesign of σxy determined by the variation of the angle θJ is not fixed only by theorientation of the FS (related to the nature of the carriers) but also to the variationof the direction of the J(k) vector respect with the e‖. For example, it easy toverify that also for an electron-like band we can have a negative dθJ/dk‖ in someportion of the FS as illustrated in Fig. (2.10). As discussed in [35, 36] by Kontaniand coworkers this effect could be relevant in cuprate systems. Indeed, due to thestrong band anisotropy, the renormalized current J in cuprates is found not parallelwith respect to v in some portions of the FS. This could explain the anomaliesobserved in the Hall effect of underdoped cuprates [35, 36]. We will review thistopic in the introduction (Section 4.1) of the fourth Chapter.

k

k+δk

Jk+δk

Jk

θk

θk+δk

FS

k

k+δk

Jk+δk

Jk

θk

θk+δk

FS

(a) (b)

Figure 2.10

Until now we discussed the case of a single-band system in which only a kindof carriers is present. In a general multiband system in which electrons and holesinteract with each others, the situation could be more involved. It is possible todemonstrate that also in a multiband system in presence of vertex corrections dueto interband interactions one recovers the same expressions Eq.s (2.119), (2.120) for

2.4 Vertex Corrections and Boltzmann Theory 55

the conductivities. In this case the integral equation for the vertex part Eq. (2.91)connects the two kind of carriers (that have opposite velocities) thus also assumingcircular FS (so that J stays always parallel to the bare velocity) we can found that Jcan even have opposite direction with respect to the bare vertex v. In this case fromthe definition Eq. (2.119) it follows that, since J · v < 0, the contribution of oneband to the total σxx becomes negative. Of course, while in a single-band systemthis can never happen, in a multiband case such a negative contribution is alwayscompensated by a (larger) positive contribution in some band, so that the overallconductivity is always positive defined, as physically expected. Concerning the σxy

conductivity let us notice that J, even if opposite in direction with respect to thebare velocity v, is always along the same direction, i.e. perpendicular to the FS. Inthis case we have θJ = θv + π, so that dθJ/dk‖ = dθv/dk‖ still holds. Despite thechange of sign of the renormalized current, the sign of the σxy conductivity is againdecided only by the orientation of the FS, i.e. by the character of the carriers. Theconsequence of these effects on the Hall transport in pnictides will be the subjectof Chapter 4.

2.4 Vertex Corrections and Boltzmann Theory

We have already given the definition of transport scattering time τtr as due to therenormalization of self-energy and of the vertex part in the evaluation of the currentresponse functions. Here we want to make an analogy between the Kubo formalismand the usual Boltzmann approach. In particular we aim to highlight the differencebetween the quasiparticle and the transport scattering time as it comes out from aKubo analysis and to discuss the correspondence of these with the known results ofthe Boltzmann theory.

In order to do this comparison we will consider as an example the case of scatter-ing by impurities. Concerning the Boltzmann theory we will summarize the mainpoint of this approach and we will discuss the implication in what concerns thedefinition of the transport scattering time. Concerning the Kubo formalism we willconsider the results of the precious Section using the definition of the self-energyand vertex part for the impurity scattering that we will recall briefly. For moredetails we refer to [89].

In Boltzmann theory electrons are described by a classical distribution functionf(k, t). Consider the system in the presence of a weak external electric field andlet us assume the impurities very diluted. In this case the rate of change of thisdistribution function df/dt is is given by

(df

dt

)

collision

= ∂tk · ∂kf, (2.123)

where we assumed (df/dt)collision = (df/dt)externalfield. The acceleration ∂tk isgiven by the Lorentz Force and we find

(df

dt

)

collision

= −eE · ∂kf, (2.124)


where we assumed no magnetic field. The rate of change of the distribution functionf is given by the so-called collision integral

(df

dt

)

collision

= 2πni

∫dk′

(2π)3|Tkk′ |2(f(k)− f(k′))δ(εk − εk′), (2.125)

where ni is the density of impurities and Tkk′ = Tk′k is the matrix element forscattering between k to k′. Here we assumed elastic scattering for electrons byimpurities (i.e. |k| = |k′|). Now, since

J = −en

∫dk

(2π)3f(k)vk, (2.126)

with n is the density of electrons, the conductivity can be found from Eq.s (2.124),(2.125), (2.126). The solution of this system of equations can be carried out easilyunder the so-called Relaxation-Time Approximation (RTA).

Let us define f0 as the equilibrium configuration in the absence of an electricfiled. In the RTA it is assumed that the rate of change of f is proportional to thedegree of difference between f and f0

(df

dt

)

collision

=f(k)− f0(k)

τrel(k), (2.127)

τrel is the relaxation time defined by the collision integral. The equation for thedistribution function now reads

f(k) = f0(k) + eτrel(k)E · vk∂εkf0(k), (2.128)

where the the electric field is supposed to be small so that in the r.h.s. f hasbeen replaced by f0. Now from Eq.s (2.126), (2.128) the conductivity can be easilyexpressed in terms of τtr. At low temperature, where ∂εk

f0(k) = δ(εk), it is foundto be given by

σxx =ne2

mτrel(k), (2.129)

that is the usual Drude expression for the conductivity with the scattering timegiven by τrel. Now we need only to define the relaxation time τrel.

In the RTA we have

f(k)− f(k′)τrel

= 2πni

∫dk′

(2π)3|Tkk′ |2(f(k)− f(k′))δ(εk − εk′). (2.130)

The quantity f(k) − f(k′) differs only in the angular part of the vectors since|k| = |k′|. It is possible to demonstrate [89] that writing this difference in terms ofangular variables one finds

1τrel

= 2πni

∫dk′

(2π)3|Tkk′ |2(1− cos θ′)δ(εk − εk′). (2.131)

The weighting factor 1 − cos θ′ = 1 − k · k′ accounts for the fact that small anglescattering events are relatively unimportant to transport, since they do not impede

2.4 Vertex Corrections and Boltzmann Theory 57

the flow of electrons. It makes the relaxation time different from the usual life-timeτ which is the time between scattering events

1τ

= πni

∫dk′

(2π)3|Tkk′ |2δ(εk − εk′). (2.132)

Let us compare now the standard results of the Boltzmann theory to whatfound by the Kubo formalism in the previous sections. In a self-consistent Bornapproximation one take into account multiple scattering events by several impuritiesbut without any crossing diagram. This approximation omits many terms, howeverthe omitted terms are not important in the low density of impurities limit ni →0. It is possible to verify that under this approximation the self-energy is purelyimaginary [89] and is given by

Σ′′(k) = −πni

∫dk′

(2π)3|Tkk′ |2δ(εk − εk′). (2.133)

As usual the imaginary part of the self-energy defines the quasiparticle scatteringrate Γ = 1/|Σ′′| and the relative scattering time 1/τ = 1/Γ. It easy to recognizein this definition the same scattering time defined in Eq. (2.132). This becauseEq. (2.133) represent just the average time between scattering events. If we nowevaluate the current-current response function in the bare-bubble approximation(Eq. (2.28) but with dressed Green’s functions) we obtain for the conductivity theusual Drude formula (see Eq. (2.113)) with τtr = 1/Γ0

tr = 1/2Γ.The prediction of Boltzmann theory Eq.s (2.129), (2.131) can be recovered only

by carrying out a conserving approximation. Indeed, we should consider the dressedbubble Eq. (2.33) with the vertex part evaluated within the same approximationsused for the self-energy. At this level the integral equation for the current vertexreads

Jα(k+, k−) = vα(k) +∑

k′

J(k′+, k′

−)ni|Tkk′ |2(iωn, iωn + iΩm)G(k′+)G(k′

−). (2.134)

One can verify that the estimate of Eq. (2.33) using Eq.s (2.133), (2.134) leadsto the correct definition of the σxx conductivity. In particular, as shown in theprevious section, it is possible to include the effect of the vertex corrections, Eq.(2.134), in the redefinition of a transport scattering time τtr, that in the case ofimpurity scattering is defined by the same Eq. (2.131) of the Boltzmann relaxationtime [89]. Scattering by impurity is a typical example that shows how the inclusionof vertex corrections is needed in order to recover the correct results in the estimateof a response function. In particular in this case it allowes for the correct definitionof the scattering time relevant in transport problems τ−1

tr ∼ 〈1− cos θ′〉.

Finally let us comment about the generality of this result and of the Boltzmannpicture. As we discussed in the previous section the possibility to recast vertexcorrection in a redefinition of a transport scattering time (see Eq.s (2.113)(2.116))relies crucially on the possibility to find for the dressed currents a solution pro-portional to the velocity Eq. (2.103). As far as this is the case, one recovers a


result equivalent to the Boltzmann approach (where this is meant the RTA for theBoltzmann equation).

Despite the common belief this is not always the case. For example, as we alreadymentioned, it has been demonstrated by Kontani et al [35] that in cuprates thesolution for the vertex is not proportional to the bare vertex i.e. J× v 6= 0 becauseof the strong anisotropy of the FS. In this case vertex corrections cannot be simplyrecast in a ridefinition of τtr. In the Boltzmann language this means that in theBoltzmann equation Eq. (2.124) the collision integral Eq. (2.125) that correspondin the Kubo formalism to all the renormalization due the interaction (represented byboth the self-energy and the vertex part) has to be evaluated without the relaxationtime approximation Eq. (2.127), i.e. Eq. (2.128) is not more valid [36].

Thus what stays always valid is the general Eq. (2.124) and the possibility towrite down the the rate of change (df/dt)collision as the collision integral Eq. (2.125),while it is not always possible to use the RTA Eq.s (2.127), (2.128). The failure ofthis Boltzmann approach is observed not only in cuprates, i.e. single-band systemwhere the strong band anisotropy leads to a change of direction in the dressedcurrent with respect to the bare one, but also in pnictides, i.e. multiband systemsin which electrons and holes interact with each others. As above mentioned bymixing holes and electrons the renormalized current J of one band can even changesign respect with the bare velocity v. Also in this case vertex corrections cannot besimply recast in a renormalization of the transport scattering times of each band.This can be seen also starting from the Boltzmann approach. Let us consider thesimple case of a two band system with one electron-like band and one hole-like.Inthis case one has to write down Eq. (2.124) for each distribution f e, fh. Sincewe are interested in the case of interacting holes and electrons we should evaluatethe collision integral in the same form of Eq. (2.125) where now |Tkk′ | can scatterelectron and hole mixing f e and fh. Thus we should solve a system of coupledequations that now connect the distributions of the different bands (see [106]). Atthis point one can try to assume the RTA for each band, but it is easy to verifythat in this case it is impossible to solve the equations separately for each band and,as a consequence, a definition of a transport scattering time for each band is notallowed. We will discuss this topic within the Kubo formalism in Chapter 4 wherewe demonstrate that when dealing with interacting holes and electrons the trial ofrecasting vertex corrections in a redefinition of transport scattering time τh

tr, τtre isnot only meaningless, but it also leads to wrong results concerning the analysis ofthe Hall effect in pnictides.

Chapter 3

Multichannel Interaction Effects

on ScF in Cuprates

We discussed in Section 1.2.2 a puzzling aspect of the physics of underdoped cuprates:the ScF seem to contribute differently to diamagnetism and conductivity above Tc.In particular it has been shown that the paraconductivity turns out to be two or-ders of magnitude smaller than the fluctuating diamagnetism in the same system[14]. In order to understand the origin of this discrepancy, we aim to analyze theinfluence of the p-h degrees of freedom on the ScF contribution to conductivity anddiamagnetism. As discussed in Sections 1.3.1 and 2.2.2, the functional integral ap-proach represents the more convenient method to do that, since it allows to easilycheck which fields couple to the em potential and to compute explicitly their contri-bution to δQµν . As a paradigmatic model of strongly correlated material we focusedon the t − J model within the slave-boson approach. In what follows, after a briefdiscussion on the outcomes of a MF analysis of the model, we will analyze the ScFby means of the functional integral approach. We will show that in proximity to aMott-insulating phase one recovers an overall suppression of the fluctuating contri-bution to the conductivity with respect to diamagnetism, in close analogy with recentexperiments on the underdoped phase of cuprate superconductors.

3.1 Preliminary on Slave Boson Approach to the t-J

model

Let us start from the t-J model Hamiltonian already presented in Section 1.4

H = −t∑

〈i,j〉 σ

(f †iσfjσ + H.c.) + J

∑

〈i,j〉

[

SiSj −14

ninj

]

− µ∑

i σ

f †iσfiσ. (3.1)

Here t is the electron hopping and J the Heisenberg exchange interaction betweenelectrons spin, f †

iσ(fiσ) is the electron creation (annihilation) operator and Si =Ψ†

iσ2 Ψi is the spin operator, σ are the Pauli matrices and Ψi = (fi↑

fi↓). ni =

∑

σ f †iσfiσ

and 〈i, j〉 denotes nearest-neighbor sites on a square lattice.To account for the strong Coulomb repulsion we forbid double occupation, i.e.

∑

σ

f †iσfiσ ≤ 1. (3.2)

59

60 3. Multichannel Interaction Effects on ScF in Cuprates

A powerful method to treat this constraint is the slave-boson method [2]. In thisformalism, one introduces an auxiliary boson operator bi to keep trace of the emptysites. In this case we can reformulate the model using the projected electron operatorf †

iσ = c†iσbi, and replacing the inequality constrain Eq. (3.2) by an equality

∑

σ

c†iσciσ + b†

i bi = 1, (3.3)

enforced by a Lagrange multiplier. Thus in the slave-boson representation theHamiltonian is

HSB = − t∑

<i,j>σ

(

bib+j c+

iσcjσ + h.c.)

+ J∑

〈i,j〉

[

SiSj −14

(1− b†i bi)(1− b†

jbj)]

− µ∑

iσ

c+iσciσ +

∑

i

λi

(∑

σ

c†iσciσ + b†

ibi − 1

)

, (3.4)

where c+iσ(ciσ) and b+

i (bi) are the spinon (fermion) and holon (boson) creation (an-nihilation) operators at site i, respectively. The non-double occupancy constraintis enforced by the last term which introduces a Lagrange multiplier field λi to theHamiltonian.

From now on the λi will be replaced by their static values and we will neglectthe boson fluctuations, so that 〈b†

i bi〉 = δ = 2x/(1 + x). The Hamiltonian Eq. (3.4)reduces then to

H = −tδ∑

〈i,j〉 σ

(c†iσcjσ + H.c.) + J

∑

〈i,j〉

SiSj , (3.5)

so that the bare dispersion ξ0k already includes the rescaling of the hopping t→ tδ

which signals the proximity to the Mott insulator

ξ0k = −2tδ(cos kx + cos ky)− µ. (3.6)

By means of standard HF factorization [107], one can decouple the interactionterm in the Hamiltonian in the p-h (χ) and p-p channel (∆)

χ0 = 〈c†i↑cj↑ + c†

i↓cj↓〉 =12〈γs(k)(c†

k↑ ck↑ + c†k↓ ck↓)〉, (3.7)

∆0 = 〈ci↑cj↓ − ci↓cj↑〉 =12〈γd(k)(ck↑c−k↓ − ck↓c−k↑)〉. (3.8)

with γs = cos kx+cos ky, γd = cos kx−cos ky, we obtain the mean field Hamiltonian

HMF =∑

k σ

ξkc†kσckσ +

3J

4N(χ+

0 χ0 + ∆+0 ∆0)

−∑

k

[

V ∗k (ck↑c−k↓ − ck↓c−k↑) + Vk(c†

−k↓c†k↑ − c†

−k↑ck↓)]

. (3.9)

Hereξk = −(2tδ +

3J

4χ0)γs − µ, Vk =

3J

8N∆0γd (3.10)

The elimination of the Lagrange multiplier enforces the chemical potential µ to bedetermined by 1− δ =

∑

σ〈c†iσciσ〉. Notice that the presence of a finite MF value of

3.1 Preliminary on Slave Boson Approach to the t-J model 61

the density order parameter leads to the HF correction of the bare dispersion ξ0,Eq. (3.6). The MF values of the density operator, the pair one and the number ofparticle satisfy the following self-consistent equations:

χ0 = − 12N

∑

k

ξk

Ek

γs(k) tanh(

βEk

2

)

, (3.11)

∆0 =3J

4N

∑

k

∆0

2Ek

γ2d(k) tanh

(βEk

2

)

, (3.12)

n0 = 1− 1N

∑

k

ξk

Ek

tanh(

βEk

2

)

, (3.13)

where Ek =√

ξ2k + 4V 2

k . The self-consistent equations above Tc simply follow fromthe relations above putting Vk = 0 and Ek = ξk.

The MF analysis summarized above can be recast in the Landau FL language.In this formalism we consider the energy of the system as a functional of δnkσ

δE =∑

kσ εqpkσδnkσ , where εqp is the quasiparticle energy. Expanding up to second

order in δnkσ we have

δE =∑

kσ

εkσδnkσ +12

∑

kk′

∑

σσ′

fσσ′(k k′)δnkσδnk′σ′ , (3.14)

where

εkσ =δE

δnkσfσσ′(k k′) =

δ2E

δnkσδnk′σ′

. (3.15)

Thus the quasiparticle energy is given by

εqp = εkσ +12

∑

k′σ′

fσσ′(k k′)δnk′σ′ , (3.16)

where εkσ defines the energy of the isolated quasiparticle, and the second termdescribes the interaction of the quasiparticle changing δnkσ .

The MF analysis Eq. (3.9) of our model Hamiltonian Eq. (3.5) allows us todefine the Landau parameter. Concerning the kinetic part of Eq. (3.9), simply byreplacing c†

kσckσ → nkσ, we find of the isolated quasiparticle

εkσ = ξk, (3.17)

where ξk is defined in Eq. (3.10). The definition of the fσσ′(k k′) function canbe found by looking at the p-h channel of the k-space interacting Hamiltonian Eq.(3.5)

Hph = −12

3J

4N

∑

kk′q

∑

σσ′

∑

α

cos(k − k′)αc†k+q/2,σck−q/2,σc†

k′−q/2,σ′ck′+q/2,σ′ , (3.18)

where α = x, y. At MF level we evaluate Hph at q = 0, so that we can replacec†

kσckσ → nkσ, and rewrite Eq. (3.18) as

HphLandau = −1

23J

4N

∑

kk′

∑

σσ′

∑

α

cos(k− k′)α nkσnk′σ′ . (3.19)


According the Landau FL theory we can define

fσσ′(k k′) = − 3J

4N

∑

α

cos(k − k′)α, (3.20)

that account for the interaction between quasiparticles. It is known that fσσ′(k k′)can be expanded in Legendre polynomials [89]. For example, the lowest harmonicdefines the F s

0 Landau coefficient, which enters in the renormalization of the com-pressibility δn0/δµ. By definition we have

δnkσ =∂nkσ

∂εqpkσ

[δεqpkσ − δµ]

=∂nkσ

∂εqpkσ

[∑

k′σ′

fσσ′(k k′)δnk′σ′ − δµ

]

, (3.21)

where we used that the variation δεqpkσ respect to the energy of the isolated quasi-

particle εkσ is given by the interaction term Eq. (3.18). Using the definition Eq.(3.20) one can verify that the above relation reduces to

δnkσ = δ(εqpkσ − εF )

[3J

8γ2

s (kF )∑

k′σ′

δnk′σ′ + δµ

]

, (3.22)

where we fix δnkσ at the FS. Using δn0 =∑

kσ δnkσ/N the definition of F s0 follows

δn0

δµ=

n(εF )1− F s

0

⇒ F s0 = −3J

8γ2

s (kF )n(εF ). (3.23)

Let us notice that, using the χ0 definition Eq. (3.7), the above result can be easilyrephrased as

F s0 = −3J

4γs(kF )χ0. (3.24)

On the other hand the current is renormalized via the F s1 Landau coefficient.

The current is defined as J = ∂εqp/∂k, thus we have

Jk =∂εkσ

∂k−∑

k′σ′

fσσ′(k k′)∂nk′σ′

∂εk′σ′

∂εk′σ′

∂k′, (3.25)

where ∂εkσ/∂k = vk, that in our model is vα = (2tδ + (3J/4)χ0) sin kα. By usingEq. (3.20), we can write the above equation as

Jk = vk

[

1 +3J

41

(2tδ + (3J/4)χ0)2

∑

k′σ′

(∂εkσ

∂k′

)2 ∂nk′σ′

∂εkσ

]

. (3.26)

Using ∂nk′σ′/∂εk′σ′ = (∂nk′σ′/∂k′)(∂k′/∂εk′σ′) and integrating by parts we find

Jk = vk

[

1− (3J/4)χ0

2tδ + (3J/4)χ0

]

, (3.27)

so that we can identify

Jk = vk

[

1 +13

F s1

]

⇒ 13

F s1 = − (3J/4)χ0

2tδ + (3J/4)χ0. (3.28)

3.2 Functional Integral Analysis 63

Notice that the back-flow current represented by F s1 exactly cancels the bare

current v in the Mott-insulator limit (δ ∼ 0). Actually, the quasiparticle carrieszero current in this limit, as expected by the insulator phenomenology.

In the next Section we will analyze, by means of the functional integral approach,the ScF contribution to conductivity and diamagnetism, the difference between thevelocity and the current will turn out as the main ingredient in order to understandthe discrepancy between the two ScF contributions. As we shall see in our approachthe analogous role of Landau FL corrections will be played by the HS fields relatedto p-h fluctuations.

3.2 Functional Integral Analysis

3.2.1 p-p and p-h Channel Decomposition of the electron-electron

Interaction

Let us start again from the slave-boson version of the t − J model Eq. (3.5) andlet us introduce the operators corresponding to the decoupling of the interaction inthe p-p and the p-h channel

Φcα(q) =

∑

kσ

cos kα c†k+q/2,σck−q/2,σ , (3.29)

Φsα(q) =

∑

kσ

sin kα c†k+q/2,σck−q/2,σ, (3.30)

Φ∆(q) =∑

k

γd(k) c−k+q/2,↓ck+q/2,↑ (3.31)

where α = x, y and γd(k) = cos kx − cos ky, so the interaction term reads

∑

〈i,j〉

SiSj = −g∑

q

(12

∑

α

Φcα qΦc

α q + Φsα qΦs

α q

)

+ Φ∆q

∗Φ∆

q , (3.32)

with g = 3J/4. The r.h.s. of Eq. (3.32) represents the interaction in the p-h density(Φc), p-h current (Φs) and p-p (Φ∆) channel, respectively. Notice that such adecomposition could be done in principle with arbitrary relative weights in the p-hor p-p channel, since each term can be transformed into the other one by allowinglarge q values. However, to have the MF results as saddle-point value we attributethe same weight to each of the two channels, while introducing a cutoff in themomentum space to prevent overcounting. As we shall see below the Φs field doesnot contribute to the MF Hamiltonian: this explains why we did not introduce itin the previous section. However its fluctuations play a crucial role in the transportproperties. This point, that has been often overlooked in the literature, explainsinstead the anomalies of the ScF observed in cuprates.

Now we can decouple (3.32) both in the p-p and in the p-h channel by means ofthe HS transformation discussed in Chapter 2. After the integration of the fermionsthe action reads:

S =∑

q

(∑

α

|φcα q|22g

+|φs

α q|22g

)

+|∆q|2

g− Tr log Akk′ . (3.33)


Here φc, φs and ∆ are the HS fields, the trace acts over momenta, frequencies andspins, k ≡ (k, iεn), q ≡ (q, iωm), and εn, ωm are the Matsubara fermion and bosonfrequencies, respectively. The Ak,k′ matrix is defined (in the usual Nambu notation)as

Ak′k = −[iωn + sin(k+k′

2)αφs

α k−k′

]τ0

+[

ξ0k − cos(k+k′

2)αφc

α k−k′

]

τ3 −[

∆k−k′γd(k+k′

2)]

τ1, (3.34)

where the sum over α is implicit and ξ0k is the bare dispersion

ξ0k = −2tδ(cos kx + cos ky)− µ. (3.35)

The Eq. (3.34) can be decomposed as Ak,k′ = −G−10 δk,k′ + Σk−k′. Here G−1

0 δk,k′

contains the q = 0 saddle-point values (φc0, φs

0, ∆0) of the HS fields obtained byminimization of the MF action

SMF =∑

q

(∑

α

|φcα q|22g

+|φs

α q|22g

)

+|∆q|2

g− Tr log(G−1

0 ), (3.36)

while Σk−k′ contains the fluctuating parts of the HS fields. The standard GL func-tional will then be given by the expansion of Eq. (3.33) around the MF action (3.36)as

SF L =∑

n

1n

Tr[G0Σkk′]n. (3.37)

3.2.2 Mean-Field Results

First of all let us check that the saddle-point equations for the MF value of theHS field coincide with the self-consistent equations Eq.s (3.11), (3.12) obtained bymeans of the MF analysis Section 3.1.

The coupled self-consistent equations are given by

∂SMF

∂∆0= 0 ⇒ 2∆0

g= Tr[G0 · γdτ1], (3.38)

∂SMF

∂φsα0

= 0 ⇒ φsα0

g= Tr[G0 · sin(kα)τ0], (3.39)

∂SMF

∂φcα0

= 0 ⇒ φcα0

g= Tr[G0 · cos(kα)τ3]. (3.40)

The MF value for the current field is always vanishing, unless at δ = 0. By substi-tution of φs

α0 = 0, we have below Tc

φc0 = − g

2N

∑

k

ξk

Ek

γs(k) tanh(

βEk

2

)

, (3.41)

∆0 = g∑

k

∆0

2Ek

γ2d(k) tanh

(βEk

2

)

. (3.42)


where we omitted the α index for the density field since φcx0 = φc

y0≡ φc

0, and we

used that 〈cos kα〉 = 〈γs(k)/2〉. In Eq.s (3.41), (3.42), Ek =√

ξ2k + (∆0γd(k))2, and

ξk is the quasiparticle dispersion, given by the τ3 term of Eq. (3.34):

ξk = −(2tδ + φc0)(cos kx + cos ky)− µ. (3.43)

As one can see, the φc0 value corresponds thus to the standard HF correction of the

quasiparticle dispersion.Obviously the saddle point Eq.s (3.41), (3.42) are completely equivalent to those

obtained by means of a HF analysis. As a matter of fact, Eq. (3.42) reproduces thestandard BCS equation Eq. (2.70), already obtained in Eq. (3.12). On the otherhand, by comparison with Eq. (3.11), we find the correspondence between the MFvalue of the HS density field φc

0, Eq. (3.41), and χ0, Eq. (3.7), φc0 = (3J/4)χ0.

Since we are interested in the analysis of the ScF contributions above Tc, inwhat follows we will consider

∆0 = 0

φc0 =

2g

N

∑

k

cos kαf(ξk), (3.44)

where we used that Ek = ξk. In Eq. (3.44) f(x) = (1+eβx)−1 is the Fermi function,β = 1/T , and we used that tanh(x/2) = 1− 2f(x).

For the sake of completeness let us explicitly check that φsα0 = 0. The self-

consistent equation reads

φsα0 =

g

N

∑

k

sin ka[f(ξk −∑

l

φsl 0 sin kl)− f(ξk +

∑

l

φsl 0 sin kl)]. (3.45)

We can analyze the equation above by expanding around ξk for small φsl 0

φsα0 = F(ξk −

∑

l

φsl 0 sin kl). (3.46)

At the zero-th order the condition is trivially fulfilled by φsα = 0. At the first order

we find

φsα0 =

g

N

∑

k

sin kα(−2∂ξkf)∑

l

φsl 0 sin kl. (3.47)

Let us consider for simplicity α = x

φsx0 = −2g

N

∑

k

(∂ξkf)(sin2 kxφs

x0 + sin kx sin kyφsy0

)

= −2g

N

1(2tδ + φc

0)2

∑

k

(∂ξkf)[

(∂kxξk)2φs

x0 + (∂kxξk)(∂ky

ξk)φsy0

]

= −2g

N

1(2tδ + φc

0)2

∑

k

(∂kxf)[

(∂kxξk)φs

x0 + (∂kyξk)φs

y0

]

,

(3.48)


where we used sin kα = ∂kαξk/(2tδ + φc

0) with ξk defined in Eq. (3.43). Integrationby parts gives

φsx0 =

2g

N

1(2tδ + φc

0)2

∑

k

(∂2k2

xξk)f(ξk)φs

x0

1 =2g

N

1(2tδ + φc

0)

∑

k

cos kxf(ξk), (3.49)

where we used ∂2k2

xξk = (2tδ + φc

0) cos kx and ∂2kxky

ξk = 0 since the dispersion islinear in coskα. By using the saddle-point equation (3.44) for the density field φc

0,Eq. (3.49) reduce to

φsα0 =

φc0

(2tδ + φc0)

, (3.50)

that is fulfilled only for δ = 0. We check this result at the first order. Howeverit easy to write the complete expansion for F(ξk −

∑

l φsl 0 sin kl), in fact the even

orders don’t give contribution since [∂2nξ f(ξ)− ∂2n

ξ f(ξ)] = 0 and we are left with

φsα0 = −2g

N

∑

k

sin kα(2∂2n+1ξk

f)

(∑

l

φsl 0 sin kl

)2n+1

, (3.51)

so that the above argument can be implemented at any order.The occurrence of a finite value of the current field is connected to an instability

of the FS. In isotropic Fermi liquids this occurs for strongly negative Landauparameters (notice in fact that φs

α0 ∼ −F s1 > 0). This instability is also known as

Pomeranchuk instability [108]. A detailed investigation of this topic is beyond ourpurpose, since we aim to analyze the behavior of ScF in underdoped system (whereδ 6= 0).

3.2.3 Beyond MF : Contributions to the Current-Current

Response Function

In order to compute the contribution of ScF to the em response function we need tointroduce in the effective action also the em potential A. For the model (3.5) thiscan be done via the Peierls substitution Eq. (2.11), which modifies the Akk′ matrixwith two additional contributions

− 2tδ sin(k+k′

2)α Aα

k′−k τ0 (3.52)

+ tδ cos(k+k′

2)αAα

k′−k+sAα−s τ3. (3.53)

Eq. (3.52) corresponds to the usual term −J ·A, where

Jα ≡ ∂kαξ0

k = 2tδ sin kα (3.54)

is the quasiparticle current. Notice that in the presence of HF corrections to thequasiparticle dispersion (3.43) the quasiparticle current Jα is different from thequasiparticle velocity

vα ≡ ∂kαξk = (2tδ + φc

0) sin kα. (3.55)


Notice that we are using Jα, vα according the usual notation used in the Landau FLlanguage. Indeed, these two quantities are related by the Landau FL F s

1 corrections,Eq. (3.27) discussed in the previous Section.

Following the strategy described in Section 2.2.2, in order to compute the AL-like ScF contribution to the em kernel δQµν , we need to compute the term orderA∆2 in the effective action S(A, ∆) in the form of Eq. (2.79), that we repeat herefor convenience

S(A, ∆) = −12

QMFαβ Aα

q Aβ−q + Aα

q F α(∆), (3.56)

where F(∆) ∼ p∆2p if we are considering only the p-p channel. By functional

derivative of the partition function Z[A] =∫D∆e−S(A,∆) of the model, the current-

current response function can be computed from Eq. (2.73), i.e.

Qαβ =δ2 ln Z[A]

δAαq δAβ

−q

|Aα

q =Aβ−q=0

= QMFαβ (q) + δQαβ(q), (3.57)

where δQαβ(q) is given by

δQαβ(q) = 〈F α(∆)F α(∆)〉 ∼∑

p

pαpβL(p)L(p + q). (3.58)

However in the presence of many relevant degrees of freedom we need to account forseveral interaction channels. In particular we have to retain the contributions of thefields which couple both to the gauge field A and ti ∆2 since they will contributeto modify the F function in Eq. (3.58). In our case such contributions will begenerated by the φs field. Indeed from Eq. (3.34) and (3.52) we see that the φs

α

field appears in the actions with the same structure of the em potential Aα, i.e. itis coupled to the fermionic current

φs

2tδ· J↔ A · J. (3.59)

Thus, we expect to find also terms ∼ φs∆2 that will modify the structure of theScF.

We notice that the MF em kernel of Eq. (3.56) QMFαβ contains the standard dia-

magnetic contribution, (see Eq. (2.16)), obtained by the first order of the fluctuatingaction expansion, Eq. (3.37), by means of insertion of Eq. (3.53), i.e.

ταα =1N

∑

k,σ

tδ cos kαnk,σ =2N

∑

k

(∂2kα

ξ0k)f(ξk) (3.60)

(see Table 3.1). Once again by comparing Eq.s (3.34) and (3.52), the φcα field

appears in the actions with the same structure of the diamagnetic term AαAα. Asa consequence all the terms obtained by insertion of the density field will givescorrection to the diamagnetic term of the em kernel, but they will not lead to amodification of the AL ScF contribution.

Effective Action

The effective action at the leading order in A, S(A, ∆) can be found by computingEq. (3.37) at various orders. Since we are interested in fluctuation effects above Tc,


we can put ∆0 = 0 in G0:

G−10 = −

[

iωn + 2tδ sin kα Aα0

]

τ0 +[

ξk + tδ cos kαAαs Aα

−s

]

τ3,

Σkk′ = −[

sin(k+k′

2)α(φs

α k−k′ + 2tδ Aαk′−k

)]

τ0 −[

∆k−k′γd(k+k′

2)]

τ1

−[

cos(k+k′

2)α(φc

α k−k′ − tδ Aαk′−k+sA

α−s

)]

τ3. (3.61)

We will consider only the terms relevant for the following discussion.At Gaussian level, SG = Tr[G0Σkk′G0Σkk′]/2 , we find four relevant contribu-

tions. The first one is proportional to ∼ ΠJαJβAαAβ , where ΠJαJβ

is the MF part ofthe current-current bubble, i.e. the paramagnetic term of the em kernel Eq. (2.24).Notice that, after the Matsubara summation, the MF part of the current-currentbubble ΠJαJβ

, Eq. (2.33), is explicitly given by

ΠJαJβ(q) = − 2

N

∑

k

(∂kαξ0)(∂kβ

ξ0)f(ξk+ q

2)− f(ξk− q

2)

iωn + ξk+ q

2− ξk− q

2

. (3.62)

The second contribution comes from the insertion of the current-field φs. As ex-pected looking at the way in which A and φs enter in the action, this term isanalogous to the paramagnetic bubble. We can write this fluctuating contributionΠs

JαJβφs

αφsβ in terms of the current-current bubble since Πs

JαJβ= ΠJaJβ

/(2tδ)2. Thethird contribution requires an insertion of A and one of φs so that the associatedbubble is found to be proportional to ΠJaJβ

/(2tδ). The last contribution comingfrom the Gaussian order gives the standard p-p bubble Eq. (2.45), Π(q)∆q∆∗

q. Atthe third order in SF L, Eq. (3.37) we find two contribution that coupled the p-pfield to the paramagnetic vertex A, i.e. A′

qα∆q−q′∆∗

q and the analogous term forφs.

The sketch of the complete list of the several fluctuating contribution up to thequartic order is reported in Table 3.1.

Collecting the relevant contributions described above we have

SGL(A, φs, ∆) = −12

ΠJaJβAα

q Aβ−q −

12

ΠαβJαJβ

2tδ

(Aα

q φs β−q + φs α

q Aβ−q

)+

12

[

g−1 −ΠJaJβ

(2tδ)2

]

φs αq φs β

−q +

[

g−1 −Π(ω) + cq2

]

∆∗q∆q

−c′q

(

Aq′ +φs

q′

2tδ

)(∆∗

q∆q−q′ + ∆∗q+q′∆q

), (3.63)

where summation over repeated indices is implicit and we already performed theexpansion of the p-p bubbles.


Table of the Fluctuating Contributions

First order: Tr[G0Σ]

∼ 12ταα(A

α)2 ∼ 121tδτααφ

cα

Second order:Tr[G0ΣG0Σ]2

∼ 12

1(2tδ)2ΠJαJβφ

sαφ

sβ

∼ 12

1(2tδ)ΠJαJβφ

sαA

β

∼ 12ΠJαJβA

αAβ∼ 12Πτατα(A

α)2(Aα)2

∼ 12

1(tδ)2Πταταφ

cαφ

cα

∼ 121tδΠταταφ

cα(A

α)2

∼ ∆2Π

Third order: Tr[G0ΣG0Σ]3

∼ Dα∆2(Aα)2 ∼ Pα∆2Aα

∼ 1tδDα∆2φc

α ∼ 12tδP

α∆2Aα

Fourth order: Tr[G0ΣG0Σ]4

∼ Mαβ∆2AαAβ ∼ Hαβ∆2AαAβ

∼ 12tδM

αβ∆2Aαφsβ ∼ 1

2tδHαβ∆2Aαφs

β

∼ 1(2tδ)2M

αβ∆2φsαφ

sβ ∼ 1

(2tδ)2Hαβ∆2φs

αφsβ


Figure 3.1 Previous page:. Fluctuating contributions obtained by Eq.s (3.37), (3.61).The solid lines are the Green’s functions G forming the fermionic loop. Wavy lines arethe gauge fields: the single lines represent insertions of the paramagnetic term Aα (asso-ciated to Jα), while pairs of wavy lines ending in a single vertex represent diamagneticinsertions. Insertions of the HS fields are represented by: solid straight lines – densityfield φc, zigzagged line – current field φs, dashed line – pairing field ∆. By direct in-spection of Eq. (3.61) one can see that an insertion of φc is equivalent to a diamagneticinsertion but a prefactor 1/tδ, while the current field insertion correspond to a param-agnetic insertion apart a factor 1/2tδ, Eq. (3.59).From the linear fluctuating order, n = 1 in Eq. (3.37), we find the standard diamagneticcontribution ταα Eq. (2.16), as well as the analogous terms coming from the densityfield.At the second order we have seven finite contributions. On the left are shown the threediagrams obtained by insertions of diamagnetic-like vertices (A2, φc). They give contri-bution proportional to a p-h bubble Πτατα

(q) = − 2

N

∑

k(∂2

kαξ0)(∂2

kαξ0)Gk+q/2Gk−q/2.

The others are already discussed in the text: on the right are shown the standardcurrent-current bubble ΠJαJβ

, Eq. (3.62), obtained from two insertions of the gaugefield, and the two analogous contributions obtained by replacing the paramagnetic in-sertions with the current field vertex: ∼ Aφs and ∼ φsφs; the last one is the p-p bubbleΠ, Eq. (2.45).The third order diagrams are corrections to the p-p bubble Π(q) with a diamagnetic-like insertion: ∼ ∆2φc, ∆2A2 (on the left), or a current one: A∆2 and φs∆2 (on theright). The last class of diagrams are the ones involved in the calculus of the AL-likecontributions to the em kernel.At the fourth order we draw down only diagrams of order ∆2 containing insertionsof both the gauge fields and the current ones. These kind of diagrams give contribu-tion beyond the leading order to the paramagnetic fluctuating part of the em kernel(contributions beyond AL).

Let us discuss explicitly the c and c′ terms of Eq. (3.63). They are obtainedfrom the expansion to leading order in q of the fermionic p-p bubbles associated tothe ∆2 term and to the A∆2 and φs∆2 terms, respectively (see Fig. 3.2).In the former case one expands

Π(q) = − T

N

∑

k

γ2d(k)Gk+q/2G−k+q/2 ≃ Π(ω)− cq2. (3.64)

where the quasiparticle Green’s function, Gk = (iωn−ξk)−1, contains the full disper-sion ξk. As a consequence, c is the ordinary stiffness (already discussed in Section2.2.1) proportional to the second-order derivative of G, which in turn scales as(∂ξkα

)2 ≡ v2α. In the case of the c′q term instead one carries out a single derivative

of the fermionic bubble which contains already a current insertion J, associated toeach A or φs field, see Eq.s (3.34) and (3.52) and Fig. 3.2. Thus we have in ashort-hand notation:

c ∝ (∂ξkα)2 ∼ v2

F , c′ ∝ ∂ξkα∂ξ0

kα∼ vF JF (3.65)

where vF , JF are the average values on the FS.


Figure 3.2. Fermionic bubbles: (a) is the Π(q) bubble associated to the ∆2 term, while(b) is the fermionic bubble relative both to A∆2 and φs∆2 terms. The solid lines arethe Green’s functions G containing the full quasiparticle dispersion ξ. The dashed linesare the pairing field ∆. Here the wavy line represents either the em potential A or thecurrent field φs, both associated to the quasiparticle current J ∝ ∂ξ0.

MF Current-Current Correlation Function and ScF Contribution

Let us compute the current-current correlation function by functional derivative ofthe effective action Eq. (3.63).

It easy to verify that if we do not consider the interactions in the p-h channelin our model, we do not have the HF correction of the band dispersion so thatξk = ξ0

k (see Eq. (3.43)). As a consequence the velocity coincides with the current,so that c = c′. Moreover by direct inspection of Eq. (3.63) one can see that theem potential A is coupled with the pairing field ∆ via a term ∼ cAp∆2

p, as wehave already discussed. The fluctuation correction to the current-current responsefunction, Eq. (3.58) is given in this case by the usual AL contribution :

δQαβ(q) = c2T∑

p

(2p + q)α(2p + q)βL(p)L(p + q) (3.66)

already defined in Eq. (2.78). As usual the ScF contribution to the physical quan-tities can be computed starting from Eq. (3.66). The paraconductivity simplyfollows from the dynamic limit (q = 0, ω → 0) after analytical continuation of theMatsubara frequency ωm to the real frequency ω

δσ = [Im δQαα(ω, q = 0)/ω]ω→0, (3.67)

while the fluctuations contribution to the diamagnetism is connected instead to thestatic limit (q→ 0, ω = 0),

δχd = −[

δQ t(q, ω = 0)/q2]

q→0, (3.68)

where δQ t is the transverse part of the fluctuation correction to the current-currentresponse function (see Eq. (3.68)). In the absence of interactions in the p-h channel,since c ∝ v2

F is constant, the ScF contribution to the diamagnetism (3.68) andconductivity (3.67) is in both cases proportional to v4

F . As a consequence the ratioδσ/δχd is expected to be of order O(1).

Such a result changes in the presence of p-h interactions. In this case it is crucialto consider the contribution of the φs field to the em response. In order to do that we


can solve the functional integration over the current field in the partition function

Z[A] =∫

D∆DφcDφse−S(A,∆,φs,φc). (3.69)

Explicitly we need to compute∫

Dφse−S(A,∆,φs,φc) =∫

Dφs exp−12

[M−1(φsα,q)2 + Bα

−qφsα,q + Bα

q φsα,−q]

= exp12

Bαq MαβBβ

q , (3.70)

where we defined from Eq. (3.63)

Mαβ =(

1g− ΠJaJb

(2tδ)2

)−1

, Bαq =

−ΠJaJb

(2tδ)Aβ

−q −c′q′

(2tδ)∆∗

q′∆q′−q. (3.71)

The elimination via Gaussian integration of the current field φs leads to

SGL(A, ∆) = −12

ΠJJ(1 + Z(q)

)Aα

q Aα−q +

(g−1 −Π(ω) + cq2)∆∗

q∆q +

−C(q)qAq′

(∆∗

q∆q−q′ + ∆∗q+q′∆q

),(3.72)

where

Z(q) =ΠJJ(q)/(2tδ)2

1/g −ΠJJ(q)/(2tδ)2, (3.73)

andC(q) = c′(1 + Z(q)). (3.74)

We notice that in Eq.s (3.72)-(3.74) we used a simplified notation valid in the q → 0limit relevant for the following discussion, so that the current-current correlationfunction ΠJJ refers to its diagonal part ΠJαJα only. At finite q the above expressionsmust be properly extended to account for the full matrix structure of the responsefunctions.

From the definition (3.72) and using Eq. (3.57), we compute all the correctionsto Qαβ . We find a correction to the MF value current-current response function(3.62) that leads to the RPA resummation of ΠJJ as

ΠRP AJJ (q) = ΠJJ(q) + Z(q)ΠJJ(q) =

ΠJJ(q)1− gΠJJ (q)/(2tδ)2

. (3.75)

From the definition, Eq. (3.62), one can easily check that ΠJJ takes different valuesin the dynamic and static limit. In the dynamic one, (q = 0, ω → 0),

ΠJJ(q = 0, ω → 0) = 0, (3.76)

as a consequence from Eq.s (3.73), (3.74) follows that

Z(q = 0, ω → 0) = 0, C(q = 0, ω → 0) = c′. (3.77)


In the opposite static limit instead we can rewrite ΠJJ in Eq. (3.62) as

ΠJJ(q → 0, ω = 0) = − 2N

∑

k

(∂kαξ0

k)2∂ξf(βξk) =2N

∑

k

Jα

vα(∂2

kαξ0

k)f(βξk). (3.78)

Since ∂2kα

ξ0k = 2tδ cos(kα), we can use the self-consistent equation for the density

field φc0, Eq. (3.44) in Eq. (3.78), so that we have

ΠJJ(q → 0, ω = 0) = 2 tδJF

vF

φc0

g, (3.79)

where we evaluated the Jα and vα at the FS. One can check that from Eq.s (3.73)we have

(1 + Z(q))−1 = 1− gΠJJ (q)(2tδ)2

, (3.80)

thus using Eq. (3.79) one finds

(1 + Z(q→ 0, ω = 0))−1 = 1− JF

vF

φc0

2tδ=

1vF

(

vF −JF

2tδφc

0

)

=1

vF((2tδ + φc

0) sin(kF )− φc0 sin(kF ))

=JF

vF, (3.81)

where we used the definition of JF and vF , Eq.s (3.54), (3.55). Thus from Eq.(3.74) we have

1 + Z(q → 0, ω = 0) = vF /JF , C(q→ 0, ω = 0) = c. (3.82)

Before analyzing the consequence of this integration on the ScF, let us checkan important properties of the current-current response function. The longitudinalpart of the em kernel must vanish in the static limit QL(q → 0, ω = 0) = 0, i.e. thestatic limit of the current-current correlation function cancels out the diamagneticcontribution ΠL

JαJα(q → 0, ω = 0) = ταα

1. For example, for a free-electron systemwe have

Π0JJ(q, ω = 0) = −2T

N

∑

k

(∂kαξ0

k)2f(βξ0

k+ q

2

)− f(βξ0k− q

2

)

ξk+ q

2− ξk− q

2

q → 0−−−→2T

N

∑

k

(∂kαξ0

k)2∂ξf(βξ0k)

= −2T

N

∑

k

(∂2kα

ξ0k)f(βξ0

k) ≡ ταα. (3.83)

Moreover, in the presence of HF corrections, the MF current-current correlationfunction Eq. (3.62) does not fulfill the condition. As one can sees form Eq. (3.78),

1Notice that this condition naturally follows from the GI condition Eq. (2.32). Indeed the staticlimit requires ω = 0 first, moreover in order to obtain the longitudinal part we have to consider(qx → 0, qy = 0). In this case Eq. (2.32) reduces to QL(q → 0, ω = 0) = 0


this is due to the different dispersion appearing in the vertex ξ0k and the one con-

tained in the Fermi function’s argument ξk. For the RPA current-current correlationfunction instead, using Eq. (3.78), we have

ΠRP AJJ (q → 0, ω = 0) = ΠJJ(q→ 0, ω = 0)(1 + Z(q→ 0, ω = 0))

=2N

∑

k

Jα

vα(∂2

kαξ0

k)f(βξk)vα

Jα

=2N

∑

k

(∂2kα

ξ0k)f(βξk) ≡ ταα. (3.84)

The RPA correction 1 + Z(q), Eq. (3.82) changing one vertex of ΠJJ leads to thecancellation of the diamagnetic term in the static limit as physically expected.

Concerning the ScF, as one can see from Eq. (3.72), the em potential A iscoupled with the pairing field ∆ via a term ∼ C(q)Ap∆2

p. As a consequence thefluctuation correction to the current-current response function, Eq. (3.58), gener-alizes the standard AL result (2.78) with a momentum and frequency dependentvertex C(q):

δQαβ(q) = T∑

p

C(q)2(2p + q)α(2p + q)βL(p)L(p + q). (3.85)

As one can see from Eq.s (3.77)-(3.82) the vertices relative to the ScF contributionto the conductivity and diamagnetism are quantitatively different:

C(q) ∝ vF JF ∼ (2tδ + φc0)2δt (q = 0, ω → 0) (3.86)

C(q) ∝ v2F ∼ (2tδ + φc

0)2 (q → 0, ω = 0). (3.87)

The difference in the two limits reflects in a difference in the overall prefactors ofthe ScF contribution to the conductivity and diamagnetism. Indeed, from Eqs.(3.67)-(3.68) one has that

δσ

|δχd|∝ ((2tδ + φc

0)2tδ)2

(2tδ + φc0)4

∝ δ2, (3.88)

i.e. the fluctuation conductivity is suppressed by the proximity to the Mott insula-tor by a factor that depends on the doping.

We notice that in the presence of HF corrections the gauge-invariant form of theGL functional for ScF is recovered in a non trivial way. Indeed, in the usual case thecoupling to the gauge field A can be obtained by the minimal-coupling substitutionq → q+A in the c q2∆2 term of the Gaussian propagator. This leads immediately tothe term linear in A, c A·q∆2, needed to compute the AL correction, see Eq.s (3.56)-(2.76). In our case by direct inspection of Eq. (3.63) one finds instead two differentcoefficients c, c′ in the q2∆2 and A · q∆2 terms, as we explained above. However,by integrating out the φs field the coupling of ScF to the gauge field is described ingeneral by a term C(q)A ·q∆2, (see Eq. (3.72)), where C(q) is given by Eq. (3.74).As a consequence, one recovers the minimal-coupling prescription only in the static


limit (3.87) where C(q) = c 2. Notice also that in order to analyze the effect of thep-h degrees of freedom on the ScF, we focused only on the paramagnetic-like termsin the effective action (coming from terms proportional to A, and φs), since theyare the only relevant in the analysis of the paraconductivity and diamagnetism.Actually the derivation of the complete leading order effective action S[A] requiresalso the inclusion of the diamagnetic term ∼ τααA2

α, of the analogous term comingfrom the density field ∼ (ταα/(tδ)2) φc

α and of the other contributions list in Table3.1. At this level one can verify that also for the quadratic term coming from theminimal coupling substitution cq2∆2 → c(q+A)2∆2, we find in our case a differentcoefficients c′′ for A2∆2. In this case the integration of the φc field is needed inorder to recover the minimal coupling prescription also for the quadratic term.

Comparison with Experiments

In Section 1.3.1, we discussed the possibility to recast the result (3.88) as an estimateof the ratio between the correlation lengths extracted experimentally from paracon-ductivity and diamagnetism. For example, using data reported in [14] for a sample ofunderdoped La2−xSrxCuO4(x ∼ 0.09) one has that at T ∼ 24 K, δσ ∼ 105(Ω m)−1,and δχd ∼ 60 A/mT. Using in Eq. (1.1) ξ0 ∼ 1 nm and d ∼ 15 Å as appropriatesfor cuprates, we obtain that kBT/dΦ2

0 = 0.053 A/mT and e2/16~dξ20 = 104(Ω m)−1,

so that δσ/δχd ∝ ξ2σ/ξ2

χd∼ 10−2. Such a strong suppression of the paraconductiv-

ity with respect to diamagnetism, that has been rephrased in [14] in terms of thevortex diffusion constant valid only in the case of KT fluctuations, can be moregenerally attributed from Eq. (3.88) to the overall δ2 factor due to the proximity tothe Mott-insulating phase. This is our main result.

We checked numerically the estimate of the ratio (3.88) within the MF solu-tion of the t − J model. We find a value quantitatively larger by about an orderof magnitude than the one experimentally found, since a prefactor ∼ 2t/φc

0 partlycompensate the δ2 suppression (see Fig. 3.3)Such a quantitative disagreement is reminiscent of analogous limitations of the MFapproach already discussed in the literature in the contexts of other physical quan-tities, as for example the scaling of the superfluid-density depletion with doping[83]. Moreover at low doping [2, 83, 107] one should also include the boson fluc-tuations neglected so far, which give a temperature TB for the boson condensationsmaller than the one for the gap opening, leading to a suppression of the criticaltemperature Tc with respect to its MF value TMF . In such a regime, the static limitof the 1 + Z correction in Eq. (3.74) above, could not be simply given by vF /JF :nonetheless, the difference between the static and dynamic limit still holds, possiblyleading only to a quantitative difference with respect to the result (3.88).

2In the generic case in order to recover the full GI one needs to introduce also the scalar potential.


0 0,05 0,1 0,15x doping

0

0,1

0,2

0,3

0,4

0,5

0,6

δ σ

/ δ

χ d

Figure 3.3. Numerical estimate of Eq. (3.88) within the MF solution of the slave bosonversion of the t-J model. We used t = 0.25 eV and J = 0.13 eV in order to reproducethe expected MF value for the gap ∆0, density field φc

0 and the chemical potential µ.The temperature is fixed just above Tc.

Chapter 4

Unconventional Hall Effect in

Pnictides

We discussed in Section 1.2.3 the anomalous outcomes of the transport experimentson the normal phase of pnictides. Within a semiclassical approach one would expectan almost vanishing Hall coefficient for slightly doped compounds, i.e. compensatedsemimetals where ne ≈ nh (see Eq. (1.2)). Striking enough the experimental sce-nario is quite different with a very large absolute value of the Hall coefficient RH ,and with a marked electron/hole character of the transport in materials which areonly slightly electron/hole doped. In addition, a strong temperature dependence ofRH is also typically found, that disappears only at very large doping away fromhalf-filling. Since µα ∝ τα, to account for these features within the Boltzmann-likesemiclassical approach one needs thus to assume a marked disparity between τe andτh. However, as widely discussed in Section 1.2.3 such a disparity has not beensupported until now by any explicit calculation. As we shall see the analysis of theHall effect in cuprates opens new questions about the possible relevance of vertexcorrections in the context of pnictides. Motivated by this precedent, we will followthe strategy defined in Section 1.3.2, by computing the DC and Hall conductivitywithin an Eliashberg approach taking into account the effect of vertex corrections.We will show that this situation gives rise to an unconventional scenario, beyondthe Boltzmann theory, where the quasiparticle currents dressed by vertex correctionsacquire the character of the majority carriers. This leads to a larger (positive ornegative) Hall coefficient than what expected on the basis of the carrier balance,with a marked temperature dependence. The present results explain the puzzlingmeasurements in pnictides and they provide a more general framework for transportproperties in multiband materials.

4.1 A precedent: Unconventional Hall Effect in Cuprates

In the first chapter we discussed several non-FL features characterizing the trans-port properties of the normal state of cuprates. The Hall transport is one of these.At high temperature RH takes a nearly constant value, and its doping dependenceis very small in agreement with the Boltzmann estimate. As the temperature de-creases RH begins to show a strong temperature dependence and its maximumvalue results very enhanced with respect to the one obtained within the Boltzmann

77

78 4. Unconventional Hall Effect in Pnictides

picture. This enhancement is more evident in the underdoped samples. In par-ticular in the hole-doped compounds RH is positive, while in the electron-dopedcompounds the sign of RH changes to negative at low temperature. This could ap-pear strange since in general the sign of σxy is opposite for hole and electron carriersdue to the opposite velocity i.e. opposite curvature of the FS (see Eq. (2.121)). Inthese electron-doped compounds, although we are adding electrons, the FS remainshole-like, so RH should stay negative and a change of sign is unexpected. However

Figure 4.1. From [35]: Temperature dependence of RH in LASCO (hole doping) andNCCO (electron doping) in the paramagnetic state. Note that in cuprates 1/ne ∼1.5× 10−3 cm3C that fits quite well the high temperature value of RH .

the situation is more involved, indeed, we have already shown that, if the currentJ(k) is no longer perpendicular to the FS the sign of σxy is no more determined bythe nature of the carrier (as we discussed in Section 2.3.2). This could be the caseof cuprates as discussed in [35, 36] where AF vertex corrections are indicated asresponsible of the anomalous angular dependence of the current. Let us summarizehere the main results of this analysis.

Kontani and co-workers derived the expression for the Hall effect within theframework of a SpF theory in a conserving approximation along the line of [90],where a gauge invariant expression for the em kernel has been derived (see Section

4.1 A precedent: Unconventional Hall Effect in Cuprates 79

2.3).Starting from the Hubbard model Hamiltonian considering interactions only in

the spin channel, the self-energy is defined as

Σ(k, εn) = T∑

q,l

Vq(ωl)G(k − q, εn − ωl), with Vq(ωl) =34

U2χ(q, ωl), (4.1)

where U is the on-site Coulomb repulsion and χ(q, ωl) is the AF propagator. Afteranalytical continuation of the Matsubara frequency ωl to the real frequency ε theimaginary part of the self-energy can be written as

Σ′′(k, 0) = −34

U2∑

q

∫ ∞

−∞

dε

2πF (ε)ImGR(k + q, ε)ImχR(q,−ε), (4.2)

where F (ε) = coth(ε/2T ) − tanh(ε/2T ). As usual the imaginary part of the self-energy defines the scattering rate Γ(k) = |Σ′′| and the correspondent scatteringtime τk = 1/Γk. While a full expression for χ(q, ωl) can be derived within theFLuctuation-EXchange (FLEX) approximation, it has been shown that most of theresults can be captured as well within a phenomenological NAFFL picture. Withinthis framework the spin propagator χ(q, ω) can be expressed for small q and ω as

χ(q, ω) =χQ

1 + ξ(T )2(q −Q)2 + iω/ωsf, (4.3)

where Q is the AF wave vector, χQ = χ0/(T + Θ) is the strength of the AFcorrelations, ωsf = ω0(T + Θ) is the frequency scale, ξ(T ) = ξ/

√T + Θ is the AF

correlation length and Θ is the Curie-Weiss temperature. In this case Eq. (4.1) isequivalent to the Eliashberg approximation that we will use in the next Section (seeEq. (4.16)).

In the case of interaction in the spin channel the scattering rate is stronglyanisotropic on the FS, with an anisotropy which becomes larger as the AF fluc-tuations grow at low temperatures. Γ(k) takes a large value around the crossingpoints with the Magnetic Brillouin Zone (MBZ)-boundary, the so-called hot spots,while it becomes small at the points where the distance from the MBZ-boundary isthe largest, which are called cold spots. Since τ cold

k ≫ τhotk , the cold spots play the

major role in the transport properties. The geometry of the FS with respect to theMBZ fixes the positions of hot and cold spots of a particular system. Looking at thegeometry of the FS in cuprates systems, one could expect the cold spot to be locatedinside the MBZ i.e. at a wave vector k < (π/2, π/2) for an hole-doped cupratessystems, while in the electron doped case one could find a major contribution bythe quasiparticle on the boundary of the BZ (outside of the MBZ).

As far as the vertex corrections are concerned, following a standard derivationwhich guarantees a conserving approximation [35, 36], the current at ω = 0 iscomputed as J(k) = v(k) + ∆J(k) with

∆J(k) =34

U2∑

q

∫ ∞

−∞

dε

2πF (ε)ImGR(k + q, ε)

ImχR(q,−ε)Γ(k− q)

J(k − q). (4.4)

In particular, it is possible to verify that for |q−Q| ≫ ξ−1, the renormalized currentcan be written as

J(k) =1

1− α2k

(v(k) + αkv(k′)), (4.5)


where αk ∼ (1 − c/ξ2(T )) < 1 with c ∼ O(1), and αk takes the maximum valuearound the hot spots. Thus J(k) shows a non trivial critical behavior which is thenatural consequence of the strong backward scattering caused by the strong AFfluctuations. In particular Eq. (4.5) means that J(k) is not parallel to v(k) (seeFig.4.2).

(0,0) (π,0)

(0,π)

Fermi Surface

(π,π)

MBZ

k//

k//

XY

BZ

BZ

Jkvk:

:

Figure 4.2. From [35]: Schematic behavior of Jk and vk. Contrary to the bare velocitythe renormalized current is not perpendicular to the FS. The strongest renormalizationoccurs on the MBZ boundary, since AF scattering is peaked at Q ∼ (π, π).

Then the conductivities σxx and σxy are evaluated from Eq.s (2.119), (2.120),(2.121), that we repeat here for convenience:

σxx = e2∑

k

δ(ξk)vx(k)Jx(k)1Γ

, (4.6)

σxy

Hz= −e3

4

∑

k

δ(ξk)A(k)Γ2

, (4.7)

where A(k) is

A(k) = |v(k)|[

J(k)× (e‖ · ∇)J(k)]

· ez = |v(k)||J(k)|2(

dθJ(k)dk‖

)

. (4.8)

Since the σxx conductivity is given by the averaged value of vx ·Jx, around the coldspots is found smaller than that given by the Boltzmann approximation, due to thevertex corrections:

vx · Jx =1

1− α2k

[|vx(k)|2 − αk|vx(k)vy(k)|] ∼ |vx(k)|21 + αk

, (4.9)

where we used that the k′ point located on the FS is related to k as (k′x, k′

y) =(−ky,−kx), and that at the cold spots |vx(k)| ∼ |vy(k)|. On the other hand, the σxy

is enhanced by vertex corrections. In particular one can see [35, 36] that the portion

4.2 Hall Transport in multiband systems 81

of FS inside (outside) of the MBZ gives rise to a positive (negative) contribution toRH in the presence of the strong AF fluctuations, as discussed in Section 2.3.2 (seeFig. 4.2). Let us stress that this result cannot be obtained within a Relaxation-Time Approximation (RTA) approximation, where the sign of RH depends onlyon the direction of v, so that an hole-like FS gives always a positive contributionwithout change of sign [35, 36](see also Section 2.4). Because of the factor τ2

k in thedefinition of σxy the sign of the Hall coefficient will be determined by the cold spots.For hole-doped system the cold spots are located inside the MBZ while in electron-doped system the cold spots stay outside. This could explain thus the sign of theHall coefficient in hole/electron-doped compounds. Concerning the temperaturedependence one can verify that as the temperature increases the SpF weaken andlose their momentum dependence since ξ(T )→ 0, so that vertex corrections vanishby symmetry ∆J(k) → 0 (see Eq.s (4.2), (4.4)), thus J(k) → v(k) and the Halleffect approach the Boltzmann value.

In [35, 36] one can find also the numerical calculation done by using the FLEXapproximation. However these calculations add only details that do not change thequalitative results discussed above.

4.2 Hall Transport in multiband systems

Motivated also by the relevance of vertex corrections highlighted by Kontani andcoworkers in the context of cuprates, the evaluation of the Hall conductivity ina conserving approximation arises as a primary need in order to understand thepossible origin of the unconventional features of the Hall effect in pnictides. Thefirst step consists in the generalization of the results obtained by Fukuyama et al.[90] (Section 2.3.1) to a multiband system with interband interactions. As we shallsee, as opposite to the case of cuprates, in the multiband case even a very simpletopology of the FS leads to a relevant role of the vertex corrections.

4.2.1 Generalization to the Multiband Systems: More is different

In the second Chapter we derived the general gauge-invariant expression for thelongitudinal and transverse conductivities on the basis of the Kubo formula for anearly-free electron system. Following this procedure for a multiband system withinterband interaction, we recover the same expression for the conductivities that arestill diagonal in the band index α (since we do not consider current Jαβ ∼ p c†

α cβ),Eq.s (2.119), (2.120). We repeat here for convenience the definitions. For thelongitudinal part we have

σαxx = e2

∑

k

δ(ξαk )vα

x (k)Jαx (k)

1Γα(k)

≃ e2

2Jα

F · kαF

2πΓαF

, (4.10)

where the vector vα denotes the band velocity in the x-y plane for the band α, Jα isthe corresponding renormalized current and Γα is the inverse quasiparticle lifetime,determined in general by electron-electron and impurity scattering processes. Dueto the symmetry of the problem (parabolic dispersions), Jα is parallel to the reducedmoment k in each band α, and Γα(k) and Jα(k) depend only on |k|, so we define


their value at the FS as ΓαF ≡ Γα(kα

F ) and JαF ≡ Jα(kα

F ). The transverse xyconductivity under a weak magnetic field H along the z axis can be written as:

σαxy

H= −e3

4

∑

k

δ(ξαk )

Aα(k)(Γα(k))2

≃ ∓ e3

8π

(JαF )2

(ΓαF )2

, (4.11)

where Aα(k) = vα[

Jα × (eα‖ · ∇)Jα

]

· ez, ez is the unit vector along the z axis andeα

‖ = (ez × vα)/|vα| is tangential to the α-th FS at k. As we discusses in Section2.3.2 for a parabolic band the overall sign in Eq. (4.11) is determined only by thecharacter of the carriers (i.e. the sign of v · k)

Once evaluated the longitudinal and transverse conductivity, the Hall coefficientRH is given by

RH =

∑

i σixy

(∑

i σixx)2Hz

. (4.12)

Eqs. (4.10)-(4.12) are quite general, since they express the conductivities in termsof the bare Fermi velocity vα and the renormalized current Jα. What makes multi-band systems peculiar is the nature of vertex corrections that determine the relationbetween vα and Jα.

Using a standard approach [89] one can establish between these two quantitiesa matrix relation

JαF = Λαβvβ

F . (4.13)

The case Λαβ = δαβ corresponds to the non-interacting system where, by usingthe 2D relation nα = (kα

F )2/2π and by identifying 1/τα = 2ΓαF , Eqs. (4.10)-(4.11)

reduce to the standard results,

σαxx =

nαe2

mτα, and σα

xy = ∓σαxxµαH

with µα = eτα/m, and the minus/plus sign holds for the electron/hole band, re-spectively. In the interacting case, the strength of the diagonal and off-diagonalcoefficients Λαβ depends on the intraband or interband interactions, respectively.In conventional materials with predominance of intraband scattering Jα

F = ΛααvαF ,

so that the effect of vertex corrections in Eq. (4.10) and (4.11) can be reabsorbedin the definition of the transport scattering time τα

tr = Λαα/2ΓαF , in analogy to the

single band case as widely discussed in the second Chapter (see discussion below Eq.(2.113)). In this case the RTA for each band is always possible due to the intrabandnature of the scattering. As a consequence, by computing the Hall coefficient bymeans of Eq. (4.12), we recover the semiclassical Boltzmann picture Eq. (1.2), thatwe repeat here for convenience

RH =1e

(nhµ2h − neµ2

e)(nhµh + neµe)2

, (4.14)

with renormalized mobilities. Things are however deeply different in multibandsystems with dominant interband interactions connecting e and h sheets, as in pnic-tides. In this case the largest elements in Λαβ are off-diagonal, leading to a mixingof the e and h characters (having ve > 0, vh < 0) and resulting in unconventional

4.3 Hall Effect in Pnictides 83

features, like a possible vanishing current JαF or a change of sign of the renormal-

ized current with respect to the bare velocity vαF . In this case the effects of vertex

corrections cannot be simply recast in a renormalization of the transport scatteringtime, so that the RTA for each band is not only unreliable, but also meaningless, asalso discussed in Section 2.4. In this situation, although the conductivities are stilldiagonal in the band index α, Eq. (4.12) cannot be reduced to the Boltzmann-likeresult (4.14) and the full gauge-invariant calculation must be carried out.

4.3 Hall Effect in Pnictides

Let us explicitly compute the Hall transport for the minimal model which containsthe main ingredients responsible for the unconventional Hall transport in pnictides.

4.3.1 The Model

We consider a two-band model with 2D parabolic electron/hole bands centered atthe Γ and M = Q = (π, π) points, with different FS areas:

ξhk = Eh

max −k2

2me− µ, ξe

k = −Eemin +

k2

2mh− µ, (4.15)

where k is the reduced momentum with respect to the Γ or M point, for the h ore band, respectively (see Fig.1.5). We take here units ~ = c = a = 1 (a being thelattice spacing). In the following we choose µ = 0, so that Ee

min and Ehmax fix the

Fermi wave vectors ke,hF in each band, and we will assume, without loss of generality,

me = mh = m.

G

M

Q

Q

Figure 4.3. Scheme of the FS for pnictides with hole pockets around the Γ point of the BZand electron ones around the M point. The ordering vector of the AF order Q = (π, π)connects the two pockets having different character.

To investigate in details this issue, in the following we compute explicitly bothΓα

F and JαF in the representative case of pnictides, where the carriers in the h and e

bands interact via SpF exchange [43]. According to neutron-scattering experiments[109], (see Fig.4.4), the SpF spectrum can be phenomenologically modeled with astandard marginal-FL spectrum, Eq. (4.3), that we repeat here for convenience

χ(q −Q, ω) =χQ

1 + ξ(T )2(q −Q)2 + iω/ωsf, (4.16)


where

χQ =χ0Θ

(T + Θ), ωsf =

ω0(T + Θ)Θ

, ξ(T ) = ξ0

√

Θ/(T + Θ).

Figure 4.4. From [109] Experimental χ′′(ω) at various temperatures.

Since the SpF are peaked around the Γ − M nesting vector q = Q, the in-teraction mediated by such a collective mode will have a predominant interbandcharacter. The crucial role of such interband retarded interaction has been alreadydemonstrated for the understanding of several spectroscopic [32, 85, 110], thermo-dynamic [85, 111] and optical [37] anomalies of pnictides. However, as we widelydiscussed in Section 1.3.2, in order to compute current vertex corrections, the ex-plicit momentum dependent of the bosonic spin spectrum (4.3), neglected so far in[32, 37, 85, 110, 111], must be taken into account. For the sake of simplicity weassume in our minimal model only interband scattering, neglecting any intrabandcoupling.

We summarized in Section 1.3.2, the main results of the Eliashberg approachfor the normal state of pnictides. The single-particle Green’s function in each bandis computed as usual by means of the Dyson equation as

Gα(k, iωn) = [iωn − ξαk − Σα(k, iωn)]−1, (4.17)

where the self-energy is given by

Σα(k, ωn) = g2T∑

q,l

χ(q, iωl)Gβ(k− q, ωn − ωl), (4.18)

that is the analogous of Eq. (1.5) but with the momentum-dependent bosonic spinspectrum (see Fig. 4.5). In Eq. (4.18) ωn, ωl are fermionic and bosonic Matsubarafrequencies, respectively, and g is the coupling to the bosonic mode χ(q, iωl) of Eq.(4.16). The α, β labels stay for e, h, with the convention α 6= β, since only interbandinteractions are considered. Let us notice that in Eq. (4.18) we accounted alreadyfor the nesting condition so that the most relevant fluctuations are around q = 0.


Thus, when for example α = h and β = e the electronic dispersion entering ther.h.s. of Eq. (4.18) is given explicitly by ξe

k+q = [(k + q)2 − (keF )2]/2m, i.e. the

momentum is measured with respect to the M = Q point according to the definitionEq. (4.15).

After analytical continuation to the real frequencies the imaginary part of theself-energy can be written as

Γα(k) = −Σ′′α(k, ω = 0)

= −g2∑

q

∫ ∞

−∞

dε

2πF (ε)ImGβ

R(k + q, ε)ImχR(q,−ε), (4.19)

where F (ε) = coth(ε/2T ) − tanh(ε/2T ) and GR is the retarded Green’s function.For the sake of a semi-analytical treatment, we approximate the spectral functionof the carriers as ImGα

R(k, ε) = −π δ(ε− ξαk ), so that we can perform explicitly the

energy integration to get

Γα(k) = −g2ωsf χQ

2

∑

q

F (ξβk+q)

ξβk+q

ω2q + (ξβ

k+q)2, (4.20)

where ωq = ωsf (1 + ξ(T )2q2).

4.3.2 Vertex corrections: Currents and Conductivities

Let us turn now to the vertex corrections. At ω = 0 the dressed current can becomputed in a conserving approximation as [35, 36]

Jα(k) = vα(k) + g2∑

q

∫ ∞

−∞

dε

2πF (ε)Gβ

R(k + q, ε)

×GβA(k + q, ε)ImχR(q, ε)Jβ(k + q) (4.21)

(see Fig. 4.5). In a compact form we can write Jα(k) = vα(k) + ∆Jα(k). By usingagain the approximated relation Gβ

RGβA = |Gα(k, ε)|2 = πδ(ε − ξα

k )/Γα(k), andEq. (4.16) we obtain

∆Jα(k) =g2ωsf χQ

2

∑

q

F (ξβk+q)

ξβk+q

ω2q + (ξβ

k+q)2

Jβ(k + q)Γβ(k + q)

. (4.22)

Σ =J = + J

(a) (b)

Figure 4.5. Diagrammatic representation of the self-energy (a) and of the dressed current(b) in our model. The solid lines represent the Green’s function and the wavy linesrepresent the spin-fluctuation propagator (4.16).

To compute the conductivities we must evaluate the above Eq. (4.20), (4.22) fork = kα

F . Moreover, by close inspection of Eqs. (4.20) and (4.22) we see that the


largest contribution to the integral comes from the q values smaller than 1/ξ(T )such that ξβ

kαF

+q = 0, i.e. such kαF + q = kβ

F . We can then further approximate theexpression (4.22) for the vertex corrections by assuming that

Jβ(kαF + q) ≈ Jβ(kβ

F ) = JβF and Γβ(kα

F + q) ≈ Γβ(kβF ) = Γβ

F . (4.23)

Taking into account also the circular symmetry of the problem we see that theJα

F is still parallel to k (and to vF ). We can check this explicitly. Indeed, usingfor example k = kα

F = (kαF , 0) in Eq. (4.22) one immediately sees that ∆Jα

y = 0

while ∆Jαx will be determined by Jβ

x (k + q) ≈ (JβF · k)(kα

F + q cos θq)/|kαF + q|.

By introducing the velocity and current projection along k, i.e. Jα ≡ JαF · k and

vα ≡ vαF · k, so that vh < 0 and ve > 0, one can then write an overall set of

self-consistent equations for Jh, Je as:

Jh =vh + λhe ve

(1− λheλeh), Je =

ve + λeh vh

(1− λheλeh). (4.24)

where the matrix Λαβ of Eq. (4.13) has been expressed in terms of the T -dependent coefficients

λαβ =g2ωsf χQ

2ΓβF

∑

q

F (ξβkα

F+q)

ξβkα

F+q

ω2q + (ξβ

kαF

+q)2

kαF + q cos θq

|kαF + q| . (4.25)

By close inspection of Eqs. (4.20) and (4.25) we see that λαβ increases as the nestingcondition is approached, since the largest contribution to the integrals comes fromvectors around q ∼ kα

F − kβF . At low T , ξ(T ) increases and only the value θq = 0

contributes to the integral (4.25), leading to λαβ → 1 and large prefactors in Eq.(4.24). Moreover, at high T ξ(T ) ≃ 0, as a consequence the propagator Eq. (4.16),and thus ωq, is independent on q, so that the integral λαβ vanishes by symmetryargument and Jα = vα. This can be easily seen by writing Eq. (4.22) in a compactform as:

∆Jα(k) =∑

k′

Imχ(k − k′)Jβ(k′)F(|k′|), (4.26)

where F(|k′|) is a function containing of ξk′ so that the k-dependence is only via|k′|. As a consequence, when ξ(T ) ≃ 0, Imχ(k− k′) ∼ 1/ω0 looses the momentum-dependence, i.e. it is constant in k′ then ∆J = 0 by symmetry since Jk′ ∝ k′.

Slightly electron-doped compounds

To elucidate the effect on the transport of such a scattering mechanism connectinge- and h-like bands we will consider a set of parameters appropriate for a slightlyelectron-doped Ba(Fe1−xCox)2As2. In particular we take 1/2m = 70 meV, and wechoose Ee

min = 90 meV and Ehmax = 66 meV (i.e kh

F = 0.30π/a and keF = 0.37π/a) to

reproduce the data at 7% doping, where long-range AF order is no more present (i.e-the system is in the normal phase) and our model applies. For the SpF propagator,Eq. (4.16), we refer to [109] by using ω0 = 15 meV, Θ = 90 K, ξ0/a = 3.6 andg2χQ = 0.8 eV. Notice that numerically J → v at very high temperature becauseξ(T ) vanishes very slowly as 1/T . Since the low-energy description (4.16) is not


expected to hold any more around a scale of the order of the room temperature,we rescaled ξ(T ) = ξ0

√

Θ/(T + Θ) exp(−T/Tcut) to account for a fast decay ofAF correlations above Tcut ≃ 300 K, so that J(k) → v(k) as T → Tcut. Finally, tomimic the residual scattering by impurities at T = 0 we added a constant (isotropic)scattering rate Γα

0 = 4 meV.The resulting currents for each band as a function of temperature, as evaluated

from Eqs. (4.20)-(4.25), are shown in Fig. 2.7 a, along with the bare Fermi velocities.Two relevant features emerge.

1. We find a strong temperature dependence of both Je and Jh, which deviatesignificantly from their bare values with lowering temperature, due to theincreasing scattering from SpF.

2. In the slightly e-doped case considered here, where |ve| & |vh|, the dominanceof λheve with respect to vh in Eq. (4.24) is reflected in a change of sign of Jh

at low T , so that for the full T -range considered |Je| ≫ |Jh|.

These features have a striking effect on the Hall transport, shown in Fig. 4.6 b,along with the Boltzmann result (4.14) computed without vertex corrections.

0 50 100 150 200 250 300T (K)

-200

-100

0

100

200

300

v, J

(m

ev)

0 50 100 150 200 250 300T (K)

0

50

100

150

200ρ(

µΩcm

)ρ0

ρ ρ (FS shrinking)

-5

-4

-3

-2

-1

0

RH(1

0-9m

3 /C)

R0H

RH RH (FS shrinking)

vh

ve

Je

Jh

ve+ λehv

h

vh+ λehv

e

(a)

(c)

(b)

Figure 4.6. (a) T dependence of the renormalized currents (filled symbols), as comparedto the bare velocities (empty symbols). We also show (dashed lines) the numeratorsof Eq. (4.24), that fix the overall sign of the currents. (b) T dependence of the Hallcoefficient RH compared to the Boltzmann result (4.14) R0

H , computed with 1/τα =2Γα

F . The units are fixed by the 2D results divided by the interlayer distance d = 6.5 A.Dashed line: RH obtained including also the effect of the FS shrinking. (c) Longitudinalresistivity as a function of T compared to ρ0 =

(∑

α e2nατα/m)

−1in the Boltzmann

approximation with 1/τα = 2ΓαF .

As expected, R0H is small and weakly T dependent, with R0

H ≃ −0.3 10−9 m3/C,as due to the almost perfect cancellation of the contributions from the h- and


e-like Fermi sheets. On the contrary RH has a strong temperature dependenceand it can attain a large negative value at low T , where the e-like renormalizedcurrent |Je| ≫ |Jh| dominates the transverse conductivity (4.11). At the sametime, the effects of the vertex renormalization are less qualitatively relevant on thelongitudinal resistivity with respect to the Boltzmann result, as shown in Fig. 4.6c. Indeed, the dependence of σα

xx in Eq. (4.10) on the sign of the renormalizedcurrent leads to a compensation between the vertex corrections in the h and ebands. Our results provide thus a consistent picture for both longitudinal andtransverse transport, in good agreement with the experiments [33, 34]. For example,at intermediate temperatures T = 150 K the experimental value [33] RH ≃ −1.810−9 m3/C is about 10 times larger than R0

H determined by the charge unbalancefrom Eq. (4.14).

For the sake of completeness let us discuss also the effect of the weakly T -dependent FS shrinking arising from the real part of the self-energy (4.18) [32, 37],already discussed in Section 1.3.2. The real part of the self-energy, χα(k) can bederived in first approximation by replacing in Eq. (4.18) Gβ with the non-interactingGreen’s function, so that

χα(k) = Σ′α(k, ω = 0)

= −g2χQωsf

π

∑

q

P∫ ∞

0dΩ

Ωω2

q + Ω2

f(−ξβ

k+q) + b(Ω)

ξβk+q + Ω

+f(ξβ

k+q) + b(Ω)

ξβk+q −Ω

, (4.27)

where P stands for principal part and b(x) is the Bose function. Notice that whenthe AF correlation length ξ(T ) = 0, so that the SpF propagator (4.3) does notdepend on the momentum, the q integration in Eq. (4.27) above can be converted inan integration over the energy ǫ, that vanishes in the usual infinite-band Eliashbergapproximation, where −∞ < ǫ <∞. However, as discussed in [32, 37], finite-bandeffects lead to a finite correction even in the case of momentum-independent spinpropagator, scaling approximately at T = 0 as:

χα ≈ −g2χQωsfNβ ln

∣∣∣∣∣

Eβmax − µ

Eβmin + µ

∣∣∣∣∣

(4.28)

where Nβ is the density of states in the β-th band. As a consequence, the r.h.s. ofEq. (4.28) is negative if β is a band having electron character (Eβ

max ≫ Eβmin), and

it is positive is β has hole character (Eβmax ≪ Eβ

min), leading to a positive/negativeshift for a α electron/hole band, respectively. These shifts at the Fermi level lead toa shrinking of the Fermi wave vectors in each band, defined in the general interactingcase as [32]:

(keF )2 = 2m[Ee

min − χeF ], (kh

F )2 = 2m[Ehmax − |χh

F |], (4.29)

where χαF ≡ χα(kα

F ). In the more general case of a momentum-dependent SpF spec-trum the quantity χα

F computed with Eq. (4.27) above preserves the sign propertydefined by Eq. (4.28), leading thus to a (weak) T -dependent shrinking of the Fermi


wave vectors [37], whose effect on the conductivities has been shown in Fig. 4.6b, c. We also checked that when ξ(T ) = 0 the expression (4.27) coincides withina 20% of accuracy with the full self-consistent solution computed in [32] for themomentum-independent SpF spectrum. Since such a FS shrinking has a negligiblerole on the Hall coefficient, as shown in Fig. 4.6 b, such a first-order approximationwill be sufficient for the purposes of the present work. On the other hand, as onecan see in Fig. 4.6 c, the shrinking of the FS contributes in part to the temperaturedependence of the longitudinal conductivity, as discussed in [37].

Moving away from the nesting condition

Let us focused now on the effects of doping. Indeed, as we mentioned above, theabsolute value of the λαβ coefficients decreases as the Fermi wave vectors move awayfrom the nesting condition ke

F = khF , realized at half-filling. We expect thus that the

effect of the vertex corrections on the Hall transport will be less relevant by furtherincreasing the Co concentration x. We investigate this issue by making a rigid-band shift of the chemical potential with doping, without changing for simplicitythe microscopical parameters of the SpF spectrum. In Table 4.1 we summarizedthe parameters used at various doping.

x µ(meV) khF (π/a) ke

F (π/a)

7% 0 0.30 0.3710% 6 0.28 0.3815% 17 0.25 0.4020% 28 0.21 0.43

Table 4.1. Parameters used to model various doping levels within a rigid-band approxi-mation. The doping is defined as x = ne − nh. Moreover the carrier densities obey tone = NF (Ee

min + µ), and nh = NF (Ehmax − µ) where NF = m/π in 2D. We started

our analysis from the study of the Ba(Fe1−xCox)2As2 x0 = 0.07 Co-doped sample, thusfor convention we fixed Ee

min = 90 meV and Ehmax = 66 meV so that µ = 0 for this

compound: x0 = n0e − n0

h = NF (Eemin − Eh

max). The chemical potential shift is givenby µ = (x − x0)/2NF and the kF vectors as usual is defined as ke

F =√

(Eemin + µ) 2m

and khF =

√

(Ehmax − µ) 2m.

By comparing two very different doping, it easy to verify the different effects ofthe vertex correction in the renormalization of the bare velocities, Fig. 4.7.In the highly doped sample we have |ve| ≫ |vh|. At low temperature, due to thestrong suppression of the λ coefficient, the renormalized current are very similar tothe bare one |Jα| ≃ |vα|. However, also when the temperature increases the renor-malization is very small. This can be easily understood looking at the temperaturedependence of the λ coefficient. Concerning the hole current for example, even ifthe λhe coefficient remains quite small in the full T -range, the correction ∼ λheve

(see Eq. (4.24)) leads to a change of sign of Jh, since |ve| ≫ |vh|. However, the


Figure 4.7. Left panel: Renormalized Currents for the Ba(Fe1−xCox)2As2at x = 0.07 andx = 0.20 Cobalt doped system. Right panel: Temperature behavior of the λeh, λhe

coefficients for the same dopings. Notice the strong suppression at low temperature forboth the λ coefficients of the 20% e-doped sample.

absolute value remains small, with |Jh| ≃ |vh|. On the other hand λeh increaseswith the temperature before to drop at higher temperature (T > 150 K) where thevertex correction vanishes. Nonetheless its contribution to Je is small since comesfrom a term ∼ λhevh (see Eq.(4.24)), but with |vh| ≪ |ve|. As a consequence boththe renormalized currents at high doping do not exhibit a strong T-dependence and|Jα| ≃ |vα|. The Hall coefficient computed by including vertex corrections for thesecompounds is expected to approach the Boltzmann value given by the simple carrierunbalance.

The map in Fig. 4.8 a, shows the absolute value at T = 100 K of the λhe as thenesting properties change. The maximum values is observed along the nesting lineke

F = khF as expected. The Hall coefficient for various doping is reported in Fig. 4.8

b, where we also mark with arrows the corresponding Boltzmann value R0H at each

doping.As one can see, the low-T enhancement of RH induced by SpF decreases withincreasing doping, and for x = 20% RH almost coincides with R0

H ≃ 1/ex, asfound experimentally [33]. The trend shown in Fig. 4.8b, where SpF spectrum iskept constant, is already in good agreement with the experiments. Nonetheless,one could also expect a decrease of the AF correlation length ξ(T ) with doping,leading to a faster suppression of vertex corrections. This effect could explain theresults in isovalent-substituted systems, as for example BaFe2(As,P/Ru)2 [71, 72]or La(Fe,Ru)AsO [73], where the change of magnitude (or even of the sign) of theHall coefficient should be attributed to a weakening of AF correlations, since nosignificant change on the FS pockets seems to occurs [112]. Finally, we notice thatfor a hole-doped system the overall temperature and doping dependence of RH

would be exactly the specular one: indeed, when the system is doped with holes,one has in general kh

F > keF , so that |Jh| ≫ |Je| at low T and the transverse

conductivity will have a predominant hole-like character, in agreement with theexperiments [74, 75]. Thus, the same mechanism accounts for the unusual Halleffect measured in pnictides both in electron and hole-doped compounds.


0 50 100 150 200 250 300T (K)

-15

-10

-5

0

RH

(1/

n Fe )

6%

10%

15%

20%

(b)

0 50 100 150 200 250 300T (K)

-6

-5

-4

-3

-2

-1

0

RH

(10

-9 m

3 /C)

x=6x=10x=15x=20

Figure 4.8. (a) Dependence of the vertex-correction coefficient λhe at T = 100 K on theFermi wave vectors in the two bands. Notice that λhe decreases as one moves away fromthe nesting line ke

F = khF . (b) RH as a function of the Co concentration x (which adds

x electrons per Fe atom) The arrows indicate the corresponding values of the T = 0Boltzmann result (4.14), which has a negligible T dependence on this scale (see Fig. 4.6b). On the right axes RH is expressed in units of the inverse number of carriers per Featom, defined as nF e = 0.32× 10−9/|RH [m3/C]|.

4.3.3 Comparison with the experiments

Even though the present work is not intended to provide a fit of the existing ex-perimental data, we show in the present section that our results are in good quan-titative agreement with the experiments, despite the use of a simplified two-bandmodel. In Fig.4.9 we show our results for RH at different doping level, along withthe experimental data of [33]. For the sake of completeness, and in analogy with[33], we also show the same data in terms of the density of carrier per Fe atomnF e = 0.32× 10−9/|RH [m3/C]|.

As one can see, our results capture the correct order of magnitude of the Hallcoefficient for slightly doped compounds. More specifically, for low doping the exper-imentally measured RH is about one order of magnitude larger than what expectedon the basis of a simple charge unbalance, i.e. of the Boltzmann estimate shown byarrows in Fig.4.9. Moreover, we reproduce also the trends observed experimentallyfor the temperature and doping variations. In particular as doping increases thetemperature variations of RH reduce considerably, and RH approaches the valueexpected on the basis of the Boltzmann approach. More specifically, at low Twhere the scattering-time anisotropy between the hole and electron band is negli-gible, RH approaches the value given by the simple carrier unbalance, with nF e ∼ x.

To further improve the quantitative agreement with the experiments severalfactors should be considered. First of all, the analysis of spectroscopic and thermo-dynamic properties of pnictides [85, 110, 111] has shown that in order to reproduce


0 50 100 150 200 250 300T (K)

-6-5-4-3-2-10

RH

(10

-9 m

3 /C)

Experiments

Theory

0 50 100 150 200 250 300T (K)

0

0,1

0,2

0,3

0,4

n Fe

x=7x=10x=15x=20

0 50 100 150 200 250 300T (K)

-3

-2,5

-2

-1,5

-1

RH

(10

-9 m

3 /C)

x=6.5x=10x=14x=20

0 50 100 150 200 250 300T (K)

0

0,1

0,2

0,3

0,4

n Fe

Figure 4.9. (Color online) Panels (a)-(b): theoretical results for the Hall coefficient (a)and the equivalent density of carrier per Fe atom (b) at various doping levels. Thearrows indicate the value of R0

H expected on the basis of a Boltzmann-like approach inthe two-band model for the corresponding doping level. Panels (c)-(d): experimentalresults for the same quantities, taken from [33]. As one can see, the experimental valuesare about one order of magnitude larger than those expected on the basis of the simplecarrier unbalance, shown by arrows in panel (a).

with high accuracy the experiments a full four-band model (two different hole bandsand two electron bands, eventually degenerate) is needed. Indeed, in this case onecan also account for the anisotropy of the interband interaction between the elec-tron pockets and the two hole ones, that have usually quite different sizes in 122compounds. This effect will be also relevant for the Hall coefficient, since the vertexcorrections in the two hole bands will acquire a different weight, contributing tothe overall magnitude of the transverse conductivity and to its temperature depen-dence. The latter one will be also affected by the exact form of the SpF propagatorand by its temperature and doping evolution, which is only partly known from theexperiments [109]. While these effects will be relevant to obtain a precise fit of thedata, they will not affect the main result shown in the present Chapter, i.e. thecrucial role played by current vertex corrections to renormalize the Hall responsewith respect to the Boltzmann value in slightly doped pnictides.

Conclusions

In this Thesis we addressed some relevant issues which arise while dealing with thecomplex phenomenology of High Temperature Superconductors (HTSC). As dis-cusses in the first Chapter, the wide variety of anomalies found in Cu- and Fe-basedsuperconductors stimulated in these years the theoretical investigation on the roleof the several degrees of freedom and their interplay, in the attempting of isolatingthe "key" mechanism for the unconventional superconductivity. In particular wefocused on the analysis of the transport properties of the normal state of cuprateand pnictide systems.

As concerns cuprates, we focalized on the analysis of the Superconducting Fluc-tuations (ScF) of underdoped phase, triggered in particular by the recent observa-tion of an overall discrepancy in the strength of the ScF contribution to conductivityand diamagnetism [14]. As a matter of fact, the paraconductivity is found smallerof about two order of magnitude with respect to the fluctuating diamagnetism in asimilarly doped compound.

The authors of [14] interpreted their data in the context of Kosterlitz-Thouless(KT) phase fluctuations within a preformed pair scenario. In this formalism, it ispossible to recast the ScF contribution to conductivity and diamagnetism in thedefinition of the vortex diffusion constant, so that the experimental findings can beseen as a signature of anomalous fast vortices in underdoped cuprates (see Section1.2.2). However, we widely discussed in Section 1.3.1 as the experimental result of[14] can be reformulated in a more general ScF language. Being the paraconductivityand the fluctuating diamagnetism connected to opposite (dynamic vs static) limitsof the ScF contribution to the current-current response function, the possibilitythat p-h interactions could affect the ScF in a way that differs in the two limitsarises. Only by using a microscopic approach one can control the contributions ofthe several interaction channels to the ScF.

The theoretical tools needed to implement this approach are summarized in thesecond Chapter. In particular in Section 2.2.1 we reviewed the standard theory ofthe ScF, while in Section 2.2.2 we show its correspondence with the Functional Inte-gral method, necessary to account for the multichannel character of the interaction.

In Chapter 3 we computed the ScF contribution to conductivity and diamag-netism in the presence of current-current interactions, by using the slave-bosonformulation for the t−J model as a paradigmatic example. By explicitly construct-ing the Ginzburg-Landau (GL) fluctuation functional in the presence of Hartree-Fock (HF) corrections, we showed that current-current interactions, needed to re-cover the gauge-invariant form of the GL functional, modify the transport coeffi-cients leading to a momentum and frequency dependence of the vertex c(q) entering

93

94 Conclusions

the Aslamazov-Larkin (AL) expression for the ScF contribution. Since different lim-its are involved in the definition of paraconductivity and diamagnetism, we obtaina different prefactor in the two cases, with a suppression of the paraconductivitybecause of the proximity to the Mott-insulating phase, as recently shown experi-mentally in cuprates [14]. We checked numerically our results within the Mean-Field (MF) analysis of the t-J model in the slave boson approach. We find a smallerdisparity between the two ScF contributions than that experimentally found. Eventhough this modeling is not satisfactory from the quantitative point of view, thedifferent strength of ScF contribution to conductivity and diamagnetism is moregeneral, since it is a consequence of the existence of a sizable difference betweenquasiparticle current and velocity caused by the HF correction. A quantitativecomparison with experiments remains an interesting theoretical challenge, that cer-tainly deserves further investigation.

Finally let us notice that the suppressed paraconductivity in the underdopedcuprates highlighted in [14] is observed in thin films (quasi-two-dimensional (2D)systems) where the KT physics is at play. The authors of [14] analyze their datain the context of a vortex unbinding picture for 2D superconductors, using theHalperin-Nelson analysis [64] of the ScF based on the Bardeen-Stephen (BS) theory[65] of the vortex motion. However the experimental outcome is independent fromthe particular modeling of the ScF, but relays only on the very general assumptionthat the ScF contribution to the several physical quantities could be expressed interms of the ScF correlation length. Thus the same results obtained within theconventional GL approach could be reasonable obtained also starting from moreexotic pictures, i.e. taking into account unconventional (non-GL) theories. A refor-mulation of the phenomenological BS theory in a more microscopic language wouldbe envisaged to account for microscopic effects due to current-current interactions.This is still an open issue, and the understanding of how the current-current inter-actions enters in the description of the ScF within the KT scheme, as already donewithin the GL approach, remains a theoretical challenge.

As regards pnictide systems, we widely showed in the first Chapter that thepresence of several sheets of the Fermi Surface (FS) makes the multiband characterof these compounds the main ingredient of any theoretical description. Indeed, theprediction of this feature by Density Functional Theory (DFT) calculations was soonconfirmed by FS sensitive experiments, such as de Haas-van Alphen (dHvA) andphotoemission spectroscopy. A much more indirect probe of such a multiband char-acter comes from transport experiments, where the contribution of carriers havinghole and electron character is unavoidably mixed. A typical example is providedby Hall effect measurements: indeed, in a (almost) compensated semimetal onewould expect an almost perfect cancellation between the hole and electron bands.Surprisingly, as we reviewed in Section 1.2.3, several measurements in both un-doped and hole and electron doped systems show a very large absolute value of theHall coefficient with a marked temperature dependence and a predominant elec-tron/hole character of the transport in electron/hole-doped materials [33, 34]. Thislead some authors to suggest the existence of a strong anisotropy between the scat-tering time of carriers having different character: however, such an interpretationdoes not have presently any support on explicit theoretical calculations for realisticmodels. In Section 1.3.2 we summarized the recent outcomes of an intermediate-

Conclusions 95

coupling Eliashberg approach to pnictides have been investigated in [32], where amomentum-independent Spin Fluctuations (SpF) propagator has been used. Onthe other hand vertex corrections could be important to deal with transport, sincethey lead to the definition of the renormalized currents. For this reason, we investi-gate the role of the momentum dependence of the SpF spectrum, which introducesvertex corrections to the quasiparticle current.

In the second chapter we widely discussed the importance of carrying out a fullgauge-invariant derivation for the response functions, including the effects of vertexcorrections. In particular, in Section 2.3, we summarized the derivation of a gauge-invariant response function by means of the Kubo formula for the Hall conductivityfor a nearly-free electron single-band system. This allowed us to discuss about thestandard Relaxation-Time Approximation (RTA) that leads usually to the inclusionof the effects of vertex corrections to the definition of a transport scattering time(see Section 2.4).

In Chapter 4 we generalized the Kubo approach to a multiband system withhole and electron carriers. We analyzed the Hall effect in a multiband modelwhere carriers interact via the exchange of SpF. We included the effects of theAntiferromagnetic (AF) vertex corrections showing the emergence of unconventionalfeatures in the transport. In particular we showed that when interactions have apredominant interband character and connect carriers of opposite electron/hole na-ture, the currents renormalized by vertex corrections are dominated by the characterof the majority carriers, and even a small asymmetry in the doping level leads toa huge renormalization of the bare velocities. This effects cannot be recast in theusual definition of a transport scattering time within the RTA approach and leadsto an Hall coefficient completely different from the one obtained through a semiclas-sical analysis. By evaluating these effects within a simplified two-band model anda phenomenological description of SpF, we were able to reproduce the main puz-zling features observed experimentally in the normal phase of pnictides, namely astrong temperature dependence of RH with a large absolute value at low T for weak(electron or hole) doping, and a more ordinary Boltzmann-like behavior at higherdoping. We also compared our results with experimental data of RH(T ) from [33]for various doping levels. Despite the use of a very minimal model, our results showa good agreements with the experiment. Moreover, the mechanism discussed here isquite general and robust: thus, an analysis based on a full self-consistent approachfor the SpF [35] within microscopic multiband models [69, 70] is expected to addonly quantitative refinements to the present results.

An open question is instead the role of vertex corrections across the AF tran-sition, where the Hall coefficient has been found experimentally to show an evenlarger T dependence [33, 34], or perhaps stronger anomalies e.g. nonlinearity againstmagnetic field [114]. Many authors speculated about the leading influence of thereconstruction of the FS occurring with the Spin Density Wave (SDW) gap openingin some parts of the FS, however until the present moment no theoretical estimatehas been calculated. On the other hand also measurements of the magnetoresistence[113, 114], both in the normal and in the broken AF phase of pnictides, reveal sev-eral anomalies, such as, for example, an unconventional linear dependence in themagnetic field. Many authors interpret this feature like a signature of the presenceof anisotropic Dirac cones in the band structure, but theoretical calculations aboutthis effect are still lacking. Certainly these issues deserve further investigations to

96 Conclusions

complete our understanding of transport properties in pnictides.

In conclusion, in this Ph.D. Thesis we demonstrated that many of the anomaloustransport properties of the HTSC can be ascribed to the multichannel characterof these classes of materials. This requires the revision of many of the standardparadigms used in the presence of a single interaction channel. It is not always aneasy task dealing with many relevant degrees of freedom: the full gauge-invariantcalculation of the transport functions must be carried out, i.e. the vertex correctionshave to be included in the analysis. This is especially crucial by dealing withtransport to account for the renormalization of quasiparticle velocity due to theinteractions.

List of Acronyms

AF Antiferromagnetic

AL Aslamazov-Larkin

ARPES Angle-Resolved PhotoEmission Spectroscopy

BEC Bose-Einstein Condensation

BCS Bardeen-Cooper-Schrieffer

BS Bardeen-Stephen

BZ Brillouin Zone

DC Direct Current

DFT Density Functional Theory

dHvA de Haas-van Alphen

DOS Density of States

em electromagnetic

FL Fermi Liquid

FLEX FLuctuation-EXchange

FS Fermi Surface

GI Gauge Invariance

GL Ginzburg-Landau

GWI Generalized Ward Identity

HF Hartree-Fock

HN Halperin-Nelson

HS Hubbard-Stratonovich

HTSC High Temperature Superconductors

KT Kosterlitz-Thouless

97

98 List of Acronyms

LDA Local Density Approximation

MBZ Magnetic Brillouin Zone

MF Mean-Field

MT Maki-Thompson

NAFFL Nearly Antiferromagnetic Fermi Liquid

NMR Nuclear Magnetic Resonance

PG Pseudogap

p-h particle-hole

p-p particle-particle

RPA Random Phase Approximation

RTA Relaxation-Time Approximation

SC Superconducting

ScF Superconducting Fluctuations

SDW Spin Density Wave

SpF Spin Fluctuations

2D two-dimensional

3D three-dimensional

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