Quantum Dynamics in One-Dimensional and Two-Leg Ladder …Sarkar, Prof. Ram Ramaswamy, Prof....

UNIVERSITÉ DE GENÈVESection de PhysiqueDepartment of Quantum Matter Physics

UPPSALA UNIVERSITYDepartment of Physics and Astronomy

FACULTÉ DES SCIENCESProfesseur Thierry Giamarchi

MATERIALS THEORY DIVI-SIONProfesseur Adrian Kantian

Quantum Dynamics inOne-Dimensional and Two-Leg

Ladder Systems

THÈSE

présentée à la Faculté des Sciences de l’Université de Genèvepour obtenir le grade de docteur ès Sciences, mention Physique

par

Naushad Ahmad Kamarde

Kushinagar (India)

Thèse n 5416

GENÈVEAtelier d’impression ReproMail

2019

This thesis is dedicated to my late mother Kamarun Nisha, my fatherKutubuddin Siddique, my late grandfather Munawwar Hussain, my allteachers, my wife Israt Jahan, and my son Mohammad Asad Qamar.

Acknowledgements

This thesis is possible because of my supervisors, teachers, labmates,friends, and family members. It is hard to mention all of them. Inparticular, I want to thank:

My main supervisor Prof. Thierry Giamarchi for giving me an oppor-tunity to work under him as a Ph.D. student, introducing the beautifulworld of one-dimensional quantum systems and introducing the theo-retical techniques to me. He also helped me at both academic and non-academic levels.

Prof. Adrian Kantian, my collaborator, and my second supervisor: in-troduced the numerical techniques to me and helped me throughout myPh.D. studies.

Dr. Christoph Berthod, for his technical support that saved my time inprogramming.

Prof. Shun Uchino and Dr. Shunsuke Furuya, for clearing doubts oftheoretical techniques and Dr. Shintaro Takayoshi for helping me withnumerical techniques.

Noam Kestin, for making the environment of the office enjoyable andmotivating me during hard times.

Fabienne Hartmeier, for helping me at administrative level.

My office mates, Dr. Laura Foini, Dr. Emanuele Coira, Dr. PjotrsGrisins, Sergio Enrique Tapias Arze, Nicolas Burgermeister, Dr. MicheleFilippone, Dr. Charles Bardyn, Dr. Sebastian Greschner, Dr. NirvanaBelen Caballero, Dr. Anne-Maria Visuri, Joao Ferreira, and Joel Schaerfor their kind supports.

My MSc supervisor late Prof. Deepak Kumar, MS(Eng.) supervisorProf. NS Vidhyadhiraja and my teachers Prof. Sanjay Puri, Prof. SubirSarkar, Prof. Ram Ramaswamy, Prof. Debashis Ghoshal, Prof. BrijeshKumar, Dr. Amit Chaubey, and Riyasat Ali for introducing the beautifulworld of Physics to me.

The Swiss National Science Foundation under MaNEP and Division II,and the financial and academic support of the Université de Genève.

And last but not least, my family members and my friends for their loveand supports.

Résumé en français

Dans cette thèse, nous avons travaillé sur trois problèmes. Dans le premier prob-lème, nous regardons un potentiel local dépendant du temps dans un bain unidi-mensionnel en interaction. Ce travail est motivé par des expériences sur les atomesultra-froids, où l’on peut moduler périodiquement le réseau optique en utilisant unpotentiel dépendant du temps. Nous utilisons la théorie de la réponse linéaire, labosonisation et le groupe de renormalisation pour calculer l’énergie déposée dans lesystème par le potentiel dépendant du temps. Nous constatons que l’énergie déposée,tant pour le régime de couplage faible que celui de couplage fort, montre un com-portement de loi de puissance en fonction de la fréquence des oscillations et montreun exposant de loi de puissance différent dans les limites de faible et fort couplage.

Dans le deuxième problème, nous travaillons sur la dynamique d’une impureté-mobile dans une échelle bosonique à deux montants. Les études théoriques et ex-périmentales, dans le contexte des atomes ultra-froids, d’une impureté mobile dansun bain motivent ce travail. Dans ce travail l’impureté est contrainte à se déplacer surune des pattes de l’échelle seulement. Nous combinons la bosonisation, le groupe derenormalisation, et le groupe de renormalisation de la matrice de densité (DMRG)pour comprendre la dynamique de l’impureté. Nous calculons la fonction de Greende l’impureté analytiquement et numériquement. Pour une faible interaction entrel’impureté et le bain, nous trouvons analytiquement que la fonction de Green del’impureté se comporte comme une loi de puissance et nos résultats numériques sonten excellent accord avec les résultats analytiques. De plus, nous donnons égalementun résultat analytique pour la fonction du Green dans le cas d’une interaction infinie.On retrouve à nouveau que la fonction de Green décroit en loi de puissance, en trèsbon accord avec les résultats numériques. Finalement, nous donnons une expressionsemi-analytique de l’exposant de cette loi de puissance, pour une impureté possé-dant une impulsion nulle, en fonction de la force de l’interaction. Ces résultats sont

i

également en accord avec le calcul numérique de l’exposant.Dans le troisiéme problme, nous étudions à nouveau la dynamique d’une im-

puretè mobile dans l’échelle bosonique à deux montants, mais dans ce cas, l’impuretéest libre de se déplacer aussi bien longitudinalement que transversalement. Nousutilisons des méthodes similaires à celles du deuxième problème. Nous calculons lafonction de Green dans les modes liants et anti-liants pour l’impureté. La fonctionde Green du mode anti-liant décroit comme une loi de puissance à faible impulsionet pour une faible interaction, tandis que la fonction de Green du mode liant décroitexponentiellement. L’analyse numérique de ce problème montre un excellent accordavec les résultats analytiques.

ii

Abstract

In this thesis, we work on three problems. In the first problem, we work on a lo-cal time-dependent potential in a one-dimensional interacting bath. This work ismotivated by cold atomic experiments, where one can shake the ultracold systemsby using a time-dependent potential. We use linear response theory, bosonization,and renormalization group to compute the energy deposition in the one-dimensionalbath by the time-dependent potential. We find that the energy deposition both forweak and strong coupling regimes shows a power-law behavior on the frequency ofoscillation, and shows a different power-law exponent in strong and weak couplinglimit.

In the second problem, we work on the dynamics of a mobile impurity in a two-leg bosonic ladder. The theoretical and experimental studies of a mobile impurityin a one-dimensional bath in context of ultracold systems motivate this work. Weconfine the impurity to move in one of the legs of the ladder. We combine bosoniza-tion, renormalization group, and density matrix renormalization group to understandthe dynamics of the impurity. We compute the Green’s function of the impurityboth theoretically and numerically. For a small interaction between the impurity andthe bath, we find theoretically that the Green’s function of the impurity decays as apower-law, and our numerical results show a good agreement with analytical result.Furthermore, we also give an analytical result of the Green’s function for an infi-nite interaction, we find again that the Green’s function decays as a power-law, andshows a good agreement with numerical results. Moreover, finally we give a semi-analytical expression for the power-law exponent at zero momentum as a function ofinteraction, again in good agreement with the numerical results.

In the third problem, we are again investigating the dynamics of a mobile impu-rity in a two-leg bosonic ladder, but in this case, the impurity also tunnels in bothlongitudinal and transverse directions. We use similar methods than for the second

iii

problem. We compute the Green’s function in the bonding and the anti-bondingmodes of the impurity. The Green’s function in the anti-bonding mode decays aspower-law at small momentum and small interaction, while the Green’s function inthe bonding decays exponentially, and our numerical results show a good agreementwith analytical ones.

iv

Contents

1 Introduction 11.0.1 Plan of the thesis . . . . . . . . . . . . . . . . . . . . . . . 3

2 Experimental Realizations 52.1 Introduction of Ultracold atoms . . . . . . . . . . . . . . . . . . . 52.2 Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Optical Lattice Geometry . . . . . . . . . . . . . . . . . . . . . . . 72.4 Realization of Quantum Hamiltonians . . . . . . . . . . . . . . . . 82.5 Experiments in connection with our Studies . . . . . . . . . . . . 9

2.5.1 Modulation of Optical Lattices . . . . . . . . . . . . . . . . 92.5.2 Mobile Impurity in Optical Lattices . . . . . . . . . . . . . 11

3 Methods 133.1 DMRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 DMRG in Conventional Form(Infinite DMRG) . . . . . . . . . . . 143.3 Matrix Product States (MPS) . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 Matrix Product Operator (MPO) . . . . . . . . . . . . . . . 193.4 Ground State Search By Variational Method . . . . . . . . . . . . . 19

3.4.1 Right Sweeping . . . . . . . . . . . . . . . . . . . . . . . . 223.4.2 Left Sweeping . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Calculation of expectation values . . . . . . . . . . . . . . . . . . . 233.6 Time Evolving Block Decimation . . . . . . . . . . . . . . . . . . 243.7 Imaginary Time Evolution as a Ground State Search . . . . . . . . . 253.8 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.9 Phenomenological Bosonization . . . . . . . . . . . . . . . . . . . 30

v

CONTENTS

4 Time dependent local potential in a Tomonaga-Luttinger liquid 334.1 Model and Method . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.2 Weak potential, bosonization . . . . . . . . . . . . . . . . . 354.1.3 Two component fermionic system . . . . . . . . . . . . . . 364.1.4 Strong coupling . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Calculation of the energy absorption . . . . . . . . . . . . . . . . . 394.2.1 Weak bare potential . . . . . . . . . . . . . . . . . . . . . . 394.2.2 Energy deposition in weak coupling limit . . . . . . . . . . 394.2.3 Crossover from weak to strong coupling limit and strong to

weak coupling limit . . . . . . . . . . . . . . . . . . . . . 404.2.4 Strong coupling renormalization of backscattering (K < 1,

ω < ω∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.5 Strong bare potential . . . . . . . . . . . . . . . . . . . . . 42

4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 444.3.1 Bosonic systems . . . . . . . . . . . . . . . . . . . . . . . 444.3.2 Fermionic systems . . . . . . . . . . . . . . . . . . . . . . 45

5 Dynamics of a Mobile Impurity in a Two Leg Bosonic Ladder 475.1 Mobile Impurity in a Two Leg Bosonic Ladder . . . . . . . . . . . 48

5.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.1.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.3 Bosonization representation . . . . . . . . . . . . . . . . . 50

5.2 Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.1 U ∆a . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.2 U ∆a . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.3 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3.2 Results at zero momentum . . . . . . . . . . . . . . . . . . 56

5.3.2.1 Interaction dependence . . . . . . . . . . . . . . 575.3.2.2 Hard core bath-impurity repulsion . . . . . . . . 59

5.3.3 Momentum dependence of the exponent . . . . . . . . . . . 595.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Mobile Impurity in a Two-Leg Bosonic Ladder with Motion in Horizon-tal and Transverse Directions 676.1 Mobile Impurity in a Two Leg Bosonic Ladder . . . . . . . . . . . 68

6.1.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.1.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2 Analytical solution for weak coupling . . . . . . . . . . . . . . . . 696.3 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.3.2 Zero momentum regime . . . . . . . . . . . . . . . . . . . 71

vi

CONTENTS

6.3.2.1 Small U . . . . . . . . . . . . . . . . . . . . . . 716.3.2.2 Hardcore bath-impurity repulsion . . . . . . . . . 71

6.3.3 Green’s function for finite momentum . . . . . . . . . . . . 756.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7 Conclusions 817.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A Calculation of Energy Deposition 85A.1 Linear response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A.2 Effective Hamiltonian for large V . . . . . . . . . . . . . . . . . . 86A.3 Calculation of correlation functions . . . . . . . . . . . . . . . . . 91

A.3.1 Fourier transformation of χR(t) . . . . . . . . . . . . . . . 91A.4 Calculation of correlation functions when V0 is relevant(K < 1) . . 93

A.4.1 Dilute instanton approximation . . . . . . . . . . . . . . . 94

B Green’s Function of the Impurity 97B.1 Linked Cluster Expansion (LCE) . . . . . . . . . . . . . . . . . . . 97B.2 DMRG procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 100B.3 Extracting exponent and error bar from numerical data . . . . . . . 102

C Green’s Function of the Impurity in Presence of Transverse Tunneling 105C.1 Green’s Function of a Mobile Impurity in a Two-leg Bosonic Ladder 105

vii

CHAPTER 1

Introduction

Cold atomic systems have proven in recent years to be a remarkable playground tostudy the effect of strong correlations in quantum systems [1, 2]. In particular theyhave allowed the realization of one dimensional quantum systems both for bosonsand for fermions. In one dimension interactions between the particles lead to avery rich set of properties, very different from their higher dimensional counterparts.The general physics goes by the description known as Tomonaga-Luttinger liquid(TLL) [3–6].

In a TLL, the interactions lead to a critical state in which correlations decreaseas power laws. This makes the system extremely sensitive to external perturbations,such as a periodic potential [7], quasi-periodic, disordered potentials [8,9], and eventhe presence of a single impurity [10, 11]. In the later case, depending on the inter-actions in the system, even a single impurity can potentially completely block thesystem. Such effects have been investigated in the condensed matter context bothin the context of carbon nanotubes [12] and also of edge states in the quantum Halleffect [13]. In such context the main probe is connected to the transport through theimpurity site, and the control parameters are either the temperature or the voltage.Theoretically, transport through a weak oscillating potential has been studied andcurrent induced by a potential oscillating in time shows a power law dependence onfrequency [14–17].

Recently ultracold systems were subjected to time-dependent potential to probethe low-energy excitations of the system. Such a technique of periodically modulat-ing potentials has been exploited with success to investigate several aspects of the

1

1. INTRODUCTION

Mott transition both in bosonic and fermionic systems [18, 19], for disordered sys-tems [20] or for superconducting systems [21]. A theoretical analysis shows thatmeasuring quantities such as the deposited energy [22,23] or the raise of the numberof doubly occupied sites [24, 25], in response to a periodic global modulation pro-vides a spectroscopic probe akin, but not identical to the optical conductivity.

Another important class of problem in quantum-many particle systems is the dy-namics of a mobile impurity in a quantum bath. In a solid, where the bath is formedby phonon excitations, this is at the heart of the polaron problem [26, 27]. For inter-acting particles, the dressing of one particle by the excitations of the density inducedby the interactions leads to the Fermi liquid description of an interacting fermiongas [28, 29]. In addition to these types of effects, it was realized by Caldeira andLeggett that the environment can deeply change the properties of a quantum degreeof freedom in a way that goes beyond the formation of a polaron with its attendantredefinition of some physical parameters of the particle, such as the mass [30, 31].

One potentially important ingredient is the dimensionality of system. It has beenshown [32] that for a one-dimensional bath the influence on the propagation of theparticle can be drastic. The particle motion may become subdiffusive with no rem-nant of quasiparticle behavior. The source for the onset of this new dynamical uni-versality class in the one-dimensional systems is a phenomenon akin to the Ander-son orthogonality catastrophe. Following this result, the phenomenon received moreextensive attention [33–38]. Combining numerical and analytical results [39] evenshowed that depending on the momentum of the particle a change in regime, fromsubdiffusive to polaronic could occur. Extensions of these results to fermionic sys-tems [40,41], driven particles [42,43] or particles coupled to several one dimensionalsystems [44] have since been performed.

How the breakdown of the quasiparticle picture and the emergence of subdif-fusive dynamics would be affected if the dimensionality of the bath increases is anopen and interesting question. Indeed, in other effects in which orthogonality mani-fests, such as the X-ray edge problem [5], it is well known that in dimensions greaterthan one the onset of the orthogonality catastrophe is suppressed as soon as the im-purity experiences recoil (i.e. impurity is mobile). This immediately gives rise tothe question as to how one actually crosses over from the singular physics of a onedimensional bath to the more conventional polaronic dynamics [45] that one couldnaively expect for two or three dimensional baths.

To address these questions, in this thesis, we worked on two different classesof problems. In the first problem, we present an analysis of the effect of a localtime-dependent potential in a one-dimensional bath and the energy deposition by thetime-depend potential.

In the second problem, we present an analysis of the dynamics of a mobile im-purity in a two-leg bosonic ladder. We have divided this problem in two parts. In the

2

first case, the impurity is confined to move in one of the legs of the ladder, and in thesecond case the impurity moves in both horizontal and transverse directions.

The main results discussed in this thesis are:

• By using TLL, renormalization group, and linear response theory, we providethe analytical results for energy deposition by a local time-dependent potentialin one-dimensional bosonic and fermionic systems.

• By using TLL and renormalization group, we provide an analytical expressionfor the Green’s function of the mobile impurity in two-leg bosonic ladder.

• By using TLL, renormalization group, and numerical density matrix renormal-ization group methods, we provide an analysis of the dynamics of a mobileimpurity in two-leg bosonic ladder by computing the Green’s function of theimpurity.

1.0.1 Plan of the thesisThe plan of the thesis is as follow:

• In chapter 2, we give a brief introduction about ultracold atoms, optical lat-tices, and the experimental realization of Bose-Hubbard and Fermi-Hubbardmodels.

• In chapter 3, we provide the analytical and numerical methods suitable forone-dimensional and quasi one-dimensional systems. We introduce densitymatrix renormalization group (DMRG) method. We discuss in details matrixproduct states, time-evolution of a matrix product states, and the variationaloptimization of a matrix product states in order to compute the ground state.We also introduce the analytical technique to describe TLL: bosonization.

• In chapter 4, we discuss the problem of the energy deposition in a one-dimensionalbath by a time-dependent potential in various regimes.

• In chapter 5, we discuss the dynamics of a mobile impurity in a two-legbosonic ladder, for which the impurity moves in one of the legs of the ladder.We compute the Green’s function of the impurity by using DMRG, bosoniza-tion, and linked cluster expansion.

• Motivated by the work in chapter 5, in chapter 6, we discuss the dynamics of amobile impurity in a two-leg ladder bath, but in this case, the impurity movesin both horizontal and transverse directions.

• The appendix A presents the calculation of the energy deposition.

• The appendices B and C present the calculation of the Green’s function of theimpurity by using linked cluster expansion.

3

1. INTRODUCTION

4

CHAPTER 2

Experimental Realizations

The physics of a quantum many-particle system is very rich and very much differ-ent from that of a non-interacting counterpart. The so-called band theory describesa non-interacting quantum system. But there is a wide range of quantum systemswhere interaction between particles is significant, such as spin systems, transitionmetal oxides, high-temperature superconductors. The quantum interacting many-particle systems are extremely challenging to solve analytically because of the expo-nential growth of the Hilbert space.

To address such complex systems, Richard Feynman proposed the idea of a quan-tum simulator [46]. A quantum simulator provides a platform to engineer complexquantum Hamiltonians such as the Bose-Hubbard and Fermi-Hubbard models, andby a precise measurement of the observables give a “solution” of these models. Ul-tracold systems allow to realize such quantum simulators [47].

To begin systematically, in this chapter, we will give a brief introduction aboutultracold atoms, optical lattices, the realization of the Bose-Hubbard model, andexperimental systems in connection with our theoretical studies.

2.1 Introduction of Ultracold atoms

Atoms move in random directions with a large velocity at room temperature, andtheir corresponding De Broglie wavelengths λB = h

mv are very small, where m,v and h are mass, the velocity of an atom, and the Planck constant, respectively.Because of short wavelengths, an ensemble of atoms does not show quantum effects

5

2. EXPERIMENTAL REALIZATIONS

at a macroscopic scale. λ′Bs can be increased by slowing the atoms. One of theways to slow an atom is by laser cooling [48]. In laser cooling, one bombards anatom by photons with the help of laser beams, and it slows the atom to nearly zerovelocity. The idea of laser cooling is as follows: suppose an atom moves toward alaser beam with a velocity ~v, and the atom absorbs a photon with a wave vector ~k,then the effective velocity of the atom will be given by ~veff = ~v − ~~k

m . Hence bymultiple successive operations, the atom can be tuned near-zero velocity, and a longDe Broglie wavelength can be achieved.

An ensemble of atoms will show emergent quantum phenomena on a macro-scopic scale if their De Broglie wavelengths are greater than the average inter-particledistance. The De Broglie wavelength λB of an atom at temperature T is given byλB = h√

2mkBTand ( VN )1/d gives the average inter-particle distance, where V , N

and d are the volume, total number of particles, and the dimensionality of the system,respectively. Thus a criterion for a system to show quantum effects at a macroscopicscale is λB ≥ (V/N)1/3 ⇒ T ≤ h2n2/3

2mkB. An ultracold system is very dilute, and

the mass of atoms is much larger than that of electrons. So, one needs a very low-temperature (of the order of nano Kelvin) to observe quantum phenomena.

Ultracold atoms play the same role in ultracold experiments as electrons playin condensed matter systems. As in quantum condensed matter systems, electronsmove in the lattice potential generated by the nuclei, we need similar lattice potentialfor ultracold atoms. For this, I will introduce optical lattices [49] in the next section.

2.2 Optical LatticesLet us consider a neutral atom, which is described by the two energy levels, |1〉and |2〉 with eigenvalues −ω0/2 and ω0/2, where ω0 is energy difference betweenlevel |1〉 and |2〉 . When the neutral atom is subjected to an oscillating electric field~E(x, t) = ~E0(x) exp(−iωt) + ~E∗0 (x) exp(iωt) this induces an electric dipole mo-

ment ~d.The potential energy of the atom in the presence of the electric field is given by

V (x, t) = − ~d. ~E(x, t) (2.1)

Because of the potential energy V (x, t), the energy levels |1〉 and |2〉 are shifted byE

(2)1 and E(2)

2 . Levels |1〉 and |2〉 experience the potential E(2)1 and E(2)

2 , respec-tively, and this effect is known as the AC Stark effect.

E(2)1 =

Ω2(x)

δ

E(2)2 = −Ω2(x)

δ

(2.2)

6

2.3 Optical Lattice Geometry

Where δ = ω − ω0, Ω(x) = d21. ~E0(x) and d12 = 〈1|d|2〉.The lattice potential experienced by the atom in the state |1〉 by the electric field

is Vlatt(x) = Ω2(x)δ . The sign of the potential can be tuned by δ = ω − ω0, and the

intensity of the electric field tunes the depth of the lattice potential. Since Vlatt(x) iscreated by the electromagnetic wave, it is known as an optical lattice.

Figure 2.1 – Ultracold atoms in one (upper panel) and two-dimensional (lower panel)optical lattices, solid spheres are the cold atoms . [Taken from Ref [50], and NIST].

2.3 Optical Lattice Geometry

As we have seen in the above section an optical lattice generated by the electric fieldis directly proportional to Ω(x)2, so the shape and the dimensionality of an opticallattice directly depend on the structure of Ω(x)2.

For example, a one dimensional optical lattice can be created by a single laserbeam Vlatt(x) =

Ω20

δ sin2(kx) where Ω(x) = Ω0 sin(kx). Similarly, a two dimen-sional optical lattice can be created by using two laser beams Ω(x, y) = Ω0 sin(kxx)x+

Ω0 sin(kyy)y and Vlatt(x, y) =Ω2

0

δ (sin2(kxx) + sin2(kyy)) and three dimensional

lattice geometry can be created by using three laser beams Vlatt(x, y, z) =Ω2

0

δ (sin2(kxx)+

7


sin2(kyy) + sin2(kzz)). Cold atoms loaded in one and two-dimensional optical lat-tices are shown in Fig. 2.1.

We have introduced the basic ingredients: cold atoms that play the role of elec-trons, and optical lattices, which play the role of lattice potential. In the next section,we will introduce the quantum Hamiltonians formed by the marriage of optical lat-tice with cold atoms.

2.4 Realization of Quantum HamiltoniansBy trapping ultracold atoms in the optical lattices, quantum Hamiltonians such asBose-Hubbard [1] and Fermi-Hubbard [2, 51] models have been realized, and manyexotic phenomena such as Bose-Einstein condensates, superfluid phase, Mott-insulatingphase, crossover from superfluid phase to Mott-insulating phase [52] , anti-ferromagneticphase [51], and many-body localized phase in random potential [53] have been stud-ied.

The important point about an ultracold system is that it provides a very control-lable quantum Hamiltonian with the desired Hamiltonian parameters. Let us recallthe Bose-Hubbard model, which was theoretically studied in the context of ultracoldsystems [54].

H = −J∑i,j

(b†i bj + h.c) +U

2

∑i

(ni(ni − 1)) (2.3)

Tunneling J and interaction U can be controlled by three parameters, light intensityΩ0, recoil energy of the atoms ER = k2

2M and s-wave scattering length as. Where kand M are the wave vector of the optical lattices and the mass of atoms respectively.b(b†) are bosonic annihilation (creation) operators. J and U are expressed in termsof V0, ER, and as as [55]

J =4ERV

1/40√π

exp(− 2

V1/20

E1/2R

)U =

√8kasE

1/4R V

3/40√

π

(2.4)

Where V0 =Ω2

0

δ . As we can see from (2.4), tunnelling J decreases as V0 increasesand onsite interaction U increases with increasing V0. Another way to control U isby changing as, which can be tuned by Feshbach resonance [56, 57].

Till now, we have introduced the necessary ingredients such as cold atoms andoptical lattices, and the loading of cold atoms in optical lattices provide the quantumHamiltonians such as the Bose-Hubbard and the Fermi-Hubbard models. In the nextsection, we will introduce different experiments on the cold atoms which would berelevant to our main work in chapters 4, 5, 6.

8

2.5 Experiments in connection with our Studies

Figure 2.2 – Schematic of modulation of cold atoms, V⊥ tunes the tunneling along x andz directions , and Vax = [Vax,0 + Amod sin(2πωt)] sin

2(ky) oscillates as a functionof time. A one-dimensional lattice is created by taking a very large V⊥ = 30ER, anda three dimensional lattice is created by V⊥ = Vax,0, and ER is the recoil energy ofatoms. [Taken from Ref [18]]

.

Figure 2.3 – Energy deposition of a one-dimensional system of interacting bosons asa function of frequency in a superfluid phase (open circles) (U/J = 2.3) and a Mott-insulating phase (solid circles) (U/J = 14) . [Taken from Ref [19]].


2.5.1 Modulation of Optical Lattices

Probing low-energy excitations is one of the fundamental issues in ultracold atomicsystems. The optical lattices were globally modulated [18–21] with the help of twolaser beams of different frequency which creates a time-dependent potential. Thedeposited energy by the time-dependent potential gives information about the low-energy excitations. The schematic of the modulation of cold atoms is depicted in Fig.2.2. The energy deposition was measured by looking at the growth of the full widthat half maxima (FWHM) of the central momentum peak [19]. FWHM as a functionof frequency in the superfluid phase and the Mott-insulating phase in one-dimension(V⊥ = 20ER) is depicted in Fig. 2.3. In the superfluid phase, FWHM shows a broadspectrum while in the Mott-insulating phase two-resonance peaks are observed. The

9


Figure 2.4 – FWHM as a function of U/J and frequency of modulation ω. Figure(a) is for the one-dimensional bosonic system; created by taking large V⊥ = 30ER,figure (b) is crossover regime from one-dimension to three-dimensions (bosonic system),which is achieved by V⊥ = 20ER, and figure (c) is for the three-dimensional case. Inthis case, V⊥ = Vax,0. In the Mott-insulating phase, FWHM shows resonance peaks.The first peak corresponds to excitations from sites with n = 1 (n is the number ofbosons at a given site) to their nearest sites with n = 1, the second peak correspondsto excitations from sites with n = 1 to their nearest sites with n = 2, and higher-orderpeaks correspond to tunneling of two atoms simultaneously from sites to their nearestneighbour sites . [Taken from Ref. [19]] .

Figure 2.5 – Density distribution of the impurity at J/U = .32 (upper panel) and ve-locity of the impurity as a function of J/U in the superfluid and Mott-insulating phasesof a one-dimensional bath, where J is tunneling amplitude of the impurity, and U is theinteraction between the impurity and the bath particles. Blue bars are experimental data,and red lines are t-DMRG results . [Taken from Ref [58]].

10


Figure 2.6 – Schematic of creation of an array of one-dimensional baths of Rb atoms, and by using species selective dipole potential (SSDP), K atoms are trapped in themiddle of one-dimensional tubes. [Taken from Ref [60]].

first peak appears at frequency ω = U , which is related to particle-hole excitationsin the Mott-insulating regime, and the second resonance peak appears at 1.91 ∗ U .For the various values of U/J measured in terms of recoil energy (ER), FWHM isshown in Fig.2.4.

Motivated by the above studies, we have studied the case of a local time-dependentpotential in a one-dimensional bath in chapter 4.

2.5.2 Mobile Impurity in Optical Lattices

Recently an impurity was created in one-dimensional bath in the ultracold gasesexperiments [58–60] in order to understand the dynamics of the impurity. Using theboson microscope [61], the local density of the impurity was measured for differenttimes [58]. The density profiles of the impurity give information about the excitationvelocity of the impurity in both Mott-insulating and superfluid phases of the bath.In Figure 2.5, the density of the impurity is depicted as a function of the position ofthe impurity for various times, showing density profile spreads. The velocity of theimpurity is extracted from the maxima of the density distribution for various times.The velocity of the impurity as a function of J/U is depicted in the lower panel ofFig. 2.5.

In another important experiment in a ultracold system, many impurities in a quan-tum bath have been realized [60]. By using a two-dimensional bath an array of one-dimensional system was created which is depicted in Fig. 2.6. At the center of onedimensional bath few impurities are trapped. The bath is made of Rubidium (Rb)atoms and impurities are made of Potassium (K) atoms. The root mean square dis-placement (σ(t) =

√X2) (where X is the position operator of an impurity) known

as axial width of the impurities oscillates as function of time which is fitted withσ(t) = σ1 + βt − A exp(−γωt) cos(

√1− γ2ωt). σ(t) for the impurities and the

bath is depicted in upper panel of Fig. 2.7 as a function of the Rb-K interaction η, by

11


Figure 2.7 – Axial width as a function of time for different impurity-bath interaction η:solid circles and triangles are axial width of the impurities and the bath. Axial width ofthe impurity is fitted with σ(t) = σ1 + βt − A exp(−γωt) cos(

√1− γ2ωt), and by

using the fit breathing frequency (ω), friction (γ) and slope (β) are depicted in the lowerpanel . Inset images are the density distribution of the impurities (blue) and the bath(red). [Taken from Ref [60]].

using the fit of σ(t), ω (breathing frequency), γ (frequency coefficient), and slope βhave been extracted and depicted in lower panel of Fig. 2.7 as a function of η.

Motivated by the experimental realization of an impurity in one-dimensionalbaths, in chapters 5 and 6, we investigate the dynamics of an impurity in a two-leg ladder bath. In chapter 5, the impurity is confined to move in one of the legs, andin chapter 6, the impurity moves in both legs.

12

CHAPTER 3

Methods

In this chapter, we present the methods used in this thesis. In particular, we willfocus on density matrix renormalization group (DMRG) method [62] in the languageof matrix product states (MPS) [63] and bosonization theory [3–6, 64].

We start with a general overview of DMRG at zero temperature, and then wemove to DMRG formulation in the MPS language, with DMRG viewed as a vari-ational method. DMRG gives the ground state and related properties (e.g., corre-lation functions, the expectation value of a local operator) in the ground state. Tocompute the time-dependent quantities, we move to a time-dependent extension ofthe DMRG; also known in the MPS language as time-evolving block decimation(TEBD) [63, 65].

Finally, we introduce an analytical low-energy description for the gapless regimeof our bosonic systems, the TLL theory [3–6, 64]. This theory has proven to behugely successful in describing the low-energy physics of one-dimensional systems.It will be used in this work to benchmark some of the numerical results and providea physical interpretation.

3.1 DMRG

DMRG [62] is a numerical method to solve the problems in a one-dimensional quan-tum system numerically. It was developed by Steve White. As we know, the Hilbertspace dimension of a quantum system grows exponentially with system size, so it ischallenging to compute the observables of a quantum many-body system for large

13

3. METHODS

system size. The idea behind DMRG is that the entanglement entropy in the groundstate of a local gapped Hamiltonian follows an area law. So, in one dimension, theentanglement entropy does not depend on the system size. Hence the ground stateof a one-dimensional system can be represented by an effective Hilbert space with asmall dimension.

3.2 DMRG in Conventional Form(Infinite DMRG)

The basic idea of infinite DMRG (iDMRG) is: we start from two small blocks,A andB of same size say l. Then we grow the system size by inserting two sites betweenblocks A and B, and system size grows from 2l to 2l + 2. Blocks A and B aredescribed by χ dimensional orthonormal spaces |al〉m and |bl〉m respectively,by symmetry |al〉m and |bl〉m are identical and m = 1, 2..χ.The Hamiltonian of system A • •B configuration is given by

H = HA ⊗ Id×d ⊗ ld×d ⊗ Iχ×χ + σeffl ⊗ σl+1 ⊗ Id×d ⊗ Iχ×χ+

Iχ×χ ⊗ σl+1 ⊗ σl+2 ⊗ Iχ×χ + Iχ×χ ⊗ Id×d ⊗ σl+2 ⊗ σeffl+3 +

Iχ×χ ⊗ Id×d ⊗ Id×d ⊗HB

(3.1)

Where HA and HB are Hamiltonians of blocks A and B, respectively, σeffl andσeffl+3 are operators defined in the block A and the block B, respectively, σ+1(σl+2)are defined in the local Hilbert space of the system and are d × d matrices, where dis the local Hilbert space, Iχ×χ is an identity matrix of size χ× χ.

Now we diagonalize (3.1) and compute the ground state energy and the groundstate wave function. The ground state of the configuration A • •B can be expressedas

|ψ〉 =∑

amσl+1σl+2bm

ψamσl+1σl+2bm |a〉m|σl+1〉|σl+2〉|b〉m (3.2)

Now let us express ψamσl+1σl+2bm in a matrix form ψ(amσl+1),(σl+2bm). The reduceddensity matrix of configuration A• is given by

ρA• = ψψ† (3.3)

ρA• is a matrix of dimension dχ × dχ, we compute the χ largest eigenvalues andcorresponding eigenvectors(|al+1〉m), these eigenvectors will be a new effectivebasis for the configuration A•. The effective Hamiltonian in the enlarged blockA• is given by O†HA•O, where O is a matrix formed by χ eigenvectors of thereduced density matrix ρA• , columns of the matrix O are eigenvectors of ρA• , andHA• = HA⊗ Id×d+σeffl ⊗σl+1. One can perform a similar transformation for theconfiguration •B to find the effective Hamiltonian in the block B. σl+1 in the new

14

3.2 DMRG in Conventional Form(Infinite DMRG)

effective basis is given by

〈al+1|σeffl+1 |a′l+1〉 =

∑alσl+1σ′l+1

〈al+1|σl+1al〉〈σl+1|σl+1|σ′l+1〉〈σ′l+1al|a′l+1〉 (3.4)

Now we know the effective basis for the left block and the right block of size l + 1,then we again add two sites between them and form the configuration A• • ••B andbuild the Hamiltonian according to (3.1), and continue the above procedures untilthe desired system size is achieved. A pictorial representation of iDMRG is depictedin Fig. 3.1.

Figure 3.1 – A pictorial representation of iDMRG. From top to bottom, we start fromsmall blocks A and B, add two sites between them, build an effective Hamiltonian,diagonalize the Hamiltonian, compute the ground state, then compute the reduce densitymatrix for A• and •B, keep the largest χ eigenvalues and corresponding eigenvectorsand they will act as an effective basis for A• and •B. We continue until a desired lengthof the system is achieved .

Once the desired system size is achieved, we move to a new version of DMRGcalled finite-DMRG.

The idea of finite-DMRG is the same as that of iDMRG, and the algorithm is asfollows.

1- By using iDMRG, we build the right and left blocks of size L− 1, so the totalsystem size is 2(L− 1).

2- Then, we do the sweeping procedure from left to right and right to left, asfollows:

a- In the first sweep, we start from the center and we add two sites (A(L − 1) ••B(L−1)) between them and compute the effective basis for the left block of size L,

15

3. METHODS

then we move to a new configuration (A(L)••B(L−2)), and compute the effectivebasis for A(L+ 1), and continue until the right block is left with one site. We shouldremember that we don’t need to compute the effective basis for the right block sinceit is already calculated in the previous steps.

b- We do the same procedure from right to left until the left block is left withone site and compute the effective basis for the right block, and the basis for the leftblock will be the same as that for the previous left to right sweep..

c- Do the right and the left sweep until the ground state energy converges. Sweep-ing from left to right and right to left in the finite-DMRG is depicted in Fig. 3.2.

Figure 3.2 – A pictorial representation of Finite-DMRG. From top to bottom, we startfrom blocks A and B, from left to right sweep: block A size is increased by one whileblock B size is reduced by one. We continue this procedure until block B has only onesite, then we do sweep from right to left until the left block has only one site. Continuesweepings until desired convergence in the ground state is achieved.

Now we will move to the recent formulation of DMRG in a new language calledmatrix product states (MPS) [63].

Before moving to the MPS formulation of DMRG, let us introduce singular valuedecomposition (SVD) [66]. SVD is a way to express any rectangular matrix as aproduct of three matrices. Let M is a n×m matrix then according to SVD

M = USV † (3.5)

Where U is a matrix of size n×min(m,n) with orthonormal columns , S is a diagonalmatrix of size min(n,m) × min(n,m) with decreasing order magnitude of diagonalentries and V † is of size min(m,n)×m with orthonormal rows.

3.3 Matrix Product States (MPS)MPS is a way to express any quantum state as a product of matrices. Let |ψ〉 be aground state of a system of size L, with a local Hilbert space of dimension d. State

16

3.3 Matrix Product States (MPS)

|ψ〉 can be expressed as

|ψ〉 =∑

σ1,σ2,..σL

cσ1σ2..σL |σ1〉|σ2〉..|σL〉 (3.6)

Let us consider cσ1σ2..σL as a tensor of rank L and regroup it as a matrix cσ1,σ2..σL ,where we consider σ1 as a row index and , σ2..σL as column index. Then accordingto SVD (3.5)

cσ1,(σ2..σL) =

r1∑a1=1

Uσ1,a1Sa1,a1

V †a1,σ2..σL

=

r1∑a1=1

Uσ1,a1ca1,σ2..σL

(3.7)

Where ca1,σ2σ3..σL = Sa1,a1V†a1,σ2σ3..σL

, now reshape ca1,σ2σ3..σL as ca1σ2,σ3..σL

and perform an SVD on ca1σ2,σ3..σL

ca1σ2,σ3..σL =

r2∑a2=1

Ua1σ2,a2Sa2,a2V†a2,σ3..σL

=

r2∑a2=1

Uσ1a1,a2ca2,σ3σ4..σL

(3.8)

Continue the SVD decompositions until the last site. we can then express cσ1,σ2..σL

as

cσ1,σ2..σL =∑

a1,a2..,aL−1

Uσ1,a1Ua1σ2,a2

Ua2σ3,a3...UaL−1,σL (3.9)

Define Uaiσi+1,ai+1 as Aσi+1ai,ai+1 , By using the property, U†U = I , matrices A’s

satisfy the following property ∑σi

Aσi†Aσi = I (3.10)

cσ1,σ2..σL in terms of A matrices can be expressed as

cσ1,σ2..σL = Aσ1Aσ2Aσ3 ....AσL

|ψ〉 =∑

σ1,σ2,..σL

Aσ1Aσ2Aσ3 ....AσL |σ1〉|σ2〉..|σL〉 (3.11)

Aσ1 and AσL are row and column matrices respectively. A pictorial representationof the decompositions of the wave function in terms of A’s matrices is depicted inFig. 3.3.

17

3. METHODS

Similarly, one can do decompositions from right to left. Let us write cσ1σ2..σL−1σL =c(σ1σ2..σL−1,σL), and perform an SVD on

c(σ1σ2..σL−1,σL) =∑aLU(σ1,σ2..σL−1,aL)SaL,aLV

†aL,σL ,

reshape U(σ1σ2..σL−1,aL)SaL,aL as c(σ1σ2..,σL−1aL), and continue SVD until lastsite, and define V † as B matrix

cσ1,σ2..σL = Bσ1Bσ2Bσ3 ....BσL

|ψ〉 =∑

σ1,σ2,..σL

Bσ1Bσ2Bσ3 ....BσL |σ1〉|σ2〉..|σL〉 (3.12)

B’s matrices satisfy∑σiBσi†Bσi = I . The dimension of matrices of A’s and B’s

are known as bond dimension.

Figure 3.3 – A pictorial representation of decomposition of a many body wave functionin terms of matrix product states. We start from tensor cσ1,σ2..σL and start decompo-sitions from the left site and move to site L, and cσ1,σ2..σL = Aσ1Aσ2Aσ3 ....AσL

.

A andB are known as left and right canonical matrices respectively. Dimensionsof A and B matrices increase from 1× d, d× d2 to dL/2−1 × dL/2 then decrease todL/2× dL/2−1, dL/2−1× dL/2−2, .., d2× 1. Dimensions of the matrices are knownas bond dimension.

As we have discussed in the previous section, two subsystems of a system canbe described by an effective Hilbert space of finite dimension, so we fix a maximummatrix dimension χ.

18

3.4 Ground State Search By Variational Method

3.3.1 Matrix Product Operator (MPO)

In the previous section, we have seen that the state |ψ〉 can be expressed as a productof matrices; similarly, any operator can be expressed in terms of a product of matri-ces, and they are known as MPO. Suppose O is an L body operator then it can beexpressed as

O =∑

σ1σ2..σLσ′1σ′2..σ′L

Oσ′1σ′2..σ′L

σ1σ2..σL |σ1σ2..σL〉〈σ′1σ′2..σ′L| (3.13)

Let us regroup Oσ′1σ′2..σ′L

σ1σ2..σL as

Oσ′1σ′2..σ′L

σ1σ2..σL = Oσ1σ′1σ2σ′2..σLσ′L

Oσ1σ′1σ2σ′2..σLσ′L

= O(σ1σ′1),(σ2σ′2..σLσ′L)

(3.14)

Now apply an SVD on O(σ1σ′1),(σ2σ′2..σLσ′L)

O(σ1σ′1),(σ2σ′2..σLσ′L) = Uσ1σ′1,β1

Sβ1,β1V †β1,σ2σ′2..σLσ

′L


′L

= O(β1σ2σ′2),(σ3σ′3..σLσ′L)

O(β1σ2σ′2),(σ3σ′3..σLσ′L) = Uβ1σ2σ′2,β2


′L

....................

Oσ1σ′1σ2σ′2..σLσ′L

= Uσ1σ′1,β1Uβ1σ2σ′2,β2

Uβ2σ3σ′3,β3...SβL,βLV

†βL,σLσ′L

(3.15)

Now define Uβi−1σiσ′i,βias Wσiσ

′i

βi−1,iand SβL,βLV

†βL,σLσ′L

as WσLσ′L

βL,1, and finally O

can be expressed as

O =∑

σ1σ2..σLσ′1σ′2..σ′L

Wσ1σ′1Wσ2σ

′2Wσ3σ

′3 ...WσLσ

′L |σ1σ2..σL〉〈σ′1σ′2..σ′L| (3.16)

Where Wσ1σ′1 is a row matrix and WσLσ

′L is a column matrix.


In this section, we will compute the ground state of a Hamiltonian H . The groundstate of a given Hamiltonian H can be expressed as

E[Mσ1 ,Mσ2 , ...MσL ] =〈ψ|H|ψ〉〈ψ|ψ〉 (3.17)

19

3. METHODS

Let us consider matrices Mσ1 ,Mσ2 , ...MσL as a variational parameters and min-imise (3.17) with respect to M’s matrices.

∂E[Mσ1 ,Mσ2 , ...MσL ]

∂Mσi= 0, i = 1, 2...., L

∂

∂M∗σjaj−1,aj

〈ψ|H|ψ〉 − E ∂

∂M∗σjaj−1,aj

〈ψ|ψ〉 = 0(3.18)

|ψ〉 =∑

σ1,σ2,..σL

Mσ1Mσ2Mσ3 ....MσL |σ1〉|σ2〉..|σL〉

=∑

σ1,σ2,..σL,ai,ai−1

(Mσ1Mσ2 ..Mσi−1)1,ai−1Mσiai−1,ai(M

σi+1Mσi+2 ..MσL)ai,1

|σ1〉|σ2〉..|σL〉(3.19)

〈ψ| =∑

σ1,σ2,..σL,ai,ai−1

〈σ1|〈σ2|..〈σL|(Mσ1Mσ2 ..Mσi−1)∗1,ai−1Mσi∗

ai−1,ai

(Mσi+1Mσi+2 ..MσL)∗ai,1

(3.20)

By using (3.19) and (3.20), 〈ψ|ψ〉 can be expressed as

〈ψ|ψ〉 =∑

σ1,σ2,..σL,ai,ai−1,a′i,a′i−1

(Mσ1Mσ2 ..Mσi−1)∗1,ai−1(Mσ1Mσ2 ..Mσi−1)1,a′i−1

Mσi∗ai−1,aiM

σia′i−1,a

′i(Mσi+1Mσi+2 ..MσL)∗ai,1(Mσi+1Mσi+2 ..MσL)a′i,1

=∑

σ1,σ2,..σL,ai,ai−1,a′i,a′i−1

(Mσi−1†...Mσ1†Mσ1 ..Mσ2 ..Mσi−1)ai−1,a′i−1

Mσi∗ai−1,aiM

σia′i−1,a

′i(Mσi+1Mσi+2 ..MσLMσL†..Mσi+1†)ai,a′i

(3.21)

Let us consider that the Mσj for j < i are left normalised and Mσj for j > i areright normalised then (Mσi−1†...Mσ1†Mσ1 ..Mσ2 ..Mσi−1)ai−1,a′i−1

= δai−1,a′i−1

and (Mσi+1Mσi+2 ..MσLMσL†..Mσi+1†)ai,a′i = δai,a′i then

〈ψ|ψ〉 =∑

σiai,ai−1

M∗σiai−1,aiMσiai−1,ai (3.22)

By using (3.22), the partial derivative of 〈ψ|ψ〉 with respect to (w.r.t.) M can beexpressed as

∂

∂M∗σjaj−1,aj

〈ψ|ψ〉 = Mσjaj−1,aj (3.23)

20


Using MPO representation of an operator O described in (3.15), similarly one canexpress a Hamiltonian H in terms of MPO, then the expectation value 〈ψ|H|ψ〉 canbe expressed as

〈ψ|H|ψ〉 =∑

a,a′,b,σ,σ′

(A∗σ11,a1

Wσ1σ′1

1,b1Aσ′11,a′1

A∗σ2a1,a2

Wσ2σ′2

b1,b2Aσ′2a′1,a

′2...

A∗σi−1ai−2,ai−1

Wσi−1σ

′i−1

bi−2,bi−1Aσ′i−1

a′i−2,a′i−1

)(M∗σiai−1,aiWσiσ′i

bi−1,biM

σ′ia′i−1,a

′i)

(B∗σi+1ai,ai+1

Wσi+1σ

′i+1

bi,bi+1Bσ′i+1

a′i,a′i+1Bσi+2ai+1,ai+2

Wσi+2σ

′i+2

bi+1,bi+2Aσ′i+2

a′i+1,a′i+2...

B∗σLaL−1,1W

σLσ′L

bL−1,1Bσ′La′L−1,1

)

=∑

bi−1,ai−1,a′i−1,σi,σ′i,bi,ai,a

′i

Lai−1,a

′i−1

bi−1(M∗σiai−1,aiW

σiσ′i

bi−1,biM

σ′ia′i−1,a

′i)R

ai,a′i

bi

(3.24)

Where a = a1, a2..aL, a′ = a′1, a′2..a′L, σ = σ1, σ2..σL, σ′ = σ′1, σ′2..σ′Land b = b1, b2..bL.

Laj−1,a

′j−1

bj−1=

∑bi,ai,a′i,σi,σ

′i,i<j

(A∗σ11,a1

Wσ1σ′1

1,b1Aσ′11,a′1

A∗σ2a1,a2

Wσ2σ′2

b1,b2Aσ′2a′1,a

′2...

A∗σi−1ai−2,ai−1

)

Raj ,a

′j

bj=

∑bi,ai,a′i,σi,σ

′i,i>j

(B∗σi+1ai,ai+1

Wσi+1σ

′i+1

bi,bi+1Bσ′i+1

a′i,a′i+1B∗σi+2ai+1,ai+2

Wσi+2σ

′i+2

bi+1,bi+2

Bσ′i+2

a′i+1,a′i+2...B∗σLaL−1,1

WσLσ

′L

bL−1,1Bσ′La′L−1,1

)

(3.25)

21

3. METHODS

In order to minimize E[Mσ1 ...MσL ], the derivative of 〈ψ|H|ψ〉 w.r.t. M can beexpressed as

∂

∂M∗σjaj−1,aj

〈ψ|H|ψ〉 =∑

bi−1,ai−1,a′i−1,σi,σ′i,bi,ai,a

′i

Lai−1,a

′i−1

bi−1(δσj ,σiδaj ,aiδaj−1,ai−1

Wσiσ′i

bi−1,biM

σ′ia′i−1,a

′i)R

ai,a′i

bi

=∑

bj−1,a′j−1,σ′j ,bj ,a

′j

Laj−1,a

′j−1

bj−1(W

σjσ′j

bj−1,bjM

σ′ja′j−1,a

′j)R

aj ,a′j

bj

=∑

a′j−1,σ′j ,a′j

Heff(σjaj−1aj),(σ′ja

′j−1a

′j)M

σ′ja′j−1,a

′j


′j−1a

′j)

=∑

bj−1,bj

Laj−1,a

′j−1

bj−1(W

σjσ′j

bj−1,bj)R

aj ,a′j

bj

(3.26)

By using (3.24) and (3.26), (3.18) can be expressed as

∑a′j−1,σ

′j ,a′j


′j−1a

′j)M

σ′ja′j−1,a

′j

= EMσjaj−1,aj

Heff (j)M(j) = EM(j)

(3.27)

Equation (3.27) shows that the optimized M(j) at site j is the lowest eigenstate ofthe Heff (j), and the ground state energy is the lowest eigenvalue of Heff (j).

The algorithms to compute the ground state energy and the ground state are givenin the subsections 3.4.1 and 3.4.2.

3.4.1 Right Sweeping1- First fix the largest possible bond dimension χ, for example let us say χ = 50 andlocal Hilbert space dimension is d = 2, then the matrix dimensions on each site willbe distributed as (1, 2), (2, 4), (4, 8), (8, 16), (16, 32), (32, 50), (50, 50),(50, 50)....(50, 50)...

, (50, 50), (50, 32), (32, 16), (16, 8), (8, 4), (4, 2), (2, 1), let us say on a given sitel, the matrix dimension is represented as (D0(l), D1(l)).

2- Start from random right canonical matrices (Bσl , l = 1, 2..., L).3- Compute R expressions from site L to 2.4- Compute Heff (1) at site 1, find the lowest eigenvalue and corresponding

eigenvector M(1), reshape M(1) vector as a matrix M(k1 ∗ D0(l) + X1, X2) =M(k1 ∗D0(l) ∗D1(l) +X1 ∗D1(l) +X2, 0), where X1 = 0, 1, ...D0(l)− 1, X2 =0, 1, ...D1(l)− 1. Perform an SVD on M(1) and store theAσ1 matrix.

22

3.5 Calculation of expectation values

5- Then move to the second site, compute L expression at site 1 using the newmatrix Aσ1 from step 4 and compute Heff (2) at site 2. By using Heff (2), computethe Aσ2 matrix, as discussed in the fourth step.

6- Continue the procedures in the fourth and fifth steps until site L − 1, andcompute L expressions, Heff (i), i = 3, 4, ..L− 1, and Aσi , i = 3, 4, ..L− 1.

3.4.2 Left Sweeping1- After the completion of the right sweeping, use L expressions from the rightsweeping procedure in order to compute the Heff (L) at site L. Compute the lowesteigenstate M(L) of Heff (L), perform an SVD on M(L), and compute the BσLmatrix.

2- By using new BσL compute R expression at site L, Heff (L − 1), computethe lowest eigenvalue and corresponding M(L− 1), perform an SVD on M(L− 1)to compute BσL−1 matrix.

3- Follow the steps 1 and 2, compute R expressions, Heff , and B(i), i = L −2, L− 2, ..2.

4- Perform the left and right sweepings until convergence in the ground state isachieved.

3.5 Calculation of expectation values

A local operator Oi acting at site i can be expressed in the local basis |σi〉 as

Oi =∑σi,σ′i

Oσi,σ′i |σi〉〈σ′i| (3.28)

〈ψ|Oi|ψ〉 in the state (3.19) can be expressed as

〈ψ|Oi|ψ〉 =∑

[Mσ1†Mσ1 ..Mσi−1†Mσi−1 ]ai−1,a′i−1Mσiai−1,ai

∗Mσ′ia′i−1,a

′iOσi,σ

′i

[Mσi+1Mσi+1†..MσLMσL†]ai,a′i(3.29)

If M ’s matrices are left canonical for j < i and right canonical for j > i, then (3.29)simplifies as

〈ψ|Oi|ψ〉 =∑

Mσiai−1,ai

∗Mσ′iai−1,aiO

σi,σ′i (3.30)

Similarly one can show that the two point correlation function 〈ψ|OiOj |ψ〉 can bewritten as

〈ψ|OiOj |ψ〉 =∑

Mσiai−1,ai

∗Mσ′iai−1,aiO

σi,σ′iMσi+1

ai,ai+1

∗Mai,ai+1...Mσj

aj−1,aj∗M

σ′jaj−1,ajO

σj ,σ′j

(3.31)

23

3. METHODS

In (3.31), we have assumed that i < j and all the matrices before site i are leftcanonical and after site j are right canonical. In (3.29,3.30,3.31) summations aretaken over the dummy indices.

3.6 Time Evolving Block DecimationTime Evolving Block Decimation (TEBD) is a numerical method to compute thetime evolution of a quantum state. It was developed by Guifre Vidal [65].

Let us say that |ψ(0)〉 is a quantum state at time t = 0 and we want to find|ψ(t)〉 at time t. In the Schrödinger picture |ψ(t)〉 = exp(-i H t)|ψ〉. The idea ofTEBD is to express a Hamiltonian H in terms of odd and even bonds representation,and express exp(−iHt) in terms of product of small matrices h by so-called Trotterdecomposition [67, 68] in order to compute the MPS at time t. Let us say that theHamiltonian of a one-dimensional quantum system is represented by H . Then wecan always express

H =∑l

hl = Hodd + Heven (3.32)

where Hodd = h1 + h3 + h5... and Heven = h2 + h4 + h6..., because of nearestneighbour coupling [hl, hl+2] = 0. But in general [hl, hl+1] 6= 0. exp(-i H t) =(exp(-i Hδt))N where δt = t/N . exp(-i Hδt)) can be expressed by so-called firstorder Trotter decomposition [67]

exp(-i Hδt)) = exp(−iHoddδt) exp(−i Hevenδt) exp(o(δt2)) (3.33)

And the fourth order Trotter decomposition as

exp(-i Hδt)) = exp(−iHoddδtθ/2) exp(−i Hevenδtθ) exp(−iHoddδt(1− θ)/2)

exp(−i Hevenδt(1− 2θ)) exp(−iHoddδt(1− θ)/2) exp(−i Hevenδtθ)

exp(−iHoddδtθ/2) exp(o(δt5))

(3.34)

where θ = 12−21/3 . In chapters 5 and 6, we have used the fourth order Trotter

decomposition [67].

exp(−iHevenδt) = exp(−ih2δt) exp(−ih4δt)).. exp(−ihL−2δt)

exp(−iHoddδt) = exp(−ih1δt)) exp(−ih3δt)).. exp(−ihL−1δt))(3.35)

For the Hamiltonian (2.3), hl can expressed as

h1 = −J(b†1b2 + h.c) +U

2n1(n1 − 1) +

U

4n2(n2 − 1)

hL−1 = −J(b†L−1bL + h.c) +U

2nL(nL − 1) +

U

4nL−1(nL−1 − 1)

hl = −J(b†l bl+1 + h.c) +U

4nl(nl − 1) +

U

4nl+1(nl+1 − 1)

(3.36)

24

3.7 Imaginary Time Evolution as a Ground State Search

The action of exp(−ihlδt)) on state |ψ〉 changes the matrices at site l and l+ 1 [65].Let us say that MPS consists of right-normalized matrices then according to Vidal’swork the updated right-normalized matrices at sites l and l + 1 are given by thefollowing procedures .

1- Form a tensor called theta tesnor as

Θσl,σl+1al−1,al+1

=∑

σ′l,σ′l+1,al

〈σl, σl+1| exp(−ihlδt))|σ′l, σ′l+1〉Sal−1,al−1[l−1]Bσlal−1,al

Bσl+1al,al+1

.

(3.37)2- Reshape Θ

σl,σl+1al−1,al+1 into a matrix, perform an SVD on

Θσlal−1,σl+1al+1=∑al

Aσlal−1,alSal,al [l]B

σl+1al,al+1 (3.38)

In (3.38), summation over al runs from al = 1 to al = dχ, but we truncate thesummation from al = 1 to al = χ, and we take the χ largest diagonal entries of Smatrix.

3- Convert A[l] matrix into B[l] matrix via relation [63] Bσl [l] = S−1[l −1]Aσl [l]S[l].

If the MPS consists of left-normalized matrices then according to Vidal’s workthe updated left-normalized matrices at sites l and l + 1 are given by the followingprocedures .

1- Form a theta tensor as

Θσl,σl+1al−1,al+1

=∑

σ′l,σ′l+1,al

〈σl, σl+1| exp(−ihlδt))|σ′l, σ′l+1〉Aσlal−1,alAσl+1al,al+1

Sal+1,al+1[l+1].

(3.39)2- Reshape Θ

σl,σl+1al−1,al+1 into a matrix, perform an SVD on

Θσlal−1,σl+1al+1=∑al

Aσlal−1,alSal,al [l]B

σl+1al,al+1 (3.40)

3- Convert B[l + 1] matrix into A[l + 1] matrix via relation [63] Aσl+1 [l + 1] =S−1[l]Bσl+1 [l + 1]S[l + 1].TEBD for real time evolution is a good method for a small amount of (simulation)time, but for the larger times, since the entanglement entropy grows linearly in time[63], one needs exponentially large bond dimension χ to capture the physics for along time, which is computationally expensive to do.

3.7 Imaginary Time Evolution as a Ground State SearchThe ground state of a Hamiltonian H can be computed using imaginary time evo-lution. Let |φ〉 be a some random state then the ground state |GS〉 of H is given

25

3. METHODS

by

|GS〉 = limτ→∞exp(−Hτ)|φ〉|| exp(−Hτ)|φ〉||

(3.41)

Equation (3.41) can be understood by expanding |φ〉 in terms of eigenstates of H ,|φ〉 = c0|GS〉+

∑i ci|Ei〉, where |Ei〉 are the excited states of the Hamiltonian.

|GS〉 = limτ→∞exp(−E0τ)[c0|GS〉+

∑i exp(−τ(Ei − E0))ci|Ei〉]

exp(−E0τ)[|c0|2 +∑i exp(−τ(Ei − E0))|ci|2]1/2

=c0

(c0c∗0)1/2|GS〉

(3.42)

Computation of the ground state using imaginary time-evolution is slow because onehas to do time-evolution for a very long time to converge to the ground state, and forevery time step one has to perform SVD, L − 1 times. On contrary in variationaloptimization, few sweeps are needed to converge to ground the state. Imaginarytime-evolution has been used in higher dimensions to compute the ground state andwe refer the reader to the relevant literature [69–71] for more details.

3.8 BosonizationBosonization [3–6, 64] is a theoretical method to express the quantum operatorsin a one-dimensional (1D) system in terms of collective excitations. For fermionsthese are particle-hole excitations created by the operators (ρ(q)† =

∑k c†k+qck,

ρ(−q)† =∑k c†kck−q). Where, ρ(q)† destroys a particle (creates a hole ) with mo-

mentum k below the Fermi energy , and creates a particle with momentum k + qabove the Fermi energy. The excitations created by the operator ρ(q)† carry a mo-mentum q, with an energy Ek(q) = ε(k + q) − ε(k), where ε(k) = −2J cos(k) isthe dispersion relation of a one dimensional non-interacting system, J is a tunnelingamplitude for a tight-binding model as discussed in (2.3). In general Ek(q) dependson both k and q, hence the excitations created by the ρ(q)† are not well defined (theyare not eigenstates of the Hamiltonian). However, in 1D, for small q [5]

Ek(q) = q (3.43)

As it is evident that the low-energy physics happens around the Fermi wavevectorkF , we linearize the dispersion relation around the momenta k = kF and k = −kF .

We also introduce two sets of fermionic creation(annihilation) operators onearound k = kF (c†Rk(cRk)) and the other around k = −kF (c†Lk(cLk)). Here Rand L stand for left and right movers. After linearization of the dispersion aroundthe Fermi wave-vectors the kinetic energy term of the system can be expressed as

H0 =∑k

[vF (k − kF )c†RkcRk − vF (k + kF )c†LkcLk] (3.44)

26

3.8 Bosonization

It is clear that (3.44) satisfies (3.43) for any value of q, k ∈ [−∞,∞]. The model(3.44) is known as Tomonaga-Luttinger liquid model. Because of k ∈ [−∞,∞],there is an infinite number of occupied states below the Fermi energy. There arethus an infinite number of states for right moving and left-moving fermions. Soin order to make the operators physical, we define the operators in normal orderform. The normal order of an operator AR for right-moving fermions is denoted by: AR :, and : AR := AR − 〈GSR|AR|GSR〉, where |GSR〉 =

∏k=kFk=−∞ c†Rk|0〉, and

the normal order for an operator AL for left moving fermion is given by : AL :=

AL− 〈GSL|AL|GSL〉, and |GSL〉 =∏k=∞k=−kF c

†Lk|0〉, where |0〉 is the vacuum and

cR/Lk|0〉 = 0,∀k, and for a finite system of size L, k = 2πnL − δb, δb = 0 for

periodic boundary condition and δb = πL for anti-periodic boundary condition, and

n ∈ [−∞,∞]. The ground state of the system is |GS〉 = |GSL〉 ⊗ |GSR〉.Let us define the commutation relations

[ρR(q1)†, ρL(q2)†] = 0 (3.45)

[ρR(q1)†, ρR(−q2)†] =∑k1,k2

[c†Rk1+q1cRk1

, c†Rk2−q2cRk2]

=∑k2

: c†Rk2−q2+q1cRk2

: +〈GSR|c†Rk2−q2+q1cRk2|GSR〉−

: c†Rk2−q2cRk2−q2 : −〈GSR|c†Rk2−q2cRk2−q1 |GSR〉

[ρR(q1)†, ρR(−q1)†] =∑k2

〈GSR|c†Rk2cRk2 |GSR〉 − 〈GSR|c

†Rk2−q1cRk2−q1 |GSR〉

=L

2π

[ ∫ ∞−∞

dk2Θ(kF − k2)−∫ ∞−∞

dk2Θ(kF − k2 + q1)]

= −q1L/2π

[ρL(q1)†, ρL(−q1)†] =∑k2

〈GSL|c†Lk2cLk2|GSL〉 − 〈GSL|c†Lk2−q1cLk2−q1 |GSL〉

=L

2π

[ ∫ ∞−∞

dk2Θ(k2 + kF )−∫ ∞−∞

dk2Θ(k2 − q1 + kF )]

= q1L/2π

(3.46)

Let us define the bosonic operators bq and b†q as

bq =

√2π

L|q|(Y (q)ρR(−q)† + Y (−q)ρL(−q)†)

b†q =

√2π

L|q|(Y (q)ρR(q)† + Y (−q)ρL(q)†)

(3.47)

27

3. METHODS

Where Y (q) is the step function.Now, we want to compute the commutation [bq, H0] relation in order to express

(3.44) in terms of the bosonic operators bq and b†q .

[bq, H0] =

√2π

L|q|∑k1,k2

[vF (k2 − kF )Θ(q)[c†R,k1−qcRk1, c†Rk2

cRk2]−

vF (k2 + kF )Θ(−q)[c†L,k1−qcLk1, c†Lk2

cLk2]]

=

√2π

L|q|[Θ(q)vF qρR(−q)† −Θ(−q)vF qρL(−q)†]

= vF |q|bq

(3.48)

The Hamiltonian H0 can be expressed in terms of bosonic operators as

H0 =∑q 6=0

vF |q|b†qbq (3.49)

One can check that bq|GS〉 = 0 [5] and the Hilbert space generated by thebosonic operator bq(b†q) is given by successive applications of b†q on |GS〉.

The single particle annihilation operators for right (ψR(x)) and left (ψL(x))movingfermions can be expressed as [5].

ψR(x) =UR√Lei(kF+

2πNRL −δb)e

−∑q e−iqx

√2πL|q|Θ(q)b†qe

∑q eiqx

√2πL|q|Θ(q)bq (3.50)

ψL(x) =UL√Lei(−kF−

2πNRL −δb)e

−∑q e−iqx

√2πL|q|Θ(−q)b†qe

∑q eiqx

√2πL|q|Θ(−q)bq

(3.51)UR and UL are known as Klein factors [5] for right and left movers, respectively. URand UL destroy a fermion at the highest occupied momentum state for right and leftmovers, respectively. NR(NL) are the number of right movers (left movers) addedon the top of |GS〉.

Now let us introduce fields θ and φ which are function of bq(b†q) as

φ(x) = −(NR +NL)πx

L− iπ

L

∑q 6=0

√L|q|2π

1

qe−α|q|/2−iqx(b†q + b−q)

θ(x) = (NR −NL)πx

L+iπ

L

∑q 6=0

√L|q|2π

1

|q|e−α|q|/2−iqx(b†q − b−q)

(3.52)

Where α is of order of 1/Λ and Λ is a momentum cut-off. For L → ∞, one canshow that

∇φ(x) =− π(ρR(x) + ρL(x))

∇θ(x) =π(ρR(x)− ρL(x))(3.53)

28

3.8 Bosonization

ψR(x) and ψL(x) in terms of θ(x) and φ(x) are given by

ψR(x) =UR√2πα

ei(kF−π/L)e−i(φ(x)−θ(x))

ψL(x) =UL√2πα

e−i(kF−π/L)e−i(−φ(x)−θ(x))

(3.54)

The fields θ(x) and φ(x) satisfy following commutation relations

[φ(x1), θ(x2)] =iπ

2sgn(x2 − x1)

[φ(x1),∇θ(x2)] = iπδ(x2 − x1)(3.55)

In terms of θ(x) and φ(x), the Hamiltonian (3.44) can be expressed as

H0 =1

2π

∫dxvF [(∇θ(x))2 + (∇φ(x))2] (3.56)

Now let us add an interaction of the form

HV =1

2

∫dxdx′V (x− x′)ρ(x)ρ(x′)

=1

2L

∑k,k′,q

V (q)c†k+qc†k′−qck′ck

(3.57)

After adding the interaction term (3.57) to (3.56), the low energy effective Hamil-tonian can be expressed as [5]

H =1

2π

∫dx[uK(∇θ(x))2 +

u

K(∇φ(x))2] (3.58)

The Hamiltonian (3.58) is known as Tomonaga-Luttinger liquid Hamiltonian. u andK are known as Luttinger parameters, which encode the effects of interactions. Theparameter K controls the decay of correlation functions in (3.58).

The correlation functions of the fields θ(x) and φ(x) in the Hamiltonian (3.58)decay as a power-law at zero temperature as

〈eiaφ(x1,τ1)e−iaφ(x2,τ2)〉 =( 1

|r1 − r2|

)a2K/2

〈eiaθ(x1,τ1)e−iaθ(x2,τ2)〉 =( 1

|r1 − r2|

)a2/2K(3.59)

Where |r1 − r2| =√

(u|τ1 − τ2|)2 + (x1 − x2)2/α. x1(x2) denotes the position,τ1(τ2) denote imaginary time, and α ∼ 1/Λ is a short distance cutoff.

29

3. METHODS

3.9 Phenomenological BosonizationIn the above section, we have seen that near the Fermi points, the excitations are welldefined and they are bosonic in nature. We have expressed the fermionic operatorsin terms of bosonic operators bq(b†q), and they capture the physics near Fermi points.In this section, we will show a general mapping between fermionic and bosonicoperators in terms of φ(x)(θ(x)). This phenomenological bosonization method wasdeveloped by Haldane [3]. The idea of the phenomenological bosonization is basedon the fact that in one dimension, the nth particle at position at xn can be labeled bya field φl(xn) and φl(xn) = 2πn. By using the properties of the delta function wecan write δ(φl(x)− 2πn) as

δ(φl(x)− 2πn) =1

|∇φl(xn)|δ(x− xn)

=1

|∇φl(x)|δ(x− xn)

(3.60)

The density ρ(x) of at position x is given by

ρ(x) =∑xn

δ(x− xn) (3.61)

Now by using (3.60), ρ(x) can be expressed in term of φ(x) as

ρ(x) =∑n

δ(φl(x)− 2πn)|∇φl(x)|

= |∇φl(x)|∑n

δ(φl(x)− 2πn)(3.62)

Now we use the Poisson summation formula∑n δ(φl(x)−2πn) = 1

2π

∑p e

ip(φl(x)−2πn) =∑p e

ip(φl(x)), where p ∈ [−∞,∞], and p takes integer values. For a perfect crys-talline lattice φl(x) = 2πρ0x, where ρ0 is the mean density. Now we define a fieldφ relative to the perfect crystalline lattice as

φl(x) = 2πρ0x− 2φ(x) (3.63)

The final expression of ρ(x) is thus

ρ(x) = [ρ0 −1

π∇φ(x)]

∑p

ei2p(πρ0x−φ(x))(3.64)

One can express the single particle creation operators for boson (ψ†B) and fermion(ψ†F ) as [5]

30

3.9 Phenomenological Bosonization

ψB(x)† = [ρ0 −1

π∇φ(x)]1/2

∑p

ei2p(πρ0x−φ(x))e−iθ(x)

ψF (x)† = [ρ0 −1

π∇φ(x)]1/2

∑p

ei(2p+1)(πρ0x−φ(x))e−iθ(x)(3.65)

In the previous section, we have derived the bosonization identities, and they cap-tures the physics of a one-dimensional system around q = 0 and q = 2kF , butthe phenomenological bosonization is an exact representation, the operators in phe-nomenological bosonization capture physics around higher momenta such as q =0, 2kF , 4kF , 6kF ....

Bosons in the continuum with a contact interaction are described by the Hamil-tonian

H =1

2m

∫dx∇ψ†(x)∇ψ(x) + U

∫dxρ(x)2 (3.66)

Using (3.64) and (3.65), one can express (3.66) as

H =

∫dx[

ρ0

2m(∇θ(x))2 +

U

π2(∇φ(x))2] + ... (3.67)

The higher order terms could renormalized the coefficients of (∇θ(x))2 and (∇φ(x))2,and for irrelevant higher order terms (in the renormalization sense), (3.67) can be ex-pressed as

H =1

2π

∫dx[uK(∇θ(x))2 +

u

K(∇φ(x))2] (3.68)

TLL parameters u and K, depend in general on the properties of the system and inparticular on the interaction between the particles.

For fermions, K = 1 for noninteracting particles while K > 1 (resp. K < 1)for attraction (resp. repulsion) between the particles. For bosons K = ∞ for non-interacting particles, and becomes smaller as the repulsion increases. For a contactinteraction only K > 1, with K = 1 being reached for infinite repulsion betweenthe bosons (the so-called Tonks-Girardeau limit). Longer range interactions allow inprinciple to reach any value of K. For special models such as the Lieb-Liningermodel or the Bose-Hubbard one for bosons or the Hubbard model for fermionsprecise relations between the microscopic parameters and the TLL parameters areknown [5, 6]. In general if the microscopic parameters are known the TLL parame-ters can be computed with an arbitrary degree of accuracy. Equation (3.68) will beused in chapters 4,5,6 in order to describe the low-energy effective Hamiltonian fora one-dimensional bath.

31

3. METHODS

32

CHAPTER 4

Time dependent local potential in aTomonaga-Luttinger liquid

In chapters 1 and 2, we have introduced the global shaking of ultracold atoms usinga time-dependent potential and discussed how the deposited energy by the time-dependent potential gives information about the excitations. In this chapter, we con-sider a local time-dependent potential in a one-dimensional bath, and we computethe deposited energy.

This chapter is structured as follows: in Sec. 4.1, we introduce the model we use,within the framework of TLL to study the energy deposition by using linear responsetheory (LRT). Depending on the interaction, two cases in which the impurity poten-tial would be relevant (resp. irrelevant) in the static case are distinguished. Sec. 4.2.2discusses the deposited energy for the case in which the static impurity potential isirrelevant. This typically corresponds to K > 1 for the TLL. The opposite case ofK < 1 is examined in Sec. 4.2.4. The results and their potential application to ex-perimental systems in cold atomic gases are discussed in Sec. 4.3. Some technicaldetails are described in the appendix A.

4.1 Model and Method

4.1.1 Model

We consider a one dimensional system, schematically shown in Fig. 4.1. The sys-tem is made of interacting bosons or fermions described by a Hamiltonian H0 which

33

4. TIME DEPENDENT LOCAL POTENTIAL IN ATOMONAGA-LUTTINGER LIQUID

x=0

Figure 4.1 – Sketch of a one dimensional system with a time dependent external localpotential V (x = 0, t) = V0 + δV cos(ωt). The rectangular box represents a staticpotential V0. The double head arrow represents the time dependent part of the potentialδV cos(ωt).

depends on the precise system under consideration (e.g. continuum or system on alattice) and will be made more precise below. Most of this chapter is concerned witha single component system, of bosons or fermions, but we also discuss briefly thecase of a two component fermionic system. This is particularly relevant in connec-tion with recent realizations of fermionic microscope [72, 73] that allow the label ofcontrol relevant to carry the time of modulation and measurements corresponding tothe present study. We consider separately the case of weak and strong potentials fora single component case and then address the two component weak potential case.

We add to this system a time dependent potential V (x, t). The total Hamiltonianwe consider is thus:

H = H0 +Himp

Himp =

∫dxV (x, t)ρ(x)

(4.1)

where ρ(x) represents the density of particles at position x. We compute here thechange in energy caused by this local modulation. This corresponds to the localversion of the shaking technique, in which a full optical lattice is modulated [22–24].

The modulation of the amplitude of the local potential is given by (see Fig. 4.1)

V (x, t) = [V0 + δV cos(ωt)]fλ(x) (4.2)

where fλ(x) is a short range function which is a regularized δ function. For simplic-ity we chose

fλ(x) =1

2λ cosh2(x/λ)(4.3)

the δ function being reproduced in the limit λ→ 0.Thus for (4.2) the impurity potential can be split in a purely static part corre-

sponding to the one of a single impurity and a purely dynamical part, of mean zero.We can thus quite generally write

H = HS + cos(ωt)O (4.4)

where HS is the purely static impurity potential, and O the operator correspondingto the modulation given by (4.2).

34


If the impurity potential V0 is weak compared to the kinetic and interaction en-ergy of the system, then one has

HS = H0 + V0

∫dxfλ(x)ρ(x) (4.5)

We consider that fλ(x) varies fast enough compared to the typical scales of theproblem and thus can be assimilated to a δ(x) function as far as the density variationof the system are concerned.

Since the perturbation is small we use linear response theory. The energy depo-sition rate is given by(see appendix A.1)

ER = −1

2ω ImχR(ω) (4.6)

where

χR(ω) = −i∫ ∞

0

dtei(ω+iδ)t〈[O(t), O(0)]〉 (4.7)

is the retarded correlation function of the operator O linked to the perturbation (seeAppendix A.1). 〈· · · 〉 denotes the quantum and/or thermal average with the Hamil-tonian HS in (4.4).

4.1.2 Weak potential, bosonizationIn order to treat the Hamiltonian (4.4) we use the bosonization technique discussedin chapter 3. The low-energy Hamiltonian of a one-dimensional bath is described bya TLL (3.68) [5]:

H0 =1

2π

∫dx[uK(∂xθ(x))2 +

u

K(∂xφ(x))2

](4.8)

The Hamiltonian HS thus describes a TLL with a single impurity, for whichphysics is well known [10]. We briefly recall the main elements of this Hamiltoniansince it will be needed to compute the effects of the dynamical perturbation.

For a perturbation weak compared to the kinetic and interaction energy, using(3.64) the impurity part becomes

H0imp = V0

∫dxfλ(x)ρ(x)

= V0

∫dxfλ(x)

[− 1

π∇φ(x) + 2ρ0 cos(2φ(x))

] (4.9)

where we have kept only the lowest (p = ±1) and the most relevant harmonics indensity (3.64).

35


The first term can be absorbed by a redefinition of the field φ

φ(x) = φ(x)− KV0

u

∫ x

0

dyfλ(y) (4.10)

The θ field is not affected by this transformation which preserves the canonical com-mutation relations (3.55). In the limit where λ→ 0 the cosine term is not affected bythis transformation since it contains only the value φ(x = 0). The static Hamiltonianis thus

HS = H0[φ] + 2V0ρ0 cos(2φ(0)) (4.11)

The ground state depends crucially on the value of K [10, 11]. A renormalizationgroup procedure based on changing the cutoff in the problem (e.g. a high energycutoff Λ) gives the flow of the parameters

dK

dl= 0

dV0

dl= (1−K)V0

(4.12)

where l is the scale of the renormalization for an effective high energy cutoff Λl =Λe−l, Λ is the maximum high energy cutoff, of the order of the bandwidth. Thus forK > 1, V0 flows to 0 and the cosine term is irrelevant. On the other hand for K < 1the cosine is relevant and V0 flows to the strong coupling limit. A schematic view ofthe RG flow is shown in Fig. 4.2. The redefinition of the field φ (4.10) can also bedone in the dynamical terms. Up to terms that are simply oscillating and thus do notcontribute to the increase of the average energy (4.6) one obtains in the limit λ→ 0

O = δV

[− 1

π∇φ(x = 0) + 2ρ0 cos(2φ(x = 0))

](4.13)

4.1.3 Two component fermionic systemIn this section, we consider a two component fermionic system, given by the Hub-bard model. We use similar techniques as for the single component case. We restrictourselves for simplicity to the weak coupling case.

The static part of Hamiltonian in bosonized language is given by [5]

HS = H0[φρ] + H0[φσ] + 2V0ρ0 cos(2φ↑(0)) + 2V0ρ0 cos(2φ↓(0)) (4.14)

HS = H0[φρ] +H0[φσ] + 4V0ρ0 cos(√

2φρ(0)) cos(√

2φσ(0)) (4.15)

Where φ↑ and φ↓ are fields for spin up and spin down fermions, φρ =φ↑+φ↓√

2,

φσ =φ↑−φ↓√

2and

H0[φρ] =1

2π

∫dx[uρKρ(∂xθρ(x))2 +

uρKρ

(∂xφρ(x))2]

(4.16)

36


Figure 4.2 – Various regimes for the energy absorption as a function of the TLL param-eterK and the strength of the backscattering potential V0. The solid line arrows indicatethe renormalization flow of the static backscattering potential V0 [10, 11]. For K < 1,V0 flows to the strong coupling limit, and for K > 1, V0 flows to zero. The green linerepresents the crossover scale ω∗ at which the renormalized potential is of order one.αb and αf represent the energy deposition exponents corresponding to the backwardand forward scattering respectively (see text). The solid blue circle represents the initialpoint for the local potential. For K < 1, even if the initial point corresponds to a smalllocal potential, it renormalizes to strong coupling when lowering the frequency ω but forK > 1, even if the initial point is a large local potential it renormalizes to weak potentialvia the lowering of ω.

.

H0[φσ] =1

2π

∫dx[uσKσ(∂xθσ(x))2 +

uσKσ

(∂xφσ(x))2]

(4.17)

uρ, uσ,Kρ,Kσ are the Luttinger parameters. For a repulsive interactionKρ < 1 andfor a spin rotation symmetric Hamiltonian Kσ renormalizes to K∗σ = 1.

The renormalization flow of V0 is given by [10, 11]

dV0

dl= [1− (Kσ +Kρ)/2]V0 (4.18)

V0 is irrelevant forKσ+Kρ > 2, and relevant forKσ+Kρ < 2. The time dependentpart of the Hamiltonian for a weak potential is given by

O = δV

[−√

2

π∇φρ(x = 0) +4V0ρ0 cos(

√2φρ(0)) cos(

√2φσ(0))

](4.19)

4.1.4 Strong couplingThe description of the previous two subsections is well adapted if the potentialV (x, t) (and the modulation itself) are weak compared to other parameters in the

37


problem, such as e.g. the kinetic energy. If the impurity potential V is the largestscale in the problem, then it is more useful to write an effective description usingan expansion in powers of 1/V . We examine such a case for a single componentfermionic system.

In order to do so, we put the system on a lattice and assume that the potential onlyexists on the site x = 0. We also assume that the interactions are onsite and nearestneighbour. We divide the full Hamiltonian in three parts. The first (resp. second)part contains all sites left (resp. right) of the origin. The third part is the impuritysite. The tunnelling and nearest neighbour interaction between sites (kinetic energy)connects the three parts:

H = H1 +H2 − J [ψ†−1ψ0 + ψ†1ψ0 + h.c] + V ψ†0ψ0 + Vnn0(n−1 + n1) (4.20)

For a large V we derive the effective Hamiltonian in powers of 1/V (see Appendix A.2).The effective Hamiltonian is

H = H1 +H2 −J2

V[(ψ†1ψ−1+

h.c.) + ψ†1ψ1 + ψ†−1ψ−1]

(4.21)

We thus see that the two half parts are coupled by an effective tunneling term

Jeff =J2

V(4.22)

and that there is an attractive potential of the same order acting on the last site ofeach half part.

For the dynamical case we simply assume that we can replace the static V by thedynamical one V = V0+δV cos(ωt), since V0 0, 1/V = 1/V0−δV/V 2

0 cos(ωt).The static part of the Hamiltonian is given by

HS = H1 +H2 − J0[(ψ†−1(x = −a)ψ1 + h.c.)+

ψ†1ψ1 + ψ†−1ψ−1](4.23)

Where 1 and 2 stand for right and left part of one dimensional chain, J0 = J2/V0

and a is lattice constant.In this case the amplitude modulation of the potential produces a small modula-

tion of the tunnelling amplitude and local potential. The operator O is

O = δt1(ψ†1(x = −a)ψ2(x = a) + h.c.)+

δt2(ψ†2(x = a)ψ2(x = a) + ψ†1(x = −a)ψ1(x = −a))(4.24)

where δt2 = − δV J2

V 20

, and δt1 = − δV J2

V 20

. Since J2/V0 is small compared to thekinetic energy we can bosonize (4.21). The expressions are derived in Appendix A.2,

38

4.2 Calculation of the energy absorption

and the final static Hamiltonian is given by

Hs =u

2π

∫ ∞−∞

dx[(∂x

˜θ(x))2 + (∂x

˜φ(x))2

]+

2a√KJ2

πV0∂2x

˜θ(0)− 2J2ρ0

V0cos(√K

2(˜θ(a)− ˜

θ(−a)))

cos(√K

2(˜φ(a)− ˜

φ(−a)))− 2J2ρ0

V0cos( 2√

K(˜φ(0))

) (4.25)

4.2 Calculation of the energy absorptionWe now compute the energy absorption for the amplitude modulation. Let us con-sider first the case for which the bare potential is weak as described in Sec. 4.1.2.

4.2.1 Weak bare potentialIf one starts with a weak initial potential both the Hamiltonian and the perturbationare described by (4.8) and (4.13) for the single component case and by (4.14) and(4.19) for the two component one. In the static part of the Hamiltonian the backscat-tering part of the potential renormalizes as described by (4.12) and (4.18). Dependingon the value of K (or Kρ for the two component case) the potential can either renor-malize to zero or flow to strong coupling, leading to a crossover to another regime.If K > 1 (resp. Kρ > 1 for the two component) the backscattering term renormal-izes to zero, and the energy absorption can thus essentially be computed with thequadratic part of the Hamiltonian (4.8), (4.16), (4.17). On the contrary if K < 1(resp. Kρ < 1) the backscattering term in (4.11) and (4.14) renormalizes to largevalues. In that case there is a crossover scale below which one cannot ignore thebackscattering term in the static Hamiltonian.

We examine sequentially these two cases.

4.2.2 Energy deposition in weak coupling limitTo compute the energy absorption rate for the single component case (4.13) we use(4.6) for a Hamiltonian H = H0 + cos(ωt)(a1O1 + a2O2) where, using (4.13)

O1 = cos(2φ(x = 0)) (4.26)

O2 = ∇φ(x = 0) (4.27)

This leads to

ER = −1

2a2

1ωIm[χR1 (ω)]− 1

2a2

2ωIm[χR2 (ω)] (4.28)

39


where χR1 (t) = −iY (t)〈[O1(t), O1(0)]〉H0 , χR2 (t) = −iY (t)〈[O2(t), O2(0)]〉H0 .Y (t) is Heaviside step function.

The explicit calculation of the correlation is performed in Appendix A.3 andgives

ER = −1

2

(δVπ

)2

ωIm[χR2 (ω)]− 1

2(2δV ρ0)2ωIm[χR1 (ω)]

=δV 2K

4πu2ω2 +

1

2(2δV ρ0)2 sin(Kπ) cos(Kπ)

(u/α)−2KΓ(1− 2K)ω2K

(4.29)

Where χR1 (ω) and χR2 (ω) are defined in (A.36) and (A.37) of the Appendix A.3, αis of order of lattice cutoff.

As we see the energy deposition rate ER shows a power law behavior as a func-tion of the frequency of the potential modulation. The forward scattering part of thepotential leads to an exponent of two, while the backward scattering part has an ex-ponent 2K. For K > 1 the energy deposition rate is dominated at small frequenciesby the forward scattering part. Note that the prefactor of the energy absorption givesdirect access to the TLL parameters of the system.

For the two component case, similar calculations lead to

ER =δV 2Kρ

2πu2ρ

ω2 +1

2(4δV ρ0)2 sin((Kρ +Kσ)π/2)

cos((Kρ +Kσ)π/2)(uρ/α)−Kρ(uσ/α)−Kσ

Γ(1− (Kρ +Kσ))ωKρ+Kσ

(4.30)

If Kρ > 1 then the dominant contribution comes also from the forward scatteringfor small frequencies.

4.2.3 Crossover from weak to strong coupling limit and strong toweak coupling limit

The behavior of the previous section remains valid only if one can compute theabsorbed energy with the quadratic part of the static Hamiltonian only. This is validforK > 1 orKρ > 1 since the bare potential of the backscattering term scales down,but becomes incorrect in the opposite limit of K < 1 (or Kρ < 1) since in that casethe backscattering part scales up in the static Hamiltonian. The RG equations (4.12)and (4.18) thus define a scale at which the backscattering term renormalizes to avalue of order one, and thus one enters a different regime to compute the absorption.As a function of the frequency of the modulation this defines a crossover frequencyω∗ below which the expressions (4.29) and (4.30) are not valid any more.

40


Let us start from a regime where K < 1, and a weak bare impurity potentialV0 < 1. V0 will flow to the strong coupling limit by lowering the frequency of themodulation. We can compute the flow parameter l∗ at which V0(l∗) ' α ω0. Using(4.12) we obtain

V0(l∗) = V0(l = 0) exp[(1−K)l∗] ' α ω0 (4.31)

exp[l∗] =( αω0

V0(l = 0)

) 11−K

(4.32)

If ω0 is the maximum frequency cutoff of the order of the bandwidth

ω∗ = ω0

( αω0

V0(l = 0)

) −11−K

(4.33)

In an opposite way, if we had started in the strong coupling regime for which thebare potential of the impurity is very strong as described in Sec. 4.1.4 but haveK > 1the weak tunnelling J0 between the two parts of the chain would flow accordingto [5, 10]

dJ0

dl= [(1− 1/K)J0] (4.34)

leading in a similar way to a crossover scale at which the tunnelling becomes of orderone

ω∗ = ω0

( ω0

J0(l = 0)

) −11−1/K

(4.35)

4.2.4 Strong coupling renormalization of backscattering (K < 1,ω < ω∗)

For frequencies of the modulation ω ω∗ one enters (for K < 1 or Kρ < 1)the strong coupling regime for which the backscattering term plays a central role inthe static Hamiltonian. Note that this regime is a priori different from the case forwhich one would have started from a strong bare potential. Indeed in that case bothforward and backward scattering would be affected. In the present case the forwardscattering potential is not affected in the static part of the Hamiltonian.

To calculate the energy deposition, we use the dilute instanton approximation [5]as described in Appendix A.4. The energy deposition due to the amplitude modula-tion in the strong coupling regime is given by

ER =δV 2K

4πu2ω2 +

1

2M2(2δV ρ0)2 cos(π/K) sin(π/K)

Γ(1− 2/K)(δ ω)2/Ke−8√

2ρ0V0(l∗)M

(4.36)

Where δ is short time cutoff. In the dilute instanton calculation, one assumes thatthe backward scattering potential is very large but near ω = ω∗, V0 is not very large.

41


The dilute instanton approximation gives the correct exponent of ER but does notprovide the correct coefficient near ω∗. ER in weak coupling and strong couplinglimit should match at ω = ω∗. Using the fact that for ω < ω∗ one has from (4.36)ER = A(ωα/u)2/K where A is a coefficient to be determined, and for ω > ω∗ wehave (4.29), the continuity equation reads

A(α/u)2/Kω2/K∗ = sin(Kπ) cos(Kπ)(u/α)−2KΓ(1− 2K)ω2K

∗ (4.37)

which determined the prefactor A

A = sin(Kπ) cos(Kπ)

(u/α)2/K−2KΓ(1− 2K)ω2K∗ /(ω∗)

2/K

= sin(Kπ) cos(Kπ) (u/α)2/K−2KΓ(1− 2K)

ω2(K−1/K)0 (αω0/V0)2(K+1)/K

(4.38)

At the frequency ω∗ the ratio between the part of ER due to the backward scat-tering and the one due to the forward scattering is

r =4πu2ρ2

0α2 sin(2Kπ)(αω0)2K

KV 20 u

2K(4.39)

Note that for a weak initial potential this ratio can become very large. Thus forfrequencies ω < ω∗ the backward scattering will continue to dominate down to alower frequency ω1 below which the energy absorption rate will be dominated bythe forward scattering. This frequency is given by

ω = ω1 =

(δV 2K

4πu2A

)1/(2(1/K−1))

(4.40)

Note that for K < 1 and ω > ω∗, ω < 1 the energy deposition is still given by(4.30), and the dominant contribution comes from the backward scattering.

The behavior of the energy deposition is depicted in Fig. 4.3 for K < 1.

4.2.5 Strong bare potentialIf the impurity potential is very large compared to the kinetic energy or interacting,one needs to start with the Hamiltonian (4.23) which corresponds to two semi-infinitechains coupled by a hopping term.

The energy modulation is thus given by

ER =δt23π

4u4ω4 +

1

2(δt4)2 cos(π/K) sin(π/K)

Γ(1− 2/K)(u/α)−2/Kω2/K

(4.41)

42


Figure 4.3 – Energy deposition as a function of frequency for a weak bare potential(see text). Depending on the frequency of the modulation three absorption regime exist.Above the crossover frequency ω = ω∗ (dashed line), the backscattering term dominateswith a powerlaw absorption ω2K . Between ω∗ and ω1 the backward scattering termdominates with an exponent 2/K leading to ω2/K behavior and below ω1 absorption isdominated by the forward scattering leading to an ω2 behavior.

Where δt3 = − 2a√KδV J2

πV 20

, and δt4 = 2J2δV ρ0

V 20

, a is the lattice spacing and J thetunnelling.

The energy deposition rate shows again a power law behavior. The exponentis the same than the one corresponding to the renormalized backward scattering,discussed in the previous section. Note however that the part corresponding to theforward scattering is now different. In this case the forward scattering contribute toan absorption going as ω4 instead of ω2 for a weak potential and thus the energyabsorption is dominated by the backscattering potential, as long as 0.5 < K < 1.When K < 0.5 the system is again dominated by the absorption coming from theforwards scattering.

In the case K > 1 the tunnelling between the two half system will scale upas described the flow (4.34). There is thus again a scale ω∗ at which the tunnellingbecomes of order one and then the Hamiltonian switches back to the one with a weakbackscattering term. In that case the exponent becomes similar to the one obtainedin Sec. 4.2.2 and is 2K. So for 1 < K < 2 the backscattering absorption dominatesas very small frequencies, while the forward scattering dominates for K > 2.

Note that since the pre factors in the strong coupling limit for both backward andforward scattering are proportional to 1/V 2

0 , the energy deposition will be very smallin comparison to the weak coupling case.

43


4.3 Results and discussionOur results show that the time dependent modulation of a single impurity in an in-teracting bath of bosons or fermions allows to probe the TLL nature of the bath, thefrequency of the modulation and the interactions between particles serving as con-trol parameters. Each regime is essentially characterized by a powerlaw behavior ofthe absorbed energy. As detailed in the previous sections the precise power dependson: i) the forward and the backward scattering parts of the impurity potential; ii) theinitial strength of the impurity potential compared to the energy scales of the bathHamiltonian; iii) the value of the interactions between the particles in the bath. Theresults are summarized in Fig. 4.2.

Experiments in ultracold gases would thus be prime candidates to test the effectspredicted there. Indeed, as discussed in Chapter 2 several experiments with bulkmodulation of the amplitude of the optical lattices have already shown [18,19,74] theinterest of this technique. We here propose to do this modulation locally. Since themodulation is local this has the advantage to be less sensitive to perturbations suchas a confining harmonic potential. On the other hand the effect of the perturbationis now much smaller of the order of 1/L where L is the size of the system, and thusmore difficult to measure.

Let us discuss separately the two cases of bosons and fermions.

4.3.1 Bosonic systemsAs mentioned in Chapter 2, realizing interacting one dimensional bosonic systems inwhich modulation is possible has been already demonstrated [18,21]. Putting a localpotential is possible either with a light blade [60] or in boson microscopes by locallyaddressing a single site [61]. Measurement of the energy deposited in the system canbe obtained either by measuring the momentum distribution [18] or in a microscopeby looking at the number fluctuations [61]. Especially in the last case, realizing boxpotentials should also be possible.

For bosons with local interactions we have K > 1 and thus are on the rightpart of the Fig. 4.2. A weak potential should thus show an absorption dominated bythe forward scattering potential, scaling essentially as ω2. It could be interesting iffeasible to engineer a local potential so that the forward part of the potential wouldbe much smaller than the backward one. For example for a system with one bosonevery two site it would require a potential of the form V0 on x = −1 and −V0 onx = 0. This would reinforce strongly the backscattering part and thus show the ω2K

absorption that would correspond to this component of the potential.Alternatively one could get rid of the forward scattering component by going

to a very strong on-site potential as detailed in Sec. 4.1.4. In such a case onewould start with absorption scaling as ω2/K which for weakly repulsive bosons forwhich K can be large would correspond to a nearly constant absorption rate. At thecrossover ω∗ the absorption would drop brutally since it would behave as ω2K . The

44

4.3 Results and discussion

crossover scale ω∗ corresponding to the renormalization of the potential in the staticpart, should thus be visible with this method.

4.3.2 Fermionic systemsShaking in fermionic systems has already been used for Hubbard-like Hamiltoni-ans [74] . Instead of measuring the deposited energy, the growth of double occupancyallows for a more sensitive detection technique [24,25,74]. Fermionic systems allowalso potentially to reach the regime K < 1. However single component fermionicsystems would require to have also finite range interactions which is for the mo-ment still not too common in experiments. Reaching such a regime with long rangeinteractions can of course be also done with bosonic systems. For fermions it is how-ever simpler to use two component systems. In particular recent experiments havedemonstrated the possibility to have fermionic microscopes [72,73,75] in which suchtwo component fermionic systems can be manipulated. These experiments would beideal testbeds for the effects described in the previous sections.

One could then explore potentially the same effects that the ones for bosons,namely the scaling ω2K and ω2/K of the absorbed energy. In particular for a verylarge repulsion U and a weak bare local potential one would expect an absorptionER ∼ ω3/2 since the limit of the exponent Kρ for the Hubbard model is Kρ = 1/2.

45


46

CHAPTER 5

Dynamics of a Mobile Impurity in a Two Leg BosonicLadder

In chapters 1 and 2, we have introduced the dynamics of a mobile impurity in aone-dimensional bath. The impurity in the one-dimensional bath shows a divergentbehaviour, and the Green’s function of the impurity decays as a power-law below acritical momentum and exponentially above a critical momentum. How the trans-verse extension of the bath will affect the dynamics of impurity is an interestingquestion. In this chapter, we address this issue by looking at an impurity propagatingin a ladder. We confine the impurity to move one-dimensionally, but the bath hasnow a transverse extension. We mostly focus on the case of a two leg ladder but willalso discuss some of the consequences of increasing the number of legs. We analysethis problem using a combination of analytical and numerical techniques in a spiritsimilar to the one used for the single chain [39], and compare the results of the ladderbath to the known results for the single chain.

The plan of the chapter is as follows: Sec. 5.1 presents the model, the variousobservable studied and the bosonization representation that will be at the heart ofthe analytical analysis. Sec. 5.2 presents the two main analytical approaches that areused, namely a field theory representation based on the bosonization technique anda linked cluster expansion representation. Sec. 5.3 presents the numerical DensityMatrix Renormalization Group (DMRG) [76, 77] analysis of this problem, and theresults for the Green’s function of the impurity. Sec. 5.4 discusses these results bothin connection with the single chain results and in view of the possible extensions.Technical details can be found in the appendix B.

47

5. DYNAMICS OF A MOBILE IMPURITY IN A TWO LEG BOSONICLADDER

Figure 5.1 – Impurity in a two leg bosonic ladder. Magenta solid circles represent thebath particles and the red circle represents the impurity. The bath particles move alongthe legs (between the legs) with hopping tb (t⊥) and have a contact interaction U1 (U2)between themselves (see text). The impurity motion is restricted to the upper leg, itsamplitude is timp (see text). The impurity and the bath particles interact by a contactinteraction U .

5.1 Mobile Impurity in a Two Leg Bosonic Ladder

5.1.1 ModelWe consider a mobile impurity moving in a two-leg Bosonic ladder. In this studywe restrict the motion of the impurity to be strictly one-dimensional, other cases willbe considered in chapter 6 [78]. This model is thus the simplest basic model forexploring the dynamics of a one-dimensional impurity in an environment that is nolonger purely 1D itself. The model we consider is depicted in Fig. 5.1. The fullHamiltonian is given by

H = HK +Hlad + U∑j

ρ1,jρimp,j (5.1)

U is the interaction strength between the particles in the ladder and the impurity.The impurity kinetic energy is given by the tight-binding Hamiltonian

HK = −timp

∑j

(d†j+1dj + h.c.) (5.2)

where dj (d†j) are the destruction (creation) operators of the impurity on site j. Thedensity of the impurity on site j is

ρimp,j = d†jdj (5.3)

The ladder Hamiltonian Hlad is given by

Hlad = H01 +H0

2 − t⊥∑j

(b†1,jb2,j + h.c.) (5.4)

where ba,j (b†a,j) are the destruction (creation) operators of a boson of the bath onchain a and site j. The b-operators obey the standard bosonic commutation relation

48


rules. The single chain Hamiltonian is the Bose-Hubbard one

H0i = −tb

∑j

(b†j+1bj + h.c.) +Ui2

∑j

ρi,j(ρi,j − 1)

− µi∑j

ρi,j

(5.5)

In the following, we use the tunneling rate tb of the bath bosons as the unit of energy.The form (5.1) is convenient for the numerical study. In order to make easily

connection with the field theory analysis we can also consider the same problem ina continuum. In this case the Hamiltonian becomes

H =P 2

2M+Hlad + U

∫dxρ1(x)ρimp(x) (5.6)

where P and M are respectively the momentum and mass of the impurity. Thedensity of the impurity is

ρimp(x) = δ(x−X) (5.7)

where X is the position of the impurity, canonically conjugate to P so that [X,P ] =i~. We set from now on ~ = 1.

In the continuum, the ladder Hamiltonian (5.4) becomes

Hlad = H01 +H0

2 − t⊥∫dx(ψ†1(x)ψ2(x) + h.c.) (5.8)

and the single chain Hamiltonian is

H0i =

1

2m

∫dx|∇ψi(x)|2 +

Ui2

∫dxρi(x)2

− µi∫dxρi(x)

(5.9)

m is the mass of the bosons, µi is the chemical potential and Ui is the intrachaininteraction of leg i = 1, 2. ψ(x)† (ψ(x)) is the creation (annihilation) operator atposition x.

5.1.2 ObservablesTo characterize the dynamics of impurity in the ladder we mostly focus on theGreen’s function of the impurity. We study it both analytically and numerically viaDMRG, each time considering the zero temperature case.

The Green’s function of the impurity is defined as

G(p, t) = 〈dp(t)d†p(t = 0)〉 (5.10)

49


where 〈· · · 〉 denotes the average in the ground state of the bath, and with zero impu-rities present. O(t) denotes the usual Heisenberg time evolution of the operator

O(t) = eiHtOe−iHt (5.11)

and the operator dp is the operator destroying an impurity with momentum p givenby

dp =∑j

eiprjdj (5.12)

with rj = aj on the lattice and the corresponding integral

dp =

∫dxeipxd(x) (5.13)

in the continuum.

5.1.3 Bosonization representationTo deal with the Hamiltonian defined in the previous section, we first consider thelow-energy effective Hamiltonian (see chapter 3), and each leg is described by TLL

H0α =

1

2π

∫dx

[uαKα(∂xθα)2 +

uαKα

(∂xφα)2

](5.14)

where α = 1, 2, uα and Kα are the so-called Tomonaga-Luttinger liquid (TLL)parameters.

Using the bosonization framework, the low-energy approximation of inter-legtunneling is

−2t⊥ρ0

∫dx cos(θ1(x)− θ2(x)). (5.15)

This makes it convenient to use the symmetric and antisymmetric combinations ofthe fields

θs,a =θ1 ± θ2√

2(5.16)

and analogous expressions for the fields φ. The new fields remain canonically con-jugate and allow to re-express the Hamiltonian of the bath as

Hlad = Hs +Ha (5.17)

with

Hs =1

2π

∫dx[usKs(∂xθs)

2 +usKs

(∂xφs)2]

Ha =1

2π

∫dx[uaKa(∂xθa)2 +

uaKa

(∂xφa)2]−

2ρ0t⊥

∫dx cos(

√2θa(x))

(5.18)

50

5.2 Analytical solutions

Because of the presence of the cosine term the antisymmetric part of the Hamiltonian(the so-called sine-Gordon Hamiltonian) will be massive when Ka > 1/4. As aresult the field θa is locked to the minima of the cosine indicating the phase coherenceacross the two legs of the ladder. The phase being locked, the density fluctuation inthe antisymmetric sector have fast decaying correlations instead of the usual powerlaw of the TLL. The symmetric sector remains massless with power law correlations.A numerical calculation of the TLL parameters for the massless phase can be foundin [79].

5.2 Analytical solutionsLet us now investigate the full Hamiltonian (5.6) (or (5.1)) to be able to compute theGreen’s function of the impurity (5.10). The interaction termHcoup with the impurityis expressed, as a function of the bosonized variables as

Hcoup =−U√

2π

∫dx (∇φa(x) +∇φs(x)) ρimp(x)+

2Uρ0

∫dx cos(

√2(φa(x) + φs(x))− 2πρ0x)ρimp(x) (5.19)

where we have retained only the lowest harmonics in the oscillating terms, higherorder terms being a priori less relevant.

Since the antisymmetric sector is gapped, with a gap ∆a due to the ordering inthe field θa, the correlation functions involving φa decrease exponentially at largedistance or time interval. We thus a priori need to distinguish the cases for whichU ∆a, for which the interaction with the impurity is not able to create excitationsin the antisymmetric sector, from the case U ∆a. We examine these two cases inturn.

5.2.1 U ∆a

In this case the coupling with the impurity cannot create excitations in the antisym-metric sector. The backscattering term containing cos(

√2(φa(x) + φs(x))) is thus

irrelevant. This term can potentially generate a cos(2√

2φs(x)) term by operatorproduct expansion (OPE) but such term would only become relevant for Ks < 1which is outside the range of allowed TLL parameters for the bosonic problem weconsider here.

Thus, as for the case of a single impurity [32] the dominant contribution is givenby the forward scattering term

−U√2π

∫dx∇φs(x)ρimp(x) (5.20)

51


in which we have kept only the part corresponding to the massless sector.In this regime the impurity behaves in the ladder system as it would be in a single

chain with effective parameters (us,Ks), but with a renormalized coupling

U2 =U√

2(5.21)

Note that for a microscopic interaction U1 = U2 the TLL parameters (us,Ks) arealso in general not equal to the one of a single chain with the corresponding interac-tion due to the presence of the irrelevant operators coming from the t⊥ term.

The Green’s function of the impurity (5.10) can thus be readily computed bythe same techniques used for the single chain case [32, 39]. Using a linked clusterexpansion (LCE), a re-summed second-order perturbation series applicable at smallbath-impurity interactions, as detailed in the appendix B.1, we find that when p <us

2timp, the difference of energies of the impurity εp−εp+q intersects the bath dispersion

us|q| only at q = 0. As shown in Appendix B.1 this leads to a power law decay ofthe Green’s function (5.10)

|G(p, t)| = e− KsU

2

4π2u2s

(1+12t2impp

2

u2s

) log(t)(5.22)

This case will be discussed in details in Sec. 5.3.2.On the contrary for p > us

2timp, εp − εp+q intersects the bath dispersion us|q| at

non-zero q′s. In that case the Green’s function decay exponentially, and enter intothe so called quasiparticle (QP) regime [39], where εp = −2timp cos(p). This casewill be discussed in Sec. 5.3.3.Dynamical structure factor S(k, ω) of one of leg of the ladder is given by

S(x, ω) = 〈GSb|ρ1(x)1

ω + iη −Hladρ1(0)|GSb〉

S(k, ω) =

∫dxS(x, ω) exp(−ikx)

(5.23)

Where η → 0+, ρ1(x) is the density operator in leg 1 of the ladder at position x.To go beyond the bosonized evaluation of the structure factor of the bath used in theappendix B.1, we also show in Fig. 5.2 the imaginary part of dynamical structurefactor define in Eq. (5.2) of one of leg of a hard core boson ladder computed usingDMRG at t⊥ = tb = 1, χ = 600 and zero temperature, together with δε(q) = εp −εp+q , for different p values at timp = 1. We find that for small q the dispersion of bathis well fitted with the linear dispersion used in the TLL representation us|q| (greenline, us = 1.8). We find, in agreement with the LCE, that for p = 0, 0.1π, 0.2π,δε(q) intersects the dynamical structure factor only at q = 0, while for large p =0.3π, 0.4π, 0.5π, 0.7π, π , δε(q) intersects the dynamical structure factor at q 6= 0.The resulting change of regime for the impurity will be discussed in more details inSec. 5.3.3 and Fig. 5.8.

52

5.2 Analytical solutions

Figure 5.2 – Dynamical structure factor of leg 1 of the two-leg hardcore boson ladderat one third filling (shaded area). Superimposed colored curves show the change in theimpurities kinetic energy when emitting an excitation with momentum q into the bath,δε(q) = εp− εp+q , for different values of p. In ascending order by maximal value theseare: red (p = 0), green (0.1π), cyan (0.2π), magenta (0.3π), red (0.4π), cyan (0.5π),magenta (0.7π) and yellow (π) . Dotted lines (p = 0, 0.1π, 0.2π) denote where δε(q)intersects with areas of finite weight of the dynamical structure factor only at q = 0.Solid lines and dotted line at p = 0.3π denote the higher p-values for which it intersectsalso at non-zero q’s. This change in regime causes the impurity to go from subdiffusiveto quasi-particle.

5.2.2 U ∆a

We now consider the more complicated opposite case for which the interaction withthe impurity, which couples to the field φ1, can potentially induce excitations acrossthe antisymmetric gap for the field θ1. In order to deal with this case we essentiallyfollow the method introduced by Lamacraft for the case of the single chain [36].

We first make a transformation to a frame of reference which moves with theimpurity, and this can be imposed by an unitary transformation UX = ei(P1+P2)X ,where P1 , P2 and X are total momentum operator of the first, second leg of theladder and position operator of the impurity respectively. In bosonized languageP1 =

∫dx∇θ1∇φ1, P2 =

∫dx∇θ2∇φ2.

By using (5.6,5.7), the effective Hamiltonian in the new frame of reference isgiven by

Heff = UXHU†X

=(P − P1 − P2)2

2M+Hlad + Uρ1(0) (5.24)

53


Where ρ1(0) is the density of leg 1 at x = 0. In the new effective Hamiltonian theposition operatorX has disappeared and P is thus a conserved quantity. As U →∞,U is replaced by an effective forward scattering potential Uφ (roughly the phase shiftcorresponding to the potential U ) [36] and ρ1 = −∇φs+∇φa√

2π. The forward scattering

can thus be absorbed in the quadratic term by creating a discontinuity in the fields φof the form

φs|0−

0+ =KsUφ√

2us

φa|0−

0+ =KaUφ√

2ua

(5.25)

The dominant term in the total current is given by

P1 + P2 =

∫dxρ0

√2∇θs(x) (5.26)

where in the above formula the origin must be excluded if the field has a discontinu-ity. For P = 0 the minimization of H imposes that the field θs remains continuousat x = 0. These two set of conditions for the fields φ and θ can be imposed onotherwise continuous fields by the unitary transformation

UP=0 = exp(iθs(0)KsUφ√

2πus) exp(iθa(0)

KaUφ√2πua

) (5.27)

The impurity Green’s function at zero momentum is thus given by

|G(0, t)| = 〈U†P=0,tUP=0,0〉

= 〈exp(−iθs(t)KsUφ√

2πus) exp(iθs(0)

KsUφ√2πus

)〉

= 〈exp(−iθa(t)KaUφ√

2πua) exp(iθa(0)

KaUφ√2πua

)〉

= |t|−Ks/4(Uφπus

)2

(5.28)

To compute the above correlation function corresponding to field θa we haveexpanded cos(

√2θa) upto second order, and exploited that the correlation function

of the anti-symmetric mode saturates to a finite value for time greater than the inverseof gap in the anti-symmetric sector. The symmetric mode decays as a power law withan exponent Ks4 (

Uφπus

)2. For hard core bosons in one dimension Uφπus

= 1, and thusthe overall exponent in (5.28) is Ks/4.

54

5.3 Numerical solution


5.3.1 MethodIn order to obtain the Green’s function of the impurity for the ladder problem quan-titatively, we compute a numerical solution of the lattice model defined in Sec. 5.1.1,by a method analogous to the one used for the single chain [39]. We use DMRG tocompute the ground state of the bath and TEBD to compute the Green’s function ofthe impurity as discussed in chapter 3.

We use a supercell approach to map the three species of bosons (A,B,C) intoa one dimensional chain. Leg 1 and leg 2 of the ladder are represented by speciesA and B respectively and the impurity is represented by C. The total number ofquantum particles in leg A and leg B is conserved, and the total number of particlesin species C is conserved separately and equal to 1. In the supercell approach, thelocal Hilbert space is of size 2 × 2 × 2 = 8 for the case of hard core bosons and of3× 3× 2 = 18 for soft core bosons if the maximum allowed occupancy per speciesA,B is 2. We thus restrict ourselves to relatively large repulsions (of the order ofU1 = U2 = 10 for softcore bosons and U1 = U2 = ∞ for hardcore ones) in eachleg for the bath so that the restriction of the Hilbert space is not a serious limitation.The interaction between the bath and the impurity can however take any value.

We consider a system for which the density in each leg is ρ0 = 1/3, to avoid thepossibility of entering to a Mott insulating state in the ladder in which the symmetricsector would be gapped as well. Transverse and impurity hopping are taken of equalvalue, t⊥ = timp = 1. We fix the size of the system to L = 101 sites per leg.

As the local Hilbert space can become large for soft-core bosons, we are limitedto moderate values of χ to maintain reasonable computational times. We further usebond dimensions χ = 300, 500, 400, 600 for hard core boson and χ = 400 for softcore boson. Further details are provided in Appendix B.2.

The ground state of the bath |GSb〉 is computed by using DMRG. We then addan impurity in the center of leg 1, at time t = 0

|ψ(t = 0)〉 = d†L+12

|GSb〉 (5.29)

We then evolve the state |ψ(t = 0)〉 as a function of time (t) with the full Hamiltonian(5.1) by using TEBD and compute

|ψ(t)〉 = e−iHtd†L+12

|GSb〉 (5.30)

We then take the overlap with the state

d†L+12 −x

|GSb〉, (5.31)

which yields the sought-after impurity Green’s function in time and space, up to aphase factor of eiEGSb t, where EGSb is ground state energy of the bath. A Fouriertransformation then yields the Green’s function in time and momentum.

55


5.3.2 Results at zero momentum

We examine first the Green’s function of the impurity G(p, t) for the case of zeromomentum p = 0. For a single chain this limit is known to lead to a power lawdecay of the Green’s function [32, 36, 39].

Examples of the decay of |G(0, t)| are shown in Fig. 5.3, both for softcore bosonsand hard core bosons. The numerical data shows a decay of the correlation function

0 1 2 3 4 5 6 7 8 9 10t

0.96

0.97

0.98

0.99

1

G(0

, t)

U=1

0 1 2 3 4 5 6 7 8 9 10t

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

G(0

, t)

U=2

0 1 2 3 4 5 6 7 8 9 10t

0.95

0.96

0.97

0.98

0.99

1

G(0

, t)

U=1

0 1 2 3 4 5 6 7 8 9 10t

0.85

0.9

0.95

1

G(0

, t)

U=2

Figure 5.3 – Modulus of the Green’s function of the impurity (see text) |G(p = 0, t)|at zero momentum of the impurity. Parameters for the interchain hopping , impurityhopping, and impurity-bath interaction in the ladder are respectively t⊥ = 1, timp = 1.Left (resp. right) column corresponds to U = 1 (resp. U = 2). The upper panel shows|G(0, t)| of softcore bosons with U1 = U2 = 10 and the lower panel hardcore bosonsfor χ = 400.

with time. Such a decay is expected from the general arguments of Sec. 5.2. Asdiscussed in Appendix B.2 the bond dimension controls the maximum time at whichthe decay of the correlation can be computed reliably with the TEBD procedure. Inour case, time of the order t ∼ 7 for soft-core and t ∼ 7 for hard-core bosons presentthe limit for reliable data.

56


5.3.2.1 Interaction dependence

To analyze the data we use the analytic estimates of Sec. 5.2 which suggest a powerlaw decay of the Green’s function.

|G(p = 0, t)| ∝(

1

t

)α(5.32)

We fit the numerical data as detailed in Appendix B.2, which confirms the powerlaw decay of the correlations and allows extracting the exponent α. This exponent isshown in Fig. 5.4 for the case of hard core bosons.

0.2 0.4 0.6 0.8 1U

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

α

fit Ks=.8145 us=1.8fit Ks=.835 us=1.86DMRG data

Figure 5.4 – Green’s function exponent at p = 0 for hard core bosons as function ofsmall U at t⊥ = timp = 1. The black curve is the exponent extracted from correlationfunctions computed from TEBD results with χ = 600. The green and red curves are theLCE exponent extracted from (5.22), αLCE = KsU

2

4π2u2s

, for the pair of values (Ks = .835,us = 1.86), (Ks = .8145, us = 1.8) respectively. These values correspond well to theTLL parameters for a ladder of hard core bosons (see text).

The good agreement between the numerical exponent of Fig. 5.4 with the LCE-formula in (5.22) confirms the analytic prediction of Sec. 5.2 that one can indeedview the ladder with a small impurity-bath interaction U as a single TLL with aneffective Ks and us but with an effective interaction U/

√2. The simple decoupling

of the Hamiltonian into symmetric and antisymmetric sector of (5.18) would naively

57


suggest that (Ks, us) = (K, v) of a single chain, but for large U1 = U2 interactionsand sizeable t⊥, irrelevant operators can lead to a sizeable renormalization of theparameters. So in general the parameters (Ks, us) for the ladder are not identical tothe ones of a single chain with the same interaction, leading also to a modificationof the exponent. This is in particular the case for hard core bosons, for which thesingle chain parameter is K = 1 [5], while for the ladder [79] one has Ks < 1 Ourcalculations yield Ks = 0.8145 and us = 1.8. Given the accuracy of determinationof the TLL parameters this compares well with the values obtained in Ref. [79]which are Ks = 0.835 and us = 1.86 (c.f. also Fig. 19 of [79]). We note that forthe single chain a value of K < 1 would have meant that the backscattering on theimpurity terms could become relevant. For the ladder case however , because theantisymmetric sector is gapped, such terms remain irrelevant even for Ks < 1.

For larger values of the impurity-bath interaction U one cannot rely on the LCEexpression anymore. The numerically computed exponent is shown in Fig. 5.5.

0 5 10 15 20U

0

0.05

0.1

0.15

0.2

α

DMRG dataKs =.8145, us=1.8Ks=.835, us=1.86

Figure 5.5 – Exponent α controlling the power-law decay of |G(p = 0, t)| as a functionof the interaction between the bath and the impurity U . The bath has t⊥ = timp = 1 andis made of hard core bosons at 1/3 filling for each leg. The black circle are the numericaldata. The red line is a fit to (5.33) (see text) with f = 0.39. The good agreement betweenthe data and the formula (5.33) shows that, as for a single chain, this formula correctlydescribes the behavior of the impurity in the ladder for a wide range of interactions.

The quadratic growth of the exponent with interaction of Fig. 5.4 is replaced by

58


a more complex behavior and a saturation of the exponent at large U . For the singlechain the interaction dependence of the exponent could be captured by an analyticexpression [39]. In order to adapt this expression to the case of ladder we use themodified interaction U/

√2 suggested by the bosonization formula of Sec. 5.2 and

the TLL parameters of the ladder which leads to

α =2f

π2

(arctan

[2√

2fu2s

U√Ks

]− π

2

)2

(5.33)

Here, f = 2α[U → ∞], and we use the independently calculated exponent at U =∞ (see next section) to obtain f = 0.39 for plotting the analytical curves in Fig. 5.5.We stress that the excellent agreement between DMRG data and analytical predictioneq. (5.33) shown in Fig. 5.5 is thus without fitting parameters, unlike that in Ref. [39],which had been lacking an independent way of obtaining f quantitatively.

5.3.2.2 Hard core bath-impurity repulsion

Let us now turn to the case for which the repulsion between the impurity and particlesof the bath is very large U →∞. Various decays of the Green’s function are shownin Fig. 5.6. As it is clear from the numerical data the powerlaw decay is still presentin this limit and persists for all the measured values of t⊥ in the ladder. A fit of thenumerical data provides access to the exponent as a function of interchain hopping asshown in Fig. 5.7. We first note that the value of the exponent for U =∞ at t⊥ = 1,α = .195 is in good agreement with the one given by the formula (5.33). Largervalues of t⊥ show a marked increase of the exponent as is obvious from Fig. 5.4.This trend of the exponent with t⊥ is surprising and is a priori not compatible withthe simple extrapolation of the formula KsU

2

4π2u2s

in which the interaction U would bereplaced by its phase shift as for the single chain [36] where U → Uφ = vπ. In sucha limit the exponent would be proportional toKs which decreases when t⊥ increasesat variance with the numerical data. This shows that there is a dependence on t⊥ ofthe exponent besides the one hidden in the t⊥ dependence of the TLL parameters.

5.3.3 Momentum dependence of the exponentWe now turn to the momentum dependence of the Green’s function, which, as forthe single chain, is much more difficult to obtain analytically.

We show in Fig. 5.8 the decay of the Green’s function for various momenta p ofthe impurity. The right panel, for small momentum of the impurity, shows a powerlaw decay of the Green’s function, similar to the one for zero momentum, albeit witha renormalized exponent. However, above a momentum threshold around 0.3π thenumerical data no longer exhibits a power-law decay, and is fitted better with anexponential decay, as shown in the right panel of the figure. This is similar to thebehavior observed for an impurity coupled to a single chain [39].

59


0.1 1 10 100t

1

|G(0

,t)|

t⊥=1t⊥=2t⊥=3t⊥=4

Figure 5.6 – Green’s function of the impurity at zero momentum on a log-log scalefor the ladder of hard core bosons and for infinite repulsion between impurity and bathparticles. t⊥ = 1, 2, 3, 4 and tb = timp = 1. The TEBD has been performed forχ = 600. The powerlaw decay of the correlation is present for all measured values oft⊥.

60


0 1 2 3 4 5 6 7t⊥

0.1

0.15

0.2

0.25

0.3

0.35

0.4

α

Figure 5.7 – Exponent of the Green’s function of an impurity with a hard core repulsionwith a bath of hard core bosons at filling 1/3 as a function of t⊥ at timp = 1, U → ∞and p=0. Circles are the numerical data for χ = 600 and the line is a guide to the eyes.

61


0.1 1t

1|G

(p,t)

|

p=0p=0.1 πp=0.2 π

1 2 3 4 5 6 7t

1

|G(p

,t)|

p=0.3πp=0.4πp=0.5πp=0.6πp=0.7πp=0.8πp=0.9πp=π

Figure 5.8 – Green’s function of the impurity in a ladder of hard core bosons, timp = 1,and transverse hopping(t⊥ = 1), U = 1, for different momenta and χ = 600.Left panel: Green’s function on log-log scale. The momenta are p = 0, 0.1π, 0.2π.On a log-log scale the Green’s functions show a linear behavior which reflects thepower law decay. Right panel: Green’s function on semi log scale. Momenta arep = 0.3π, 0.4π, 0.5π, 0.6π, 0.7π, 0.8π, 0.9π, π. On the semi log scale the Green’sfunctions show a linear behavior which reflects the exponential decay.

For small momentum, a momentum dependent exponent can be extracted by thefitting methods described in Appendix. B.3. The results are shown in Fig. 5.9. Thegrowth of the exponent with momentum can be expected since on general groundsone finds [32, 39] that the exponent varies as

α(p) = α(p = 0) + βp2 (5.34)

The coefficient β can be computed for small interactions (see Appendix B.1) and isgiven by

β =3KsU

2

π2u2s

t2imp

u2s

(5.35)

Comparison between the numerics and the LCE shows a very good agreement be-tween the two for small interactions.

This good agreement with the numerical exponent and the LCE one suggeststhat, as for the single chain we can use the LCE to estimate the critical momentump∗ = us

2timpwhich separates the ID regime from the polaronic one. The crossover

depends on the TLL characteristics of the bath, namely the velocity of sound in theladder. Using the values extracted from [79] we get

p∗ = 0.93 (5.36)

62


0.2 0.4 0.6 0.8 1U

0

0.002

0.004

0.006

0.008

α

Ks=.8145 us=1.8Ks=.835 us=1.86DMRG data

0.2 0.4 0.6 0.8 1U

0

0.005

0.01

0.015

0.02

α

Ks=.8145 us=1.8Ks=.835 us=1.86DMRG data

Figure 5.9 – Green’s function exponent of the impurity in a ladder of hard core bosons,as a function of U for p = 0.1π (left panel), 0.2π (right panel) timp = 1, transversehopping (t⊥ = 1) and χ = 600. The black curves represent the DMRG data and thegreen and red curves are fitted curves of the linked cluster result with the TLL parametersare mentioned in inset.

which is in reasonably good agreement with the observed change of behavior inFig. 5.8. Beyond the LCE we also see that the change of behavior is in good agree-ment with the criteria of intersection discussed in Fig. 5.2 when one uses properlythe structure factor of the ladder.

Beyond p = p∗ and for small U , the Green’s function decays exponentially, theimpurity behaves like a quasi-particle, and the Green’s function of the impurity interm of life time τ(p) is given by

|G(p, t)| = exp(−t/τ(p)) (5.37)

In Fig. 5.10, we plot the inverse of life time 1/τ(p), defined in eq. (5.37) of the QP asa function of p for different interactions U = 0.3, 0.4, 1. As can be expected 1/τ(p)increases with increasing interaction. Let us now turn to the limit of infinite repulsionbetween the impurity and the bath. The numerical Green’s functions are given inFig. 5.11. In a similar way than for the small interaction limit, the numerical datashows a crossover from a powerlaw regime to another type of decay upon increasingthe momentum. The transition occurs for after p around p∗ = 0.9π (analytically it ishard to compute p∗ for infinite interaction ).

63


0.4 0.5 0.6 0.7 0.8 0.9 1 1.1p/πa

0

0.1

0.2

0.3

0.4

0.5

0.6

1/τ(

p)

U=0.3U=0.4U=1

Figure 5.10 – Inverse life time of the impurity as function of the momentum for a bathof hard core bosons at one third filling. The parameters are timp = tb = t⊥ = 1,U = 0.3, 0.4, 1, a = 1 is lattice spacing.

5.4 Discussion

Let us now discuss the various results for the ladder systems in comparison withthe case for which the bath is a single chain. The motion of the impurity being stillstrictly one-dimensional all differences are directly related to the change of nature ofthe bath.

Let us first turn to the weak coupling between the bath and the impurity. Thenumerical results fully confirm that in such limit the field theory description (5.22)gives an excellent approximation of the exponent at zero momentum. The fact thatonly the symmetric mode is massless roughly leads to the replacement of the inter-action between bath and impurity by U/

√2. This would lead, if the TLL parameters

were identical for the symmetric modes and a single chain, to a reduction of the ex-ponent. This trend can be easily extended to a bath consisting of an N -leg bosonicladder. In that case all modes, except the global symmetric mode, are gapped and theinteraction with the impurity would become U/

√N . Note that such a ladder with a

large number of legs is very close to an anisotropic two-dimensional superfluid [80].We, however, see that some of the peculiarities of the ladder, which would likely

disappear when increasing the number of legs, manifest themselves in the exponent

64

5.4 Discussion

0.1 1 10t

0.1

1

|G(p

,t)|

p=0p=0.1πp=0.2πp=0.3πp=0.4π

0.1 1 10t

0.01

0.1

|G(p

,t)|

p=0.7πp=0.8π

0 3 6 9 12 15 18 21t

0.01

0.1

1

|G(p

,t)|

p=0.9πp=π

Figure 5.11 – Green’s function of the impurity as a function of time for p =0, 0.1π, 0.2π, 0.3π, 0.4π, 0.7π, 0.8π, 0.9π, π for hard core boson on log-log (p =0 − 0.8π) scale and semi-log scale (p = 0.9π, π), t⊥ = 1.5, U=∞, timp = 1, forbond dimensions χ = 600.

65


in addition to the above mentioned effect. The ladder itself is more affected by theinteractions within the bath than a single chain. This is readily seen by the fact thata ladder of hard core bosons at a filling of one boson per rung would be an insulator,while a single chain remains a superfluid at half filling. As a consequence the TLLparameterK can reach values below one [79] corresponding to “enhanced” repulsioncompared to the case of a single chain. This effect contributes also to the decay andto a further reduction of the exponent compared to the effect on U itself.

On the other hand increasing t⊥ leads to an opposite trend, particularly visible inthe limit of large t⊥ (see Fig. 5.7), in which a marked increase of the exponent canbe seen. Naively one could consider that increasing t⊥ brings back the system to asingle mode for the rung and thus pushes the system back to a more one-dimensionalbehavior. However the impurity couples still to both modes making the calculationof the exponent delicate. How such effects would be modified upon increasing thenumber of legs is of course an interesting and open question.

As can be seen from Fig. 5.5 the general expression that was introduced in [39]provides a good description of the decay exponent at zero momentum for the lad-der as well. Note that in the present case we have a full numerical evaluation ofthe infinite repulsion case between impurity and the bath, making it an essentiallyparameter-free formula and allowing to check that the large impurity-bath repulsionlimit is indeed correctly reproduced by this formula. This confirms that for a systemin which the impurity couples to a single massless mode the generalization of thefree fermion solution (5.33) captures the essential physics, and we would expect itto hold to the case of N chains as well. As for the single chain, the finite momen-tum case remains more difficult to interpret. We recover for the ladder the same tworegimes that were observed in the single chain, namely a regime dominated by in-frared divergences (ID) for which we have the powerlaw decay of the correlations,and a different regime, which seems, within the accuracy of the numerical solutionalso correspond to a faster decay of the correlations, and thus to a more conventionalpolaronic regime. As for the single chain the separation between these two regimesseems to be reasonably well given by the possibility to excite real particle-hole exci-tations in the bath, as shown by Fig. 5.2.

66

CHAPTER 6

Mobile Impurity in a Two-Leg Bosonic Ladder withMotion in Horizontal and Transverse Directions

In chapter 5, we have investigated the dynamics of a mobile impurity in a two-legladder bath, and the impurity was confined to move in one of the legs of the ladder.One of the questions of interest is what would happen if the impurity also movesin the transverse direction. This would provide a understanding of the dimensionalcrossover of the dynamics of the impurity from one-dimension to two-dimension.In this chapter, we address the dynamics of a mobile impurity in a two-leg bosonicladder bath, the impurity moves in both legs, and it also tunnels from one leg to otherleg.

The plan of this chapter is as follows: In Sec. 6.1, we describe the model onthe lattice and in the continuum limit and various observables. In Sec. 6.2, we giveanalytical expressions for the observables by using bosonization and linked clusterexpansion. In Sec. 6.3 we present the DMRG [76, 77] analysis of this problem, andthe results for the Green’s function of the impurity. Sec. 6.4 discusses these resultsin connection with chapter 5. The analytical expression of the Green’s function isgiven in the Appendix C.

67

6. MOBILE IMPURITY IN A TWO-LEG BOSONIC LADDER WITHMOTION IN HORIZONTAL AND TRANSVERSE DIRECTIONS

Figure 6.1 – Impurity in a two-leg bosonic ladder. Magenta solid circles represent thebath particles and the red circle represents the impurity. The bath particles move alongthe legs (resp. between the legs) with hopping tb (resp. t⊥)(see text). The impuritymoves in both longitudinal and transverse directions with amplitudes timp and t⊥imp (seetext) respectively. The impurity and the bath particles interact by the contact interactionsU1imp and U2imp in leg 1 and leg 2 respectively.


6.1.1 ModelWe consider a mobile impurity moving in a two-leg Bosonic ladder in both horizontaland transverse directions. The model we consider is depicted in Fig. 6.1. The fullHamiltonian is given by

H = HK +Hlad + U1imp

∑j

ρ1,jρimp,1j + U2imp

∑j

ρ2,jρimp,2j (6.1)

Where U1imp, U2imp are the interactions strength between the particles in the leg 1and in the leg 2 with the impurity. We consider U1imp = U2imp = U . Hlad is definedin (5.4).

The impurity kinetic energy is given by a tight-binding Hamiltonian

HK = −timp

∑j

(d†1j+1d1j + d†2j+1d2j + h.c.)+

t⊥imp

∑j

(d†1jd2j + h.c.)(6.2)

We diagonalize theHK by using symmetric and anti-symmetric combinations of d1j

and d2j , and HK can be re-expresed as

Himp =∑q

εs(q)d†sqdsq + εa(q)d†aqdaq (6.3)

where dγj (d†γj) are the destruction (creation) operators of the impurity on site j in

leg γ = 1, 2, dsq =d1q+d2q√

2, daq =

d1q−d2q√2

, εa(q) = −2timp cos(q)−t⊥imp, εs(q) =

−2timp cos(q) + t⊥imp. timp and t⊥imp are tunneling amplitudes of the impurity in

68

6.2 Analytical solution for weak coupling

the horizontal and transverse directions, respectively. The density operator of theimpurity on site j is

ρimp,γj = d†γjdγj (6.4)

The form (6.1) is convenient for the numerical study. In order to make easilyconnection with the field theory analysis we can also consider the same problem ina continuum. In that case the Hamiltonian becomes

H =P 2

2M+ t⊥imp[|1〉〈2|+ |2〉〈1|]+

Hlad + U [ρ1(X)|1〉〈1|+ ρ2(X)|2〉〈2|](6.5)

Where X and P are the position and momentum operators of the impurity.

6.1.2 ObservablesTo characterize the dynamics of the impurity in the ladder we mostly focus on theGreen’s function of the impurity, similar to chapter 5. We study it both analyticallyand numerically via DMRG, each time considering the zero temperature case. Com-pared to the case [81] in chapter 5 where the impurity was confined to a single chain,it is now necessary to introduce the various Green’s functions for the impurity

G11(p, t) = 〈d1p(t)d†1p(t = 0)〉

G12(p, t) = 〈d2p(t)d†1p(t = 0)〉

Gs(p, t) = 〈dsp(t)d†sp(t = 0)〉

Ga(p, t) = 〈dap(t)d†ap(t = 0)〉

(6.6)

By symmetry of the Hamiltonian, Gs(p, t) = G11(p, t) + G12(p, t), Ga(p, t) =G11(p, t)−G12(p, t). In the following sections, we will focus on Gs(p, t)(Ga(p, t))to understand the dynamics of the impurity.

6.2 Analytical solution for weak couplingLet us now investigate the full Hamiltonian (6.5) (or (6.1)) to compute the Green’sfunction of the impurity (6.6). Using the symmetric and anti-symmetric modes ofthe impurity the interaction term Hcoup can be expressed as

Hcoup =U

2

∫dx(ρ1(x) + ρ2(x))(ds(x)†ds(x) + da(x)†da(x))

+U

2

∫dx(ρ1(x)− ρ2(x))(ds(x)†da(x) + h.c) (6.7)

69


Finally, in the bosonized language, Hcoup for small U can be expressed as

Hcoup =−U√

2π

∫dx∇φs(x)(ds(x)†ds(x) + da(x)†da(x))

−U√2π

∫dx∇φa(x)(d†sda(x) + h.c) (6.8)

One should note that we have consider only the relevant terms that are the forwardscattering parts of the symmetric and anti-symmetric modes of the bath, similar tochapter 5.

The Green’s functions are computed by the same techniques used in chapter5 [39]. The calculation is detailed in Appendix C.1 and gives for the asymptoticbehavior of the impurity Green’s function (6.6) as

|Ga(0, t)| = |e−KsU

2

4π2u2s

log(t)

|Gs(0, t)| = e−aU2t

(6.9)

Where Ks = .835, us = 1.86 for tb = t⊥ = 1, U1 = U2 = ∞, ρ0 = 1/3[79] . For weak repulsion between the impurity and the bath we thus find that theGreen’s function in the anti-bonding mode decays as a power-law with time similarto chapter 5 [81], but in the symmetric mode, it decays as exponentially. And a =

ua8π2Ka

(−ua/(2timp)+f1)2

√(−ua/(2timp)+f1)2+∆2

a/u2af1

, f1 =

√u2a

4t2imp+

∆a+2t⊥imp

timp+∆2a/u

2a

, ∆a = ∆a

√ua2π√Ka

.

6.3 Numerical solutionAnalyzing the regime U ∆a is much more involved since now excitations acrossthe antisymmetric gap can be created. We thus turn to a numerical analysis of thisproblem.

6.3.1 MethodWe use a method similar to the one described in chapter 5 to compute the Green’sfunction of the impurity. For completeness let us recall the method, which is de-scribed below.

We map the ladder-impurity problem to a one-dimensional problem by a super-cell approach. We denote bath particles in leg 1 and leg 2 by B and C, the impurityin leg 1 and leg 2 by A and D and the total number of bath particles and number ofimpurity are conserved separately. The local dimension of Hilbert space for A, B, C,D is two for hardcore bosons hence the dimension of local Hilbert space of supercell(A, B, C, D) is 2× 2× 2× 2 = 16.

70


We compute the Green’s function of the impurity in the ground state of theladder . The ground state (|GSb〉) is computed using DMRG. The Green’s func-tions G11(x, t)(G12(x, t)) of the impurity in the Heisenberg picture are given byeiEGSb t〈GSb|d1,L+1

2 −xe−iHtd†

1,L+12

|GSb〉 (eiEGSb t〈GSb|d2,L+12 −x

e−iHtd†1,L+1

2

|GSb〉),

where EGSb is the ground state energy of the bath. We compute e−iHtd†1,L+1

2

|GSb〉using TEBD and 〈GSb|d2,L+1

2 −x(〈GSb|d1,L+1

2 −x) are computed using DMRG. Us-

ing G11(x, t) and G12(x, t), we compute Ga(x, t) = G11(x, t) + G12(x, t) andGs(x, t) = G11(x, t) − G12(x, t). For numerical calculation we have kept the bathat one-third filling and bath is made of hardcore bosons. We have used a bond di-mension χ = 400 to compute the Green’s function in a reasonable time. We choseHamiltonian parameters t⊥ = tb = timp = 1, U1 = U2 = ∞ and various values oft⊥imp and U .

6.3.2 Zero momentum regime6.3.2.1 Small U

We have shown the Green’s function of the impurity in the symmetric and anti-symmetric modes |Gs(p, t)|, |Ga(p, t)| at momentum p = 0 in Fig. 6.2 on the semi-log and log-log scales, we find that |Ga(0, t)| decays as a power-law which is similarto one observed in chapter 5. However, the Green’s function of the impurity in thesymmetric mode shows an exponentially decay.

For the parameters used in these two figures the gap in the antisymmetric sector is∆2a = .33tb. One is thus for the values of the impurity-bath interaction in the regime

of weak coupling for which a comparison with the analytical results of Sec. 6.2is meaningful. The numerical analysis thus fully confirms that in this regime theGs(0, t) and Ga(0, t) decay exponentially and power-law, respectively.

To further analyze the data we fit the numerical results to the form

|Gs(p = 0, t)| ∝ exp(−t/τ(0))

|Ga(p = 0, t)| ∝(

1

t

)α (6.10)

The numerical data shows a decay of the correlation function with time, and thisexpected from Sec. 6.2. In Fig. 6.3, the inverse life time and power-law exponentat p = 0 are shown, and both numerical and analytical results show an excellentagreement.

6.3.2.2 Hardcore bath-impurity repulsion

In this section, we discuss the case of U → ∞. |Ga(p, t)| as a function of timeis shown in Fig. 6.4 for different t⊥imp on the log-log scale at p = 0. |Ga(p, t)|

71


0 2 4 6t

1

|Gs(0

,t)|

U=0.5U=0.6U=0.7U=0.8U=0.9U=1

0.01 0.1 1t

1

|Ga(0

,t)|

U=0.5U=0.6U=0.7U=0.8U=0.9U=1

Figure 6.2 – The modulus of the green’s function of the impurity in symmetric (up-per panel) and anti-symmetric sectors (lower panel) as a function of time for hardcorebosonic bath at timp = tb = 1, t⊥ = 1, U = 0.5 − 1, t⊥imp = 3 and p = 0. The upperpanel is depicted on semi-log scale show a linear behaviour while lower panel shows alinear behaviour on log-log scale.

72


0.2 0.3 0.4 0.5 0.6 0.7 0.8U

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

1/τ(

0)

Life-time from DMRG resultsLCE result 1/τ(0)=.0476*U2

0.2 0.3 0.4 0.5 0.6 0.7 0.8U

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

α

DMRG ResultLCE α=.0061*U2

Figure 6.3 – Life-time of the impurity in symmetric sector (upper panel) and power-lawexponent (lower panel) in the anti-symmetric at timp = tb = 1, t⊥ = 1, t⊥imp = 3 asfunction of U at zero momentum. The black lines are numerical results and the red linesare LCE results, both numerical and analytical results show a nice agreement for smallU .

73


t

1

|Ga(0

, t)|

t⊥imp= 1t⊥imp= 2t⊥imp= 3t⊥imp= 5

Figure 6.4 – Modulus of the Green’s function of the impurity (see text) |Ga(p, t)| atmomentum p = 0 for the impurity. Parameters for the intrachain hopping, interchainhopping (for the ladder), impurity hopping, impurity interaction, impurity transversetunneling in the ladder are respectively tb = 1, t⊥ = 1, timp = 1, t⊥imp = 1, 2, 3, 5,U =∞.

74


1 2 3 4 5t⊥imp

0.2

0.3

0.4

0.5α

Figure 6.5 – Exponent of the Green’s function of the impurity with a hardcore repulsionwith the bath of hardcore bosons at filling 1/3 as a function of t⊥imp at timp = 1, U →∞and p=0. Circles are the numerical data for χ = 400 and the line is a guide to the eyes.

decays as a power-law and the power-law exponent as function of t⊥imp is depictedin Fig. 6.5.

For small t⊥imp, exponent is similar to ones observed in the one-dimensionalmotion of an impurity in a two-leg ladder bath [81], and it increases with increas-ing t⊥imp and for a large t⊥imp, exponent is similar to the one observed in the onedimensional motion of an impurity in a one-dimensional bath . These trends showthat as a function of t⊥imp, the dynamics of the impurity shows a crossover fromone-dimensional motion in a ladder to the motion in a one-dimensional bath.

6.3.3 Green’s function for finite momentumAs we have seen previously, the Green’s function decays as a power-law in the anti-symmetric mode and exponentially in the symmetric mode at p = 0, now we turnon finite momentum. The Green’s functions |Ga(p, t)| and |Gs(p, t)| for finite mo-mentum are shown in Fig. 6.6 and 6.7, respectively, for various momenta p of theimpurity. We find that the |Ga(p, t)| decays as power-law below p = 0.3π andbeyond p = 0.3π, the |Ga(p, t)| decays exponentially and the impurity in the anti-

75


0.01 0.1 1t

1

|Ga(p

,t)|

p=0.0p=0.1πp=0.2π

0 1 2 3 4 5t

1

|Ga(p

,t)|

p=0.0p=0.1πp=0.2π

0.01 0.1 1t

1

|Ga(p

,t)|

p=0.3πp=0.4πp=0.5πp=0.6πp=0.8πp=0.9πp=π

0 1 2 3 4 5t

1

|Ga(p

,t)|

p=0.3πp=0.4πp=0.5πp=0.6πp=0.8πp=0.9πp=π

Figure 6.6 – Modulus of the Green’s function of the impurity in anti-symmetric sector(see text) |Ga(p, t)| at different momentum p (shown in inset) for the impurity on log-log scale (right panel) and semi-log scale (left panel). The Hamiltonian parameters aretb = 1, t⊥ = 1, timp = 1, t⊥imp = 3 and U = 1.0. One note a linear behavior onlog-log scale for small momenta (p = 0− 0.2π), and linear behavior for large momentap = 0.3π − π on semi-log scale.

76


1t

1

|Gs(p

,t)|

p=0p=0.2πp=0.3πp=0.4πp=0.5πp=0.7πp=0.9π

1 2 3 4 5 6t

1

|Gs(p

,t)|

p=0p=0.2πp=0.3πp=0.4πp=0.5πp=0.7πp=0.9π

Figure 6.7 – Modulus of the Green’s function of the impurity in the symmetric sector(see text) |Gs(p, t)| at different momentum p (shown in inset) for the impurity. TheHamiltonian parameters are tb = 1, t⊥ = 1, timp = 1, t⊥imp = 3 and U = 1.0 onthe log-log scale (right panel) and the semi-log scale (left panel) . One can see a linearbehavior on semi-log scale for all momenta (p = 0− 0.9π).

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1p/πa

0

0.05

0.1

1/τ(

p)

0 0.2 0.4 0.6 0.8 1p/πa

0.05

0.1

0.15

0.2

1/τ(

p)

Figure 6.8 – Inverse life-time of the impurity in both anti-symmetric (left panel) andsymmetric (right panel) sectors as a function of momentum at tb = 1, t⊥ = 1, timp = 1,t⊥imp = 3 and U = 1.0.

77


symmetric mode enters into a QP regime. A similar crossover was established inthe one-dimensional motion of an impurity in a one-dimensional bath [39] and inchapter 5 for a two-leg bosonic ladder bath [81].

Beyond p = p∗ and for small U , the Green’s function decays exponentially, theimpurity behaves like a QP and the Green’s function of the impurity in term of lifetime τ(p) is given by

|Ga(p, t)| = exp(−t/τ(p)) (6.11)

In the left panel of Fig. 6.8, we plot the inverse of life time 1/τ(p) in the anti-symmetric mode of the impurity, as a function of p for U = 1, t⊥imp = 3. As can beexpected 1/τ(p) increases with increasing p.

In Fig. 6.7, we have shown |Gs(p, t)| on the semi-log and log-log scales, we findthat |Gs(p, t)| always decays as exponentially for all momenta, and the impurity inthe symmetric band always behaves like a QP. In the right panel of the Fig. 6.8, wehave shown the inverse lifetime as a function p, it shows a non-monotonic behaviorbut overall increases with increasing p.

6.4 DiscussionOur findings suggest that the impurity exhibits a very different dynamics in the ladderbecause of the motion in the horizontal and transverse directions than one observedin chapter 5. Let us discuss the weak interaction limit. Initially, both the sym-metric and anti-symmetric modes of the impurity couple to the gapless and gappedmodes of the bath, but our numerical and analytical findings suggest that in the long-time limit the gapped mode of the bath effectively couples to both symmetric andanti-symmetric modes of the impurity and the gapless mode of the bath couples tothe anti-symmetric mode of the impurity with an effective interaction U/

√2. The

Green’s function in the symmetric and anti-symmetric modes decay exponentiallyand power-law as depicted in Fig. 6.2. The power-law exponent and the inverse life-time at zero momentum are depicted in Fig. 6.3. Both the numerical and analyticalresults show an excellent agreement as depicted in 6.3.

The transverse tunneling of the impurity is the main ingredient for the exponen-tial decay of the Green’s function of the impurity in the symmetric mode. However,the Green’s function for the anti-symmetric mode, the transverse tunneling of theimpurity, gives a prefactor in the long-time limit.

Now let us turn to infinite interaction limit. The Green’s function of the impurityin the anti-symmetric mode at zero momentum is depicted in Fig. 6.4 for varioustransverse tunnelings of the impurity and it decays as a power-law. The power-law exponent as a function of the transverse tunneling of the impurity is depictedin Fig. 6.5, which increases with increasing transverse tunneling. The power-law

78

6.4 Discussion

exponent is similar to that of one-dimensional (1D) motion of the impurity in a ladderas discussed in chapter 5, for small tunneling. While for large transverse tunneling,it is similar to the 1D motion of the impurity in a 1D bath.

For large transverse tunneling, the impurity would favor the low energy anti-symmetric mode. Hence the impurity effectively moves in 1D, which coupled to thegapless mode of the bath and shows the power-law exponent similar to a 1D bath.This description contradicts our common understanding that in a ladder, the power-law exponent should be smaller than that of 1D motion in the ladder, as discussed inchapter 5.

Now let us turn to the case of finite momentum. For small interaction and smallmomentum, the Green’s function in the anti-symmetric mode of the impurity decaysas a power-law, shown in the upper panel of Fig. 6.6. Beyond a critical momentump∗, the Green’s function is depicted in the lower panel of Fig. 6.6. The Green’sfunction decays exponentially, and the impurity enters into a QP regime, which issimilar to ones observed in chapter 5. The critical momentum is precisely equal tothat of 1D motion of an impurity in a ladder and a 1D bath. The Green’s functions ofthe impurity in the symmetric mode are depicted in Fig. 6.7, and decay exponentiallyfor all momenta, and the impurity in the symmetric mode always behaves like a QP.

79


80

CHAPTER 7

Conclusions

In this thesis, we have studied two kinds of problem. In the first problem, we havestudied the energy deposition due to the periodic modulation in time of a local po-tential in an interacting one dimensional system of bosons or fermions.

In the second category, we have investigated the effect of a quantum bath madeof a bosonic ladder on an impurity confined to move in one dimension, and alsotunnel from one leg to other legs. We have computed, both analytically and byusing TEBD, the Green’s function of the impurity as a function time for variousHamiltonian parameters. The conclusions of the time-dependent potential in theone-dimensional bath are as follows:

Using linear response and bosonization we have computed the absorbed energyboth for the regimes of a weak and a strong local potential. We find deposited energyvarying as a powerlaw of the modulating frequency, with exponents that depend bothon the interactions in the system and the nature and strength of the local potential.The results are summarized in Fig. 4.2. This behavior allows to probe the TLL natureof the one dimensional interacting systems and we have discussed the experimentalconsequences of potential experiments in ultracold gases.

Quite generally this study shows that the possibility to engineer local time de-pendent potentials allows to use the frequency of the modulation probe as a scaleto explore the various renormalization regimes that the local potential would leadto. The advantage of the technique is the fact that the modulation frequency canbe precisely controlled in a large range of scales. The drawback is clearly that themeasurement of the energy deposited is an effect varying as the inverse size of thesystem and thus difficult to measure. One could thus replace such a measurement by

81

7. CONCLUSIONS

measurements of the local density, which could potentially be achieved in systemssuch as the microscopes. Studies along that line are under way.

In the second category of problems, we have investigated the dynamics of a mo-bile impurity in a two-leg ladder bath. We have investigated the dynamics of theimpurity in two cases. In the first case, the impurity is confined to move in of thelegs of the ladder, and in the second case the impurity has freedom to tunnel fromone leg to other legs.

The conclusions of the first case are as follows:

We find that, as for a single chain bath, the presence of the bath affects drasticallythe mobility of the impurity, in a way quite different than a simple renormalization ofthe mass of the impurity via a polaronic effect. One measure of such effects is givenby the Green’s function of the impurity corresponding to the creation of an impuritywith a given momentum p at time zero and its destruction at time t. We compute thisGreen’s function both analytically using a bosonization representation of the bath, alinked cluster expansion and numerically with TEBD technique.

We find that for small momentum the Green’s function decays as a power law.For weak interaction between the impurity and the bath, the exponent is smaller thanfor a single chain, due to the gap appearing in the antisymmetric mode of the ladder.This trend would increase with the number of legs of the ladder, and reflect the trendto the more two dimensional behavior. On the other hand some aspects of the ladderare manifest in the fact that for large tunnelling in the ladder one finds an increase ofthe exponent.

We studied also the dependence of the Green’s function on the momentum of theimpurity and found, in a similar way that for the single chain that there is anotherregime than the power-law decay that appears, in which one has an exponential de-cay. The transition between these two regimes is well connected with the possibilityto excite real rather than virtual particle hole excitations in the bath.

In the second case, we have computed the Green’s function of the impurity in thebonding and anti-bonding modes of the impurity. We find that for small interaction:the Green’s function in the anti-bonding mode decays as power-law like the case inthe first case, while the Green’s function in the bonding mode decays exponentially.

Cold atomic systems could provide a good realization of the systems describedthe impurity in a ladder. On one hand bosonic ladders have been realized [82] andmore generally systems made of many bosonic tubes are routinely realizable eitherin bulk [18] or in boson microscopes. Atom chips systems provide also an excellentrealization of a ladder system [83]. Systems with bath and impurities have alreadybeen realized for single chains [58–60,84] by using two species (such as K and Rb) orinternal degrees of freedom. Combination of these two aspects should be reachablein a very near future.

82

7.1 Outlook

7.1 OutlookOur work on the local time-dependent potential could be extended for many localtime-dependent potentials in a one-dimensional bath. Another extension is for atwo-leg ladder bath to probe the excitations and gap in the anti-symmetric sector ofthe ladder.

Our work on the dynamics of the impurity in the ladder bath has several exten-sions. One of the extensions is to go from a two-leg ladder bath to a two-dimensionalbath. In chapters 5 and 6, we have considered only the bosonic ladder. What wouldhappen if the bath consists of fermions? In chapters 5 and 6, for small interaction,we have only considered the forward scattering parts of the density operator andneglected the higher harmonic. However, in the case of fermions, the first harmonicwill be more relevant than the forward scattering term. The backward scattering termwill make the Green’s function calculation difficult analytically, and in this case, ourlinked cluster expansion would not be valid. So, another possible extension is tounderstand the dynamics of the impurity in a two-leg fermionic bath. One of thepossible extension is what would happen if the impurity moves in 1D but interactwith all the modes of the bath, realization of such system could be possible in thebath formed by 173Yb Fermi gas [85]. Another possible extension is to understandthe dynamics of the impurity in the disordered ladder system.

83

7. CONCLUSIONS

84

APPENDIX A

Calculation of Energy Deposition

A.1 Linear response

We give here a brief reminder of the energy deposited in a system by a periodic mod-ulation of the form (4.4) The rate of energy deposition in a system of HamiltonianH = H0 + a0 cos(ωt)O, is given by

E =∂H

∂t

= −ωa0sin(ωt)O(x = 0, t)

(A.1)

Where H0 is a static part of Hamiltonian, a0 is the amplitude of the time dependentoscillation cos(ωt) and O is an operator coupled to the time dependent oscillationcos(ωt), within LRT, O is given by

O(t) = 〈O(t)〉H0− ia0

∫ ∞0

dt1 cos(ω(t− t1))

〈[O(t1), O(0)]〉H0

(A.2)

85

A. CALCULATION OF ENERGY DEPOSITION

By using (A.2), O is given by

O(t) = 〈O(t)〉H0− ia0

∫ ∞0

dt1 cos(ω(t− t1))

〈[O(t1), O(0)]〉H0

O(t) = 〈O(t)〉H0+ a0

∫ ∞0

dt1 cos(ω(t− t1))XR(t1)

O(t) = 〈O(t)〉H0 +a0

2[eiωtχR(−ω) + e−iωtχR(ω)]

(A.3)

where χR(t) = −iY (t)〈[O(t), O(0)]〉H0and Y (t) is the Heaviside function.

By using (A.3), (A.1) can be expressed as

E(t) = −ωa0sin(ωt)〈O(t)〉H0

− 1

2a2

0ω sin(ωt)[eiωtχR(−ω) + e−iωtχR(ω)](A.4)

We now calculate the cycle average of E(t) over the period of T = 2π/ω.

ER =1

T

∫ T

0

dtE(t)

= − 1

4ia2

0ω[−χR2 (−ω) + χR(ω)]

= −1

2a2

0ωIm[χR(ω)]

(A.5)

A.2 Effective Hamiltonian for large V

In this section, we derive the effective Hamiltonian for a system of hard core bosonsfor a large V, in power of 1/V . We consider, the site at x = 0 is connected to sites atx = −1 and x = 1 via tunnelling J , and nearest neighbour interaction Vn.

H = H1 +H2 +H3 (A.6)

Where, H1 and H2 represent the left and right parts of the Hamiltonian. H3 is givenby

H3 = −J [ψ†−1ψ0 + ψ†1ψ0 + h.c] + V ψ†0ψ0 + Vnn0(n−1 + n1) (A.7)

86


H3 in the basis of |0, 0, 0〉, |0, 1, 0〉, |0, 0, 1〉, |1, 0, 0〉, |1, 1, 0〉, |1, 0, 1〉|0, 1, 1〉, |1, 1, 1〉

H3 =

0 0 0 0 0 0 0 00 V −J −J 0 0 0 00 −J 0 0 0 0 0 00 −J 0 0 0 0 0 00 0 0 0 V + Vn −J 0 00 0 0 0 −J 0 −J 00 0 0 0 0 −J V + Vn 00 0 0 0 0 0 0 V + 2Vn

(A.8)

H3 is composed of two block diagonal matrices depending on the total number ofparticles. We examine the part with one particle and two particles.

H =

V −J −J 0 0 0−J 0 0 0 0 0−J 0 0 0 0 00 0 0 V + Vn −J 00 0 0 −J 0 −J0 0 0 0 −J V + Vn

(A.9)

H = Ht + HV , where Ht contains only tunnelling, and HV = H − Ht. In orderto find the low energy effective Hamiltonian, we make a canonical transformation ofH .

H = WHW † '(

1 + iS +(i)2

2S2

)H

(1− iS +

(−i)2

2S2

)' H + i[S, H] +

i2

2[S, [S, H]] +O(S3)

whereW = eiS . The canonical transformation does not change the energy but rotatethe state |ψ〉 to eiS |ψ〉 and we chose S such that Ht + i[S, HV ] = 0

S =

0 iJV

iJV 0 0 0

− iJV 0 0 0 0 0− iJV 0 0 0 0 0

0 0 0 0 iJVn+V 0

0 0 0 −iJVn+V 0 −iJ

Vn+V

0 0 0 0 iJVn+V 0

(A.10)

H = H1 ⊕ H2 (A.11)

87


H =

2J2

V + V 4J3

V 24J3

V 2 0 0 04J3

V 2−J2

V−J2

V 0 0 04J3

V 2−J2

V−J2

V 0 0 0

0 0 0 d+ J2

Vn+V4J3

(Vn+V )2J2

Vn+V

0 0 0 4J3

(Vn+V )2−2J2

Vn+V4J3

(Vn+V )2

0 0 0 J2

Vn+V4J3

(Vn+V )2 d+ J2

Vn+V

(A.12)

Where d = Vn + V .

H1 =

2J2

V + V 4J3

V 24J3

V 2

4J3

V 2−J2

V−J2

V4J3

V 2−J2

V−J2

V

(A.13)

H2 =

Vn + V + J2

Vn+V4J3

(Vn+V )2J2

Vn+V4J3

(Vn+V )2−2J2

Vn+V4J3

(Vn+V )2

J2

Vn+V4J3

(Vn+V )2 Vn + V + J2

Vn+V

(A.14)

Since V is very large, hence 1/V 2 ' 0

H1 '

2J2

V + V 0 0

0 −J2

V−J2

V

0 −J2

V−J2

V

(A.15)

H2 =

Vn + V + J2

Vn+V 0 J2

Vn+V

0 −2J2

Vn+V 0J2

Vn+V 0 Vn + V + J2

Vn+V

(A.16)

In the original Hamiltonian (A.8), the n0 = 0 sector is coupled to n0 = 1 sector viaJ but after the canonical transformation the n0 = 0 sector is completely decoupledfrom the n0 = 1 sector. The sector n0 = 0 represents the low energy effectiveHamiltonian. It is thus given by

H = −J2

V[|100〉〈100|+ |001〉〈001|+ |100〉〈001|+ |001〉〈100|−

2J2

V + Vn|101〉〈101|]

(A.17)

The full low energy Hamiltonian is thus

H = H1 +H2 −J2

V[|100〉〈100|+ |001〉〈001|+

|100〉〈001|+ |001〉〈100| − 2J2

V + Vn|101〉〈101|]

(A.18)

88


In second quantization (A.18) can be written as

H = H1 +H2 + b(ψ1(−a)†ψ2(a) + h.c)− c(n1(−a) + n2(a))+

Vnn1(−a)n2(a)(A.19)

Using (A.18) and (A.19) we find b = −J2

V , c = J2

V , Vn − 2c = − 2J2

V+Vn, Vn ' 0.

The bosonized form of H is given by

H =1

2π

∫ 0

−∞dx[uK(∂xθ1(x))2 +

u

K(∂xφ1(x))2

]+

1

2π

∫ ∞0

dx[uK(∂xθ2(x))2 +

u

K(∂xφ2(x))2

]+

J2

πV∂xφ1(−a) +

J2

πV∂xφ2(a)

− 2J2ρ0

V(cos(2φ1(−a)) + cos(2φ2(a)))−

2J2ρ0

Vcos(θ1(−a)− θ2(a))

(A.20)

At x = 0, the number of particles is zero, which is imposed by φ1(0) = φ2(0) =0. Now let us define H as

H = H10 +H20 +J2

πV∂xφ1(−a) +

J2

πV∂xφ2(a)

− 2J2ρ0

V(cos(2φ1(−a))− cos(2φ2(a)))−

2J2ρ0

Vcos(θ1(−a)− θ2(a))

(A.21)

Where H10 = 12π

∫ 0

−∞ dx[uK(∂xθ1(x))2 + u

K (∂xφ1(x))2]

and H20 = 12π

∫∞0dx[uK(∂xθ2(x))2 + u

K (∂xφ2(x))2].

We define the fields θ = θ√K

and φ =√Kφ. Using these field H10 and H20 can

be written as

H10 =1

2π

∫ 0

−∞dx[u(∂xθ1(x))2 + u(∂xφ1(x))2

](A.22)

and

H20 =1

2π

∫ ∞0

dx[u(∂xθ2(x))2 + u(∂xφ2(x))2

](A.23)

If we define φ, θ in term of the chiral fields φL = φ+ θ, φR = θ − φ, the constraintφ1(0) = φ2(0) = 0, imply that φL(x) = φR(−x) [5]. H10 and H20 are given in

89


term of the new fields φL and φR by

H10 =u

4π

∫ ∞−∞

dx(∂xφR1(x))2

H20 =u

4π

∫ ∞−∞

dx(∂xφL2(x))2

(A.24)

In terms of φR1 and φL2 the term ∂xφ1(−a) + ∂xφ2(a) becomes

∂x(φ1(−a) + φ2(a)) =√K∂x[φ1(−a) + φ2(a)]

=

√K

2∂x[φL1(−a)− φR1(−a) + φL2(a)− φR2(a)]

=

√K

2∂x[φR1(a)− φR1(−a) + φL2(a)− φL2(−a)]

'√K

22a∂x[∂xφR1(0) + ∂xφL2(0)]

= a√K∂x[∂xφR1(0) + ∂xφL2(0)]

(A.25)

In the same way

cos(θ1(−a)− θ2(a)) = cos( 1√

K(θ1(−a)− θ2(a))

)= cos

( 1

2√K

(φL1(−a) + φR1(−a)− φL2(a)− φR2(a)))

' cos( 1√

K(φR1(0)− φL2(0))

) (A.26)

and

cos(φ1(−a)) = cos(√K

2(φR1(a)− φR1(−a))

)cos(φ2(a)) = cos

(√K2

(φL2(a)− φL2(−a))) (A.27)

Finally we define ˜θ = φR1+φL2

2 and ˜φ = φL2−φR1

2 , and by using (A.24),(A.25),(A.26), (A.27), the Hamiltonian (A.21) can be written as

H =u

2π

∫ ∞−∞

dx[(∂x

˜θ(x))2 + (∂x

˜φ(x))2

]+

2a√KJ2

πV∂2x

˜θ(0)− 2J2ρ0

Vcos(√K

2(˜θ(a)− ˜

θ(−a)))

cos(√K

2(˜φ(a)− ˜

φ(−a)))− 2J2ρ0

Vcos( 2√

K(˜φ(0))

) (A.28)

90

A.3 Calculation of correlation functions

Since V = V0 + δV cos(ωt), and V0 δV , this means that 1/V = 1/V0(1 −δV/V0 cos(ωt)) = 1/V0 − δV/V 2

0 cos(ωt).

A.3 Calculation of correlation functionsIn this section, we compute response functions that are useful to obtain the energydeposited in the system. Two types of correlation functions are required and denotedby χR1 (ω) and χR2 (ω) . These correlation functions are defined below.

χ1(τ) = −〈Tτ cos(2φ(τ)) cos(2φ(0))〉H0

= −〈Tτ sin(2φ(τ)) sin(2φ(0))〉H0

= −1

2〈Tτe2iφ(τ)e−2iφ(0)〉H0

= −1

2e−2K log(u|τ |/α)

= −1

2(u|τ |/α)−2K

(A.29)

Where Tτ is time ordering operator and τ = it is imaginary time, t is real time.Correlation function χ1(τ) in real time is given by

χT1 (t) = −1

2(uit/α)−2K

= −1

2(i)−2K(ut/α)−2K

= −1

2(ut/α)−2K(cos(πK)− i sin(πK))

(A.30)

The retarded correlation function in real time is defined as

χR1 (t) = −iY (t)〈[cos(2φ(t)), cos(2φ(0))]〉H0

= −Y (t)[2 ImχT1 (t)]

= −Y (t) sin(πK)(ut/α)−2K

(A.31)

A.3.1 Fourier transformation of χR(t)

Fourier transformation of χR1 (t) is expressed as

χR1 (ω) =

∫ ∞−∞

dt eiωtχR(t)

= −∫ ∞

0

dt eiωt sin(πK)(ut/α)−2K

(A.32)

91


The above integral can be evaluated as∫ ∞0

dt eiωt sinh(πt/β)−2K =

22K β

2πB(−iβω

2π+K, 1− 2K) (A.33)

As β →∞, sinh(πt/β) = πt/β ⇒ β/π sinh(πt/β) = t∫ ∞0

dt eiωt(β/π)−2K sinh(πt/β)−2K =

22K−1(βπ

)−2K+1

B(−iβω

2π+K, 1− 2K

)(A.34)

Equation (A.33) can be expressed as∫ ∞0

dt eiωt(t)−2K = 22K−1(βπ

)−2K+1

B(−iβω

2π+K, 1− 2K

)' 22K−1

(βπ

)−2K+1

Γ(1− 2K)(−iβω2π

)2K−1

= (−iω)2K−1Γ(1− 2K)

= e−iπ(K−1/2)Γ(1− 2K)ω2K−1

(A.35)

Equation (A.32) can be written as

χR1 (ω) = − sin(πK)(u/α)−2Ke−iπ(K−1/2)

Γ(1− 2K)ω2K−1

Im[χR1 (ω)] = sin(πK)(u/α)−2K sin(π(K − 1/2))

Γ(1− 2K)ω2K−1

(A.36)

Response function of ∂xφ(0, 0) is given by

χR2 (t) = −iY (t)〈[∂xφ(0, t), ∂xφ(0, 0)]〉H0

χR2 (ω) = −uK2

∫dk

k2

−(ω + iδ)2 + (uk)2

Im[χR2 (ω)] = −uKπ4ω

∫dkk2δ

(−ω2 + (uk)2

2ω

)Im[χR2 (ω)] = −Kπω

2u2

(A.37)

92

A.4 Calculation of correlation functions when V0 is relevant(K < 1)

A.4 Calculation of correlation functions when V0 isrelevant(K < 1)

In this section, we will derive the two point correlation function 〈φ(q1)φ(q2)〉 in termof a local field φ(0, τ). We define the local field φ0(τ), acting at x = 0 as

φ(0, τ) = φ0(τ) (A.38)

By using (A.38), R(q1, q2) = 〈φ(q1)φ(q2)〉 can be written as

R(q1, q2) =

∫DφDφ0Dλφ(q1)φ(q2)e(−S−i

∫dτλ(τ)[φ0(τ)−φ(0,τ)])∫

DφDφ0Dλe(−S−i∫dτλ(τ)[φ0(τ)−φ(0,τ)])

(A.39)

S is action of the system, and λ is a Lagrange multiplier. Now define−S−i∫dτλ(τ)[φ0(τ)−

φ(0, τ)] in terms of the frequency(ω) and momentum(k).

I = S + i

∫dτλ(τ)[φ0(τ)− φ(0, τ)]

=1

2πKu

1

βΩ

∑q

(ω2n + u2k2)φ∗(q)φ(q)+

i[ 1

β

∑ωn

λ∗(ωn)φ0(ωn)− 1

βΩ

∑q

λ∗(ωn)φ(q)]+

2V0ρ0

∫dτ cos(2φ0(τ))

=1

2πKu

1

βΩ

∑q

(ω2n + u2k2)(φ∗(q)− iπuK

(ω2n + u2k2)

λ∗(ωn))

(φ(q)− iπuK

(ω2n + u2k2)

λ(ωn)) +πK

4β∑ωn

1

|ωn|(λ∗(ωn) +

2i|ωn|πK

φ∗0(ωn))(λ(ωn) +2i|ωn|πK

φ0(ωn))

+1

β

∑ωn

|ωn|πK

φ∗0(ωn)φ0(ωn) + 2V0ρ0


(A.40)

93


By using (A.40), (A.39) can be written as

R(q1, q2) =πKβΩu

ω2n1

+ u2k21

δq1,−q2−

π2u2K2

(ω2n1

+ u2k21)(ω2

n2+ u2k2

2)

2β|ωn1 |πK

δωn1 ,−ωn2+

π2u2K2

(ω2n1

+ u2k21)(ω2

n2+ u2k2

2)

4|ωn1||ωn2

|π2K2∫

Dφ0φ0(ωn1)φ0(ωn2

)e−1

βπK

∑ωn|ωn|φ∗0(ωn)φ0(ωn)−SI∫

Dφ0e− 1βπK

∑ωn|ωn|φ∗0(ωn)φ0(ωn)−SI

(A.41)

Where SI = 2V0ρ0


−〈∇φ(0, τ)∇φ(0, 0)〉 = −( 1

βΩ

)2 ∑q1,q2

(−k1k2)e(−iωn1τ)

〈φ(q1)φ(q2)〉(A.42)

By using (A.40), (A.42) can be written as

−〈∇φ(0, τ)∇φ(0, 0)〉 = −( 1

βΩ

)2 ∑q1,q2

(−k1k2)e(−iωn1τ)

πKβΩu

ω2n1

+ u2k21

δq1,−q2

(A.43)

Fourier transformation of (A.43) can be written as

−〈∇φ(0, τ)∇φ(0, 0)〉(ωn) = −( 1

Ω

)∑k

(k2)πKu

ω2n + u2k2

− Im(〈∇φ(0, τ)∇φ(0, 0)〉(ω)) = −Kπω2u2

(A.44)

A.4.1 Dilute instanton approximationBy using (A.40), the effective action in term of the local field φ0 is given by

S =1

βπK

∑ωn

|ωn|φ∗0(ωn)φ0(ωn) + 2V0ρ0

∫dτ cos(2φ0(τ)) (A.45)

The action diverges for the large frequency, and to overcome this problem, we add amass term 1

2M(∂τφ)2 [5, 10] to the action.

S =1

βπK

∑ωn

|ωn|φ∗0(ωn)φ0(ωn) + 2V0ρ0


+

∫dτ

1

2M(∂τφ)

2

(A.46)

94

A.4 Calculation of correlation functions when V0 is relevant(K < 1)

For a very large V0, the partition function is dominated by a trajectory, which min-imizes the action 2V0ρ0

∫dτ cos(2φ0(τ)) +

∫dτ 1

2M(∂τφ)2, and solution is given

by [5, 11]

M

2

(dφ(τ)

dτ

)2

= cos(2φ(τ))− 1 (A.47)

φ(τ) = π/2 + 2 tan−1[tanh[√

2V0ρ0/Mτ ]] (A.48)

For V0 >> 0, ∂τ φ(τ) ' δ(τ), where δ(τ) is delta function. The general solution ofthe field φ0(τ) is given by the linear combination of φ(τ),

φ0(τ) =∑i

εiφ(τ − τi) (A.49)

Where εi = ±1, and∑i εi = 0. By using (A.49) and (A.46), partition function of

the system is given by [5].

Z =

∞∑p=0

∆2p0

∑ε1=±1...ε2p=±1

∫ ∞0

dτ2p

∫ τ2p

0

dτ2p−1....

∫ τ2

0

dτ1

e2/K∑i>j εiεj log(|τi−τj |/δ)

(A.50)

Where ∆0 = e−4√

2ρ0V0M , δ is short time cutoff. Since, in the strong couplinglimit V0 >> 1, so the large number of instantons will give small contribution to thepartition function because of the pre-factor ∆0 = e−4

√2ρ0V0M , and the dominant

contribution is given by one instanton and one anti-instanton.For our calculation, we consider one instanton(ε1 = 1) and one anti-instanton(ε2 =−1). The field φ is given by

φ0(τ) = φ(τ − τ1)− φ(τ − τ2) (A.51)

R1(τ) = 〈cos(2φ0(τ)) cos(2φ0(0))〉

R1(τ) =(

1 + e−8√

2ρ0V0M

∫ β/2

−β/2dτ1∫ β/2

−β/2dτ2 cos(2φ(τ − τ1)− 2φ(τ − τ2))

cos(2φ(−τ1)− 2φ(−τ2))(|τ1 − τ2|/δ)−2/K)

(1 + e−8

√2ρ0V0M

∫ β/2

−β/2dτ1∫ β/2

−β/2dτ2(|τ1 − τ2|/δ)−2/K

)−1

' −M2

2e−8√

2ρ0V0M (|τ |/δ)−2/K

(A.52)

95


Where β = 1/T , T is temperature. By using appendix B, imaginary part of〈cos(2φ0(τ)) cos(2φ0(0))〉(ω) is given by

Im[R1(τ)](ω) = −M2e−8√

2ρ0V0M cos(π/K)

sin (π/K)Γ(1− 2/K)(δ ω)2/K−1(A.53)

96

APPENDIX B

Green’s Function of the Impurity

B.1 Linked Cluster Expansion (LCE)

We have shown in the main text that for small interaction between impurity and lad-der compared to the gap in the antisymmetric sector, the impurity effectively couplesto the forward scattering part of the symmetric sector. We give in this appendix theLCE calculation of the Green’s function of the impurity for a weak interaction U .We consider the symmetric part of (5.18) as the bath Hamiltonian, and represent itin term of the usual bosonic operators [5].

In second quantized notation (5.20) is described by

H = Hs +Himp +Hcoup

Hs =∑q

us|q|b†sqbsq

Hcoup =∑q,k

V (q)d†k+qdk(bsq + b†s−q)

Himp =∑q

ε(q)d†qdq

(B.1)

Where b† and d† are the creation operators for the bath and the impurity respectively,Himp is the tight binding Hamiltonian of the impurity , and Hcoup is interaction be-

97

B. GREEN’S FUNCTION OF THE IMPURITY

tween impurity and bath.

V (q) =U√

2

√Ks|q|2πL

exp(− |q|

2qc

)(B.2)

where qc is an ultraviolet cutoff of the order of the inverse lattice spacing.The Green’s function of the impurity is defined as

G(p, t) = −i〈dp(t)d†p(0)〉 (B.3)

By using LCE, (B.3) can be written as

G(p, t) = −ie−iεpteF2(p,t) (B.4)

Where F2(p, t) is defined as

F2(p, t) = eiεptW2(p, t) (B.5)

W2(p, t) is given by

W2(p, t) = −1

2

∫ t

0

dt1

∫ t

0

dt2

〈Tτdp(t)Hcoup(t1)Hcoup(t2)d†p(0)〉(B.6)

By employing Wick’s theorem W2(p, t) is given by

W2(p, t) = −∑q

V (q)2

∫ t

0

dt1

∫ t

0

dt2Y (t1)e−iε(p)t1

Y (t2 − t1)e−iε(p+q)(t2−t1)Y (t− t2)

e−iε(p)(t−t2)Y (t2 − t1)e−i(us|q|(t2−t1))

(B.7)

Where Y (t) is a step function, which is zero for t < 0 and one for t > 0. Y (t)changes the limit of integration of t2 and t1, and F2(p, t) is modified as

F2(p, t) = −∑q

V (q)2

∫ t

0

dt2

∫ t2

0

dt1e−iε(p)t1

e−iε(p+q)(t2−t1)

e−iε(p)(t−t2)e−i(us|q|(t2−t1))

(B.8)

98

B.1 Linked Cluster Expansion (LCE)

F2(p, t) = −∑q

∫duV (q)2

∫ t

0

dt2

∫ t2

0

dt1

e−it1ueit2u

δ(u− (ε(p)− ε(p+ q)− us|q|))(B.9)

After performing an integration over t1 and t2, F2(p, t) is given by

F2(p, t) = −∑q

∫duV (q)2

1 + iut− eitu

u2

δ(u− (ε(p)− ε(p+ q)− us|q|)) (B.10)

which can be rewritten as

F2(p, t) = −∫du

1 + iut− eitu

u2R(u) (B.11)

For small momentum ε(p) ' timpp2,

R(u) =∑q

V (q)2δ(u− (ε(p)− ε(p+ q)− us|q|)) (B.12)

We evaluate R(u) as

R(u) =1

2π

∫dqV (q)2δ(u− (ε(p)− ε(p+ q)− us|q|)) (B.13)

q’s roots inside the delta function are given by

q>± = −(p+

us2timp

)±

√(p+

us2J

)2

− u

timp(B.14)

and

q<± = −(p− us

2timp

)±

√(p− us

2timp

)2

− u

timp(B.15)

q> corresponds to q > 0 and q< corresponds to q < 0. For (p− us2timp

) < 0 ∧ u > 0,

99


R(u) = 0 while for (p− us2timp

) > 0 ∧ u > 0 one has

R(u) =1

2timp

[( p− us/(2timp)√(p− us/(2timp))

2 − u/timp

− 1)

e−|q<+ |/(2qc) +

( p− us/(2timp)√(p− us/(2timp))

2 − u/timp

+ 1)

e−|q<− |/(2qc)

](B.16)

For u < 0.

R(u) =1

2timp

[(1−

p+ us/(2timp)√(p+ us/(2timp))

2 − u/timp

)e−|q

>+ |/(2qc)+( p− us/(2timp)√

(p− us/(2timp))2 − u/J

+ 1)e−|q

<− |/(2qc)

]If (p− us

2timp) < 0,∧u < 0

R(u) ' u (B.17)

Re[F2(p, t)] ' − log(t) (B.18)

For small |(p− us2timp

)|

Re[F2(p, t)] ' −KsU2

4π2u2s

(1 +12t2impp

2

u2s

) log(t) (B.19)

leading to the Green’s function decay

|G(p, t)| = e− KsU

2

4π2u2s

(1+12t2impp

2

u2s

) log(t)(B.20)

B.2 DMRG procedureIn Fig. B.1, we show the Green’s function of the impurity for χ = 300, 400, 500, 600at zero momentum. We find that the Green’s function upturns after a certain time.This is because entanglement entropy linearly scale with time, so one needs expo-nentially large χ to get a good result. We find that with increasing χ, the upturnwhich is a numerical artifact that signals the growth of entanglement entropy withtime evolution, is pushed down.

100

B.2 DMRG procedure

10t

0.1

1

|G(0

,t)|

χ=300χ=400χ=500χ=600

Figure B.1 – Green’s function as a function of time at p=0 for hard core boson, t⊥ =1.5, U=∞, tb = 1, timp = 1 for bond dimensions χ = 300, 400, 500, 600. Withincreasing χ the upturn in Green’s function pushed down.

101


B.3 Extracting exponent and error bar from numeri-cal data

The absolute value of Green’s function of the impurity decays monotonically as afunction of time but has also oscillations. To extract the exponents, we first select allthe data (up to a maximum time) where Green’s function is maximum. We computethe slope between two neighboring data and take the average of all the slopes toobtain the exponent.

If on a log-log scale, log |G(0, tl)|′s are maxima of log |G(0, t)|, at time tl,where l = 1, 2, ..n, then the slope between time tl and tl+1 is given by βl =log(|G(0,tl))−log(|G(0,tl+1))

log(tl+1)−log(tl), and the exponent α = β1+β2+.....βn−1

n−1 . The error bar

is given by |α−β1|+|α−β2|+.....+|α−βn−1|n−1 .

To distinguish between a power-law decay and an exponential one, and determinethe critical momentum p∗ at which such a change occurs, we show in Fig. B.2 theGreen’s function of impurity at U = 1, p = 0.2π on both log-log scale and semi-logscale.

The comparison both on a log-log scale and a semi-log one of the these twobehaviors allow us to determine p∗ ∼ 0.3π at which the change of behavior frompowerlaw to exponential occurs. At p = 0.2π, the Green’s function decays buthas oscillations. To extract the exponent we select all the points where the Green’sfunction is maximum and use the above procedure to compute the error bars andexponent.

102

B.3 Extracting exponent and error bar from numerical data

0.1 1t

1

|G(p

,t)|

0 1 2 3 4 5 6 7 8t

1

|G(p

,t)|

t

1

|G(p

,t)|

1 2 3 4 5 6 7 8t

1

|G(p

,t)|

Figure B.2 – Green’s function of the impurity as a function of time for hard core bosonsat timp = tb = 1, t⊥ = 1, U = 1. The upper panel represents the Green’s function atp = 0.2π, and the lower panel the Green’s function at p = 0.3π on log-log scale (left)and semi-log scale (right). At p = 0.2π, the Green’s function is linear on a log-log scalebut deviates on a semi-log scale from linear behavior. On the other hand at p = 0.3π,the Green’s function is linear on a semi-log scale but deviates on a log-log scale fromlinear behavior. This allows us to fix p∗ ∼ 0.3π as the critical momentum at which thedecay for the impurity goes from powerlaw to exponential.

103


104

APPENDIX C

Green’s Function of the Impurity in Presence ofTransverse Tunneling

C.1 Green’s Function of a Mobile Impurity in a Two-leg Bosonic Ladder

As we have shown in the main text (see section 6.2) that for small interaction Ubetween the impurity and the ladder bath the impurity effectively coupled to the onlyforward scattering terms of the gapped and gapless modes of the bath. In this section,we give a linked-cluster expansion (LCE) expression of the Green’s function of theimpurity similar to appendix B. We express the cos(

√2θa) = 1 − θ2

a and use thecontinuity equation ∇φa(q, t) = ∂θa(q,t)

∂t . The impurity-bath Hamiltonian in the

105

C. GREEN’S FUNCTION OF THE IMPURITY IN PRESENCE OFTRANSVERSE TUNNELING

bosonized form is expressed as

H = Hs +Ha +Himp +Hcoup

Hs =∑q

v|q|b†sqbsq

Ha =1

2π

∫dx[uaKa(∂xθa)2 +

uaKa

(∂xφa)2]−

∆2a

∫dxθ(x)2

Hcoup =∑q,k

[V (q)(d†sk+qdsk + d†ak+qdak)(bsq + b†s−q)

+ U(d†ak+qdsk + d†sk+qdak)∂θa(q, t)

∂t]

Himp =∑q

εs(q)d†sqdsq + εa(q)d†aqdaq

(C.1)

Where Where U = − U√2π

, b† and d† are the creation operators for the bath andthe impurity respectively, εa(p) = −2timp cos(p) − t⊥imp, εs(p) = −2timp cos(p) +t⊥imp,Himp is the tight binding Hamiltonian of the impurity , andHcoup is interactionbetween the impurity and the bath.

V (q) =U√

2

√Ks|q|2πL

exp(− |q|

2qc

)(C.2)

The Green’s function of the impurity in symmetric and ant-symmetric sectors aredefined by

Gs(p, t) = −i〈dsp(t)d†sp(0)〉Ga(p, t) = −i〈dap(t)d†ap(0)〉

(C.3)

By using LCE, (C.3) can be written as

Gs(p, t) = −ie−iεs(p)teF2s(p,t)

Ga(p, t) = −ie−iεa(p)teF2a(p,t) (C.4)

Where F2,s(p, t) is defined as

F2s(p, t) = eiεs(p)tW2s(p, t) (C.5)

W2s(p, t) is given by

W2s(p, t) = −1

2

∫ t

0

dt1

∫ t

0

dt2

〈Tτdsp(t)Hcoup(t1)Hcoup(t2)d†sp(0)〉(C.6)

106


By employing Wick’s theorem W2s(p, t) is given by

W2s(p, t) = −∑q

∫ t

0

dt1

∫ t

0

dt2Y (t− t1)Y (t1 − t2)

Y (t2)[V (q)2e−iεs(p)(t−t1)e−iεs(p+q)(t1−t2)

e−iεs(p)(t2)e−i(v|q|(t1−t2))

+U2ua4πKa

√u2aq

2 +2π∆2

auaKa

e−iεs(p)(t−t1)e−iεa(p+q)(t1−t2)

e−iεs(p)(t2)e−i(

√u2aq

2+2π∆2

auaKa

(t1−t2))] (C.7)

Where Y (t) is a step function, which is zero for t < 0 and one for t > 0. Y (t)changes the limit of integration of t2 and t1, and F2s(p, t) is modified as

F2s(p, t) = −∑q

∫ t

0

dt1

∫ t1

0

dt2

[V (q)2eiεs(p)t1

e−iεs(p+q)(t1−t2)

e−iεs(p)(t2)e−i(v|q|(t1−t2)) +U2ua4πKa√

u2aq

2 +∆2auaKa

eiεs(p)t1

e−iεa(p+q)(t1−t2)

e−iεs(p)(t2)e−i(

√u2aq

2+∆2auaKa

(t1−t2))]

(C.8)

107


F2s(p, t) = −∑q

∫duV (q)2

∫ t

0

dt1

∫ t1

0

dt2

eit1ue−it2u[δ(u− ε(p) + ε(p+ q) + v|q|)+δ(u− ε(p) + ε(p+ q) + 2t⊥imp + v|q|)]

F2s(p, t) = −∑q

∫du

∫ t

0

dt1

∫ t1

0

dt2[V (q)2eit1ue−it2uδ(u− ε(p) + ε(p+ q) + us|q|)+

U2ua4πKa

√u2aq

2 +2π∆2

auaKa

e−it1(u−2t⊥imp)eit2(u−2t⊥imp)

δ(u+ ε(p)− ε(p+ q)−

√u2aq

2 +2π∆2

auaKa

)]

(C.9)

Re[F2s(p, t)] = −∑q

∫du[V (q)2(1− cos(ut))

u2

δ(u− ε(p) + ε(p+ q) + us|q|)+

U2ua4πKa

√u2aq

2 +2π∆2

auaKa

(1− cos(t(u− 2t⊥imp)))

(u− 2t⊥imp)2

δ(u+ ε(p)− ε(p+ q)−

√u2aq

2 +2π∆2

auaKa

)]

(C.10)

108


Similarly one can show that

Re[F2a(p, t)] = −∑q

∫du[V (q)2δ(u− ε(p) + ε(p+ q) + us|q|))

(1− cos(ut))

u2+U2ua4πKa

√u2aq

2 +2π∆2

auaKa

(1− cos(t(u+ 2t⊥imp)))

(u+ 2t⊥imp)2

δ(u+ ε(p)− ε(p+ q)−

√u2aq

2 +2π∆2

auaKa

)]

= −∫du[ (1− cos(ut))

u2R1(u, p)+

(1− cos(t(u+ 2t⊥imp)))

(u+ 2t⊥imp)2R2(u, p)

]

(C.11)

Now define ε(p) = 2timpp2 . And R(u) is expressed as

R1(u, p) =1

2π

∫dqV (q)2δ(u− (ε(p)− ε(p+ q)− us|q|)) (C.12)

R2(u, p) =U2ua

8π2Ka

∫dq

√u2aq

2 +2∆2

auaπ

Ka

δ(u+ ε(p)− ε(p+ q)−

√u2aq

2 +∆2aua2π

Ka)]

(C.13)

R1(u, p) was computed in Ref. [39,81] and in appendix B for (p− us2timp

) < 0,∧u <0. Computation of R2(u, p) for an arbitrary p is difficult so we restrict at p = 0

R1(u, 0) ∝ u (C.14)

And for u > 0

R2(u, 0) ∝ A2 (C.15)

109


Let us define As(u, t) and Aa(u, t) as

As(u, t) =

∫ ∞0

du[1− cos(ut)

u

]+∫ ∞

∆a√

2uaπ√Ka

−2t⊥imp

du[1− cos(ut)

u2A2

]Aa(u, t) =

∫ ∞0

du[1− cos(ut)

u

]+∫ ∞

∆a√

2uaπ√Ka

+2t⊥imp

du[1− cos(ut)

u2A2

](C.16)

As(u, t) = A2πt

Aa(u, t) = − log(t)(C.17)

Where A2 = U2ua8π2Ka

(−ua/(2timp)+f1)2

√(−ua/(2timp)+f1)2+∆2

a/u2af1

, f1 =

√u2a

4t2imp+

∆a+2t⊥imp

timp+∆2a/u

2a

, ∆a =

∆a

√ua2π√Ka

By using equations (C.10, C.11, C.16, C.17), the final expression ofF2s, F2a

for 2t⊥imp >∆a

√2uaπ√Ka

in long-time limit is given by

Re[F2s(0, t)] ' −A2πt (C.18)

Re[F2a(0, t)] ' − log(t) (C.19)

For 2t⊥imp <∆a

√2uaπ√Ka

Re[F2s(0, t)] ' − log(t) (C.20)

Re[F2a(0, t)] ' − log(t) (C.21)

leading to the Green’s function decay

|Gs(0, t)| = e−A2πt (C.22)

Similarly following above procedures Ga(p, t) is given by

|Ga(0, t)| = e−KsU

2

4π2v2 log(t) (C.23)

In the symmetric mode of the impurity the Green’s function decays exponentiallybut in the anti-symmetric mode the Green’s function decays as power-law.

110

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