University of Groningen

Phonons, charge and spin in correlated systemsMacridin, Alexandru; Sawatzky, G.A

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.

Document VersionPublisher's PDF, also known as Version of record

Publication date:2003

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):Macridin, A., & Sawatzky, G. A. (2003). Phonons, charge and spin in correlated systems. s.n.

CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.

Download date: 24-06-2021

https://research.rug.nl/en/publications/phonons-charge-and-spin-in-correlated-systems(d5eafdc2-d500-4fd2-aab0-d8a0d95ad2f0).html

Chapter 3

Holstein Polaron

3.1 Introduction

The concept of a polaron was introduced by Landau [1] to describe an electron moving in apolarizable lattice carrying the lattice deformation with it. In a more general context, theterm of polaron describes a quantum particle interacting with a bosonic environment. Thebosonic modes dress the particle, changing its properties like the energy and the effectivemass. Recently, the environments and particles encountered in condensed matter physicshave become more diverse, so for example the polaron concept is also used now to describethe interaction of a hole or of an electron with spin or orbital excitations.

The polaron properties are dependent on both the particle-environment couplingstrength and polaron momentum. At zero momentum and at small coupling the par-ticle is lightly dressed, polarizing weakly the environment over a large spatial extentaround itself. Upon increasing the coupling, the particle changes its properties more orless abruptly depending on the environment modes (phonons) characteristic frequency,becoming extremely heavy and deforming the environment strongly over a small spatialextent around itself at large coupling. In the cuprates, manganates and most of the othercorrelated materials the electron-lattice interaction was found to be in the intermediateregion where the transition from light to heavy polaron takes place. The analytical cal-culations based on perturbation theory fail in this region. Therefore, recently a largevariety of non-perturbative approaches like exact diagonalization on small clusters [2–6],Quantum Monte Carlo simulations [7–11], dynamical mean field theory [12], variationaltechniques [13–16], density matrix renormalization group [17] and exact diagonalizationon a variational determined Hilbert subspace [18] have been applied, giving more or lessaccurate results for different regions of parameter space. However the transition region isnot well understood, and aside from the ground state properties little is known about thestable excited states which exists at intermediate coupling.

At small and intermediate coupling the polaron at zero momentum is fundamentallydifferent from the one at large momentum. If at zero momentum the free electron con-figuration has the most significant contribution, at large k the configuration with thehighest weight is an electron plus a phonon which carries almost the entire momentum.Consequently at large momentum the quasiparticle weight is extremely small and the dis-persion becomes flat, having the phonon characteristic shape. Perturbation theory fails

31

32 Chapter 3. Holstein Polaron

to describe the large momentum polaron even at small coupling (although it works wellfor the zero momentum state).

In this chapter we address the polaron problem using a Quantum Monte Carlo algo-rithm. We calculate the electron and the electron plus an arbitrary number of phononimaginary time Green’s functions by expressing them as a sum of integrals with an everincreasing multiplicity number and applying the general approach introduced in Chap-ter 2. Every term in the sum can be represented by a Feynman diagram, this being thereason why the algorithm is called the Diagrammatic Quantum Monte Carlo. We extractthe polaron properties from the long time behavior of the Green’s functions, where theground state is projected out. The method was developed by Prokof’ev et al. [7, 8] andapplied to the Fröhlich polaron. It addresses the problem in the thermodynamic limit(infinite lattice) and it proves to give accurate results (better than 1% accuracy) for allregions of the parameter space provided that the binding energy is not extremely small.Coupled with the analytic-continuation spectral analysis algorithm [8] information aboutthe excited states can be obtained too.

The model Hamiltonian we consider is the popular Holstein one which assumes anon-site coupling of the electrons with a dispersionless lattice vibration mode

H = −t∑

〈ij〉(c†icj +H.c) + ω0

∑i

b†ibi + g∑

i

ni(b†i + bi) (3.1)

Here c†iσ(ciσ) is the creation (annihilation) operator of an electron at site i, and analoguesb†i , bi are phonon creation and annihilation operators. The first term is the electrontight-binding kinetic energy. The second term describes the lattice degrees of freedomconsidered as a set of independent oscillators at each site, with frequency ω0. The electronscouple through the density niσ = c

†iσciσ to the local lattice displacement xi ∝ (b†i + bi)

with the strength g and this interaction is described by the last term of Eq. 3.1. In themomentum representation Eq. 3.1 becomes

H =∑

k

ε(k)c†kck + ω0∑

q

b†qbq +g√N

∑

k,q

c†k+qck(b†−q + bq) (3.2)

with

ε(k) = −2tz∑

δ=1

cos(kδ) (3.3)

where z is the dimensionality of the problem. The model is characterized by two rela-tive parameters, the dimensionless coupling constant α = g2/ztω0 (see Sec. 3.2.2 for thedefinition) and the adiabaticity ratio ω0/zt.

3.2 Analytical Approaches

Despite its apparent simplicity the model can not be solved analytically. A frequentlyused approach is the adiabatic approximation which considers the ion mass infinitelylarge (which would result in ω0 = 0), thus reducing the problem to an electron interacting

3.2. Analytical Approaches 33

one phononone electron and

k

E(k)

free electron

ω0

K

Figure 3.1: Schematic plot of the Holstein Hamiltonian states when g = 0. The full linerepresent the free electron dispersion. The dashed and the dotted lines are states withone phonon and one electron. There are N (i.e. a infinity in the thermodynamic limit)such states, one for every phonon, shifted on the horizontal axis with the correspondingphonon momentum. There are also (not shown here) states with more phonons whichwould be shifted on the vertical axis by ω0 times the number of phonons.

with a static field produced by a deformed lattice. The lattice configuration is deter-mined self-consistently by minimizing the system energy. Unrealistic results, like a firstorder transition from a mobile to a localized self-trapped state, appear as a result of suchapproximation. We are not going to discuss the adiabatic approximation. The perturba-tive calculations are analytical approaches which consider dynamical phonons but theirvalidity is restricted to the extreme cases of weak and strong electron-phonon coupling.

3.2.1 Weak-Coupling Perturbation

When the electron-phonon interaction is weak the last term of Eq. 3.2 can be treated assmall perturbation. A schematic representation of the situation is presented in Fig. 3.1.Here the dispersion of a few states is shown for g = 0. When g is switched on, thefree electron state will mix with a continuum of states situated vertically (momentumconservation) above the energy (see for example the relevant states for K marked inFig. 3.1) ≈ ω0 − ε(k) + ε(0).

At small momentum the electron will be lightly dressed by the phonons and its effectivemass will increase slightly. The lattice is deformed weakly on a large extent around theelectron and this state is called large polaron. The best perturbation theory schemedescribing the large polaron at small momentum is the ordinary Rayleigh-Schrödinger

34 Chapter 3. Holstein Polaron

+ +

+ . . . .

Σ =

+ =

Figure 3.2: Self-Consistent-Born calculation of the self-energy.

one, where the first order energy correction is

ε(1)RS(k) = −

1

N

∑q

g2

ω0 + ε(k − q)− ε(k) (3.4)

The reason why this approach works so well is related to the fact that there alwaysis a continuum of states (corresponding to zero momentum polaron plus a free phonon)starting at the energy E0+ω0. E0 is the zero momentum polaron energy. Eq. 3.4 takes thisenergy separation properly in the denominator unlike the Wigner-Brillouin (or Green’sfunction) perturbation theory scheme

ε(1)WB(k) = −

1

N

∑q

g2

ω0 + ε(k − q)− ε(k)− ε(1)WB(k)(3.5)

which always overestimates the denominator and therefore produces a smaller energycorrection [19].

However when the momentum is increased the Rayleigh-Schrödinger perturbation the-ory fails, strongly overestimating the energy correction term. This will result in an un-physical maximum in the energy dispersion as it is shown with dashed line in Fig. 3.12.The failure of perturbation theory at large momenta happens in other polaron modelstoo, as for example in the Fröhlich polaron model [7, 20, 21]. At large momentum theelectron energy is very close to or degenerate with the electron plus phonon continuum.This will cause the Rayleigh-Schrödinger perturbation theory suitable for discrete levels tofail. The Green’s function technique is more appropriate for describing interactions withcontinuum states. With this scheme the unphysical downturn in the energy dispersiondisappears. However in first order the results are still not satisfactory. Now the stateswith more than one phonon are not considered and as a result the continuum energywill start at ε(0) + ω0, containing one electron (instead of one polaron) plus one phonon.Higher order calculations are required for acceptable results. From this point of viewthe best analytical approximation is the Self-Consistent-Born-Approximation (sometimesalso called Hartree-Fock or Non-Crossing approximation) which consists in solving thefollowing system of equations:

Σ(k, ω) = 1N

∫dΩ

∑q g

2D0(q,Ω)G(k − q, ω − Ω) (3.6)

G−1(k, ω) = ω − ε(k)− Σ(k, ω) (3.7)where G(k, ω) and D0(q,Ω) are the full electron and the bare phonon propagators. Here aninfinite set of particular diagrams (the non-crossing ones, see Fig. 3.2) which can contain

3.2. Analytical Approaches 35

from one to an infinite number phonons is summed up. As a result the continuum ofstates will have the right energy.

At small momenta the Self-Consistent-Born-Approximation results are poorer thanthe Rayleigh-Schrödinger perturbation theory ones, still underestimating the energy cor-rection. One should expect a similar underestimation of the Self Consistent Born Approx-imation energy correction at large momenta and therefore an exact solution is desirable.

3.2.2 Strong-Coupling Perturbation

If the electron-phonon interaction is strong, the hopping term in the Eq. 3.1 can be treatedas a perturbation. The last two terms can be diagonalized using the Lang-Firsov canonicaltransformation [22]. This is obtained via the unitary operator eS, where

S = − gω0

∑i

ni(b†i − bi) (3.8)

Using formula

à = eSAe−S = A+ [S,A] +1

2[S, [S,A]] + .. (3.9)

the transformed operators become

b̃i = bi +g

ω0ni (3.10)

c̃i = ci egω0

(b†i − bi) (3.11)

The physical meaning of this canonical transformation is a shift of the ions equilibriumposition at the sites where the electron is present.

〈x̃i〉 = 〈b̃†i + b̃i〉 = 〈b†i + bi +2g

ω0ni〉 = 〈xi〉+ 2g

ω0〈ni〉 (3.12)

The Hamiltonian written in the new basis is

H = Ht +H0 (3.13)

with

H0 = ω0∑

i

b̃†i b̃i −g2

ω0

∑i

ñi, (3.14)

Ht = −t∑

〈ij〉,(c̃†i c̃jX

†iXj +H.c) (3.15)

and

Xi = e− g

ω0(b̃†i − b̃i) (3.16)

As can be seen from the second term of Eq. 3.14, the lattice deformation energy gaineddue to the electron presence is

Ep = g2/ω0 (3.17)

36 Chapter 3. Holstein Polaron

The dimensionless electron-phonon coupling constant should be defined as the ratio be-tween this energy and the bare electron kinetic energy which is proportional to the hoppingt and with the lattice dimensionality z. We define it as ∗ †

α =g2

zω0t(3.18)

The electron hopping is accompanied by the changing of ions equilibrium position (seethe term X†iXj in Eq. 3.15).

The Hamiltonian H0 produces a N -degenerate ground state, every state consisting ofa localized electron at a particular site. In the zeroth order the translational symmetry isbroken, the electron being “trapped” by the lattice deformation. The first order correctionlifts the degeneracy, resulting in an exponentially reduced nearest-neighbor hopping

teff = t 〈i|c̃†i c̃jX†iXj|j〉 = t e− g2

ω20 = t e−αzt

ω0 (3.19)

The second order perturbation theory has a much stronger effect. It corresponds to avirtual transition of the electron without carrying its lattice deformation to a nearest-neighbor location. The intermediate state will have an energy equal to 2Ep, because itcontains a site with deformation and without the electron and a site with the electronand without the deformation. Therefore the second order correction will be

E(2) = −2z t2

2Ep= − t

α(3.20)

Explicit calculation of the matrices involved in the first and the second order perturbationtheory can be found in [24]. Summing up Eq. 3.19 and Eq. 3.20 the polaron dispersionresults in

E(k) = −αzt− tα− 2teff

z∑i=1

cos(ki) (3.21)

The exponentially reduced teff implies an exponentially large effective mass

m∗ = m eαztω0 (3.22)

The number of phonons (in the initial basis), Nph, and the quasiparticle weight (the frac-tion of free electron configuration in the polaron state), Z0, can also be easily calculated.In zeroth order we get

Nph = 〈b†ibi〉 = 〈(b̃†i −g

ω0)(b̃i − g

ω0)〉 = g

2

ω20=αzt

ω0(3.23)

∗If the approximation N(0) ≈ 1/2W where W is the electron bandwidth is considered, the relationto the electron-phonon coupling constant defined in Section 1.3 will be α = 2λ.

†Other authors define the dimensionless coupling constant as the ratio between the deformationenergy and phonon frequency [23], which indicates the number of phonons in the polaronic cloud (seeEq. 3.23).

3.3. Diagrammatic Quantum Monte Carlo Algorithm 37

and

Z0 = |〈Xi〉|2 = e−g2

ω20 = e−αzt

ω0 (3.24)

To conclude, for large electron-phonon coupling, the polaron is a heavy particle withan exponentially large mass and an exponentially small quasiparticle weight. Within avery small range around the electron the lattice is strongly deformed, for this reason thestrong-coupling polaron is also called small polaron.

3.3 Diagrammatic Quantum Monte Carlo Algorithm

Let’s consider the equation

〈ψ|e−τH |ψ〉 =∑

ν

|〈ψ|ν〉|2e−τEν (3.25)

where |ψ〉 is a whatever state and {|ν〉} form the complete set of the eigenstates withenergy Eν . We see that at large τ time Eq. 3.25 converges to

|〈ν0|ψ〉|2e−τEν0 (3.26)

where |ν0〉 is the ground state of the system. Suppose the ground state is separated fromthe first excited state by a gap ∆. We can obtain the ground state energy and the overlapof the ground state with |ψ〉 with an accuracy better than 1% (for example) calculatingEq. 3.25 at a time τ ≈ 5/∆.

If we take |ψ〉 ≡ |k〉, where |k〉 is the free electron with momentum k state, thecalculation of Eq. 3.25 will result in the calculation of the zero temperature MatsubaraGreen’s function

G(k, τ)def= 〈0|c†k(τ)ck|0〉 = 〈k|e−τH |k〉 =

∑ν

|〈k|νk〉|2e−τEν(k) (3.27)

which at large time will provide the energy E(k) (≡ Eν0(k), the lowest energy in the kirreducible channel) and the quasiparticle weight of the polaron with momentum k

Z0(k) = |〈ν0k|k〉|2 (3.28)

As we are going to show, aside from the electron Green’s function, our algorithm allowsas to calculate at the same time the n-phonon correlation functions

P n(k, τ) =∑

q1,q2,...,qn

〈0|ck−q1−q2−...qn(τ)bq1(τ)bq2(τ)...bqn(τ)b†qn ...b†q2b†q1c†k−q1−q2−...qn |0〉

(3.29)from which we can extract

Zn0 (k) =∑

q1,q2,...,qn

|〈ν0k|b†qn ...b†q2b†q1c†k−q1−q2−...qn|0〉|2 (3.30)

i.e. the contribution of the n-phonon configurations to the polaron state.

38 Chapter 3. Holstein Polaron

τ1 τ2τ=0 τ

q

kk k−q

Figure 3.3: A typical diagram which represents a term in in Eq. 3.37. The weight ofthis diagram determined with the rules presented in Appendix 3.6.1 is D(q, τ1, τ2; k, τ) =(g dτ)2 dq e−ε(k)(τ1) e−ε(k − q)(τ2 − τ1) e−ω0(τ2 − τ1) e−ε(k)(τ − τ2).

The total number of phonons in the polaronic cloud will be

Nph(k) = 〈ν0k|∑

q

b†qbq|ν0k〉 =∑

n

nZn0 (k) (3.31)

In principle, calculating the polaron energy for many values of momentum around k =0 the effective mass (1/m∗ = ∂2E(k)/∂k2) can be determined. A more effective wayto compute the polaron mass is by making use of the energy estimator introduced inAppendix 3.6.2.(Eq. 3.61).

Now we are going to show that the one-electron Green’s function, Eq 3.27, (and similarthe n-phonon Green’s function, Eq 3.29) can be written in the general form of Eq. 2.16,therefore allowing us to apply the general Diagrammatic Quantum Monte Carlo techniquediscussed in Chapter 2. Let’s start with the Hamiltonian (3.2) and consider

H = H0 +H1 (3.32)

with H1 being the electron-phonon interaction term

H1 =g√N

∑

k,q

c†k+qck(b†−q + bq) (3.33)

The imaginary time evolution operator can be written as

e−τH = e−τH0S(τ) (3.34)with

S(τ) =∞∑

n=0

(−1)nn!

∫ τ0

...

∫ τ0

dτ1...dτnT [H1(τ1)...H1(τn)]

(3.35)

where H1 in the interaction picture is

H1(τ) = eτH0H1e

−τH0 (3.36)Using Eq. 3.34 and Eq. 3.35 the Green’s function (Eq. 3.35) will be

G(k, τ) = e−ε(k)τ ∞∑

n=0

(−1)nn!

∫ τ0

∫ τ0

...

∫ τ0

dτ1...dτnT [〈k|H1(τ1)H1(τ2)...H1(τn)|k〉](3.37)

3.3. Diagrammatic Quantum Monte Carlo Algorithm 39

q1

q1

k+q1

k−q+

q1

k

q

τ=0 τ

kk−q

Figure 3.4: A diagram which contributes to the 1-phonon Green’s function, P 1(k, τ).

which is formally similar to Eq. 2.16, i.e. we reduced the calculation of the Green’s functionto the calculation of a series of integrals with an ever increasing number of integrationvariables.

It can be easily shown that every term in Eq. 3.37 can be represented by a diagram. Asimple set of rules can be derived to determine the value (the weight) of every particulardiagram. An example is given in Fig. 3.3. The algorithm generates stochastically allthe diagrams, according to their probability. The rules used for the determination ofdiagrams weight and the practical implementation of the Diagrammatic QMC updatesare presented in Appendix 3.6.1.

In order to determine the polaron energy we need to calculate the behavior of theGreen’s function at large imaginary time, therefore we need knowledge of G(k, τ) inmany τ points. If a Monte Carlo run is necessary for every τ point the algorithm is notvery efficient. However we can generate in the same run diagrams at all times (length)between zero and a τmax chosen large enough to project out the ground state properties.Further, we notice that the one-electron Green’s function (Eq. 3.27) is a particular casecorresponding to n = 0 of n-phonon Green’s function, P n(k, τ) (Eq. 3.29). Therefore,we generate in one run all the possible diagrams with length in the interval (0, τmax) andwith the number of external phonons between 0 and Nmaxph . An example of a diagramwith one external phonon is given in Fig. 3.4.

When generating the diagrams, the momentum of the internal and external phononpropagators is chosen randomly from a continuum set of values situated in the BrillouinZone. Thus our code addresses the problem in the thermodynamic limit. The length ofthe generated diagrams can also take a continuum set of values in the interval (0, τmax).The Green’s functions, the energy, the effective mass and the phonon distribution arecomputed using the estimators introduced in Appendix 3.6.2.

In Fig. 3.5 we show a typical result of the Diagrammatic QMC algorithm applied toour problem. The sum of all possible diagrams with length τ gives

P (k, τ) =∞∑

n=0

P n(k, τ) =∑

ν

e−τEν(k) τ→∞−→ e−τE(k) (3.38)

Because of the exponentially small probability of large time diagrams, it is more efficientto generate diagrams multiplied by a factor eτµ, where µ is chosen close to the value ofthe polaron energy, E(k). We plotted logarithm of P (k, τ)eτµ and of G(k, τ)eτµ versusimaginary time. Notice that at long time convergence is reached. The extrapolation ofG(k, τ) at zero gives the quasiparticle weight Z0(k). An important remark should bemade about the strong drop seen in P (k, τ) at short time. This is due to the fact that we

40 Chapter 3. Holstein Polaron

0 10 20 30 40 50Imaginary time

-2

-1.5

-1

-0.5

0ln(P( )*exp( ))

ln(G( )*exp( ))

ln(Z

) 0ω = 0.5 tα = 2.645µ = -3.156 tE = -3.155 tZ = 0.1190

µ∗τ

µ∗τ

τ

τ

1 dimensional case

Figure 3.5: A typical example of the Diagrammatic QMC computed Green’s functions.

generate only connected diagrams (diagrams where the phonon propagators are alwaysglued to the electron propagator). The disconnected diagrams have an exponentiallysmall contribution at large time, therefore they can be neglected, but at small time theiromission will result in a strong potential drop which will not allow an efficient samplingfor both long and short time diagrams. This problem can be eliminated using a fictitiouspotential renormalization [8], which means a proper choice of function g(x) in Eq. 2.15.The neglect of the disconnected diagrams makes also possible the normalization

P (k, τ) = G(k, τ) = 1 (3.39)

3.4 Results of the QMC Calculation

Related to low-dimensional physics of strongly correlated materials we study both theone-dimensional and the two-dimensional case. We study the polaron properties as afunction of both electron-phonon coupling and its momentum and look at the influenceof phonon frequency ω0 on these properties. In the subsequent calculations we take thehopping t to be the unit of energy.

3.4.1 Ground State Properties

In this section we focus on the intermediate electron-phonon coupling physics which ischaracterized by the transition from the weak-coupling light state to the strong-couplingheavy one. It was proved mathematically [25, 26] that this transition is analytic, thus noabrupt changes in the polaron properties are expected to happen in the transition region.

3.4. Results of the QMC Calculation 41

0 1 2 3 4 5α

-10

-8

-6

-4

-2

Ene

rgy

(eV

)

1D, = 0.1 t1D, = 0.5 t2D, = 0.5 t2D, = 1.0 t

ωω

ωω

second order PT

second order PT

Figure 3.6: Ground state polaron energy versus electron-phonon coupling in the one-dimensional (diamonds) and two-dimensional (circle) cases. The solid line correspondsin both cases to smaller adiabaticity ratio ω0/zt. The dotted line is the second orderperturbation theory result (Eq. 3.21).

0 1 2 3 4 5α

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Z

1D, =0.1 t1D, =0.5 t2D, =0.5 t2D, =1.0 t

ωωωω

0

Figure 3.7: The quasiparticle weight Z0 (Eq. 3.28) versus electron-phonon coupling inthe one-dimensional (diamonds) and two-dimensional (circle) cases. The solid line corre-sponds in both cases to smaller adiabaticity ratio ω0/zt.

42 Chapter 3. Holstein Polaron

0 1 2 3 4 5α

0

2

4

6

8

10

12

14

16

18

20

1D, =0.1 t1D, =0.5 t2D, =0.5 t2D, =1.0 tω

ωωω

Nph strong-coupling PT

strong-coupling PTstrong-coupling PT

Figure 3.8: The average number of phonons in the phononic cloud versus electron-phononcoupling for the one-dimensional (diamonds) and two-dimensional (circle) cases. Thedotted line are the strong coupling perturbation theory results (Eq. 3.23)

0 1 2 3 4 5α

0

1

2

3

4

5

6

7

8

ln(

m*/

m )

1D, =0.1 t1D, =0.5 t2D, =0.5 t2D, =1.0 t

ωωωω

strong coupling PT

Figure 3.9: Logarithm of the polaron effective mass versus electron-phonon coupling inthe one-dimensional (diamonds) and two-dimensional (circle) cases. The dotted line is thestrong coupling perturbation theory result (Eq. 3.22) corresponding to both (1D,ω0 = 0.5)and (2D,ω0 = 1) cases.

3.4. Results of the QMC Calculation 43

A popular interpretation of the large to small polaron transition is the following [14, 15,27–29]. There are two variational kinds of polaron states, one having the characteristicsof the large polaron described by the weak-coupling perturbation theory and the otherbeing the heavy state which distorts the lattice strongly around itself and correspondingto the strong-coupling regime. In the intermediate coupling regime the two proposedstates are close in energy. The hybridization matrix element between these two statesis different from zero, thus the states mix with each other. Therefore the transitionphysics is a crossover between the large and the small polaron states. One consequenceof this interpretation is the existence of a second polaron state in the crossover region.For the one dimensional case one more polaron state in the intermediate coupling regionwas obtained by Bonc̃a et al. [18] using a technique based on exact diagonalization ofa variational determined Hilbert space, and by A. Mishchenko analyzing our QMC datawith a novel imaginary-time analytic continuation method [8]. The agreement of V.Cataudella et al. ground state polaron variational calculation [14], which assumes thisscenario, with other methods seems to indicate that the crossover assumption is good.However, as we discuss below, this is not the whole story, and, in order to explain thetransition region, one should either consider a crossover of more than two states or evenabandon the crossover interpretation and find something else.

The spectral calculation of the similar Rashba-Pekar model [30] shows more than onestable excited state in the crossover region. A similar situation is encountered even forthe one-dimensional Holstein polaron case when the adiabaticity ratio is small [31], thusdisputing the two states only crossover supposition. Bonc̃a et al. [18] showed that the firstexcited state of the transition region approaches asymptotically the continuum when theelectron-phonon coupling is increased. In the strong-coupling limit this state is a weaklybound state of a localized small polaron and one more phonon. Thus, the phonon numberof this first excited state at large electron-phonon coupling is (see also Eq. 3.23)

N1ph = 〈b†ibi〉 = 〈b̃†i b̃i〉+αzt

ω0= 1 +

αzt

ω0(3.40)

On the other hand, if the crossover assumption is correct this state should be the con-tinuation of the light polaron to large α and thus should be characterized by a relativelysmall number of phonons and it is evident that this is not happening.

One solution to this problem is to assume a crossover of more than two states. Besidesthe large polaron and the small polaron states some other states above the continuumthreshold can be involved too. The mixing with each other and with the large polaronand small polaron states can be strong in the transition region, resulting in more than onestable excited states. However the transition region physics is not well understood andfurther investigation is required, especially with respect to the excited states properties.One efficient way to determine them is to do the spectral analysis of the data obtainedwith Diagrammatic Quantum Monte Carlo. Nevertheless, besides showing one particularcase which proves the existence of another stable polaron state, our results are strictlyrelated to the ground state properties only.

The ground state energy of the Holstein polaron model as a function of electron-phonon coupling α for different values of ω0 is shown in Fig. 3.6. Notice that the slopechanges around αc = 2. Thus the transition takes place when the lattice deformation

44 Chapter 3. Holstein Polaron

0 5 10 15 20n

0

0.2Z(n

) =2.102 =2.25 =2.89

0 5 10 15 20

0

0.2

0.4Z

(n)

=2.00 =2.645 =3.92

1D, =0.5 t

2D, =0.5 tααα

α

α

ω

α

ωa)

b)

Figure 3.10: Phonon distribution, Z(n) (Eq. 3.30), in the transition region for 1-dimensional (a) and 2-dimensional (b) case at different electron-phonon couplings.

0.0 0.5 1.0 1.5 2.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

=2.645α= −3.145

= 0.42−

1D

ω

=0.5 tω

E

EE1

0arbr

itrar

y un

its 0

0

0

Z =0.28

Z = 0.119

10

Figure 3.11: Spectral function (Eq. 3.41) calculated by analytical continuation of QMCdata. One-dimensional case in the transition region. Two stable states exist below thecontinuum which begins at 0.5 t.

3.4. Results of the QMC Calculation 45

energy becomes approximatively equal to the bare electron bandwidth. The smaller isthe adiabaticity ratio the sharper is the transition, but it is always smooth, as it shouldbe. The quasiparticle weight Z0 (Eq. 3.28) is shown in Fig. 3.7. The one-dimensional andthe two-dimensional cases behave differently. In the one-dimensional case the electrongets dressed rapidly when α is increasing and the smaller is ω0 the faster Z0 decreases.In the two-dimensional case, at small α, the dressing is much smaller and independent ofω0. In the transition region Z0 drops rapidly. We notice again that when ω0 is small thetransition is very sharp (resembling a first order transition). Similar conclusions can bedrawn from Fig. 3.8 where the number of phonons in the polaronic cloud is presented. InFig 3.9 the polaron effective mass on a logarithmic scale is shown. Again, the sharpesttransition happens in the two-dimensional case at small phonon frequency. Notice thateven for our largest α the effective mass is not very close to the strong-coupling value,showing that we are still in the intermediate region. However the other properties (energy,average number of phonons) are much closer to the perturbation theory value.

In Fig 3.10 we show how the phononic cloud evolves with the coupling strength inthe transition region. Before the transition, the largest contribution to the polaron stateis the free electron configuration. After transition, at large α, the phonon distributionis Gaussian centered on Nph (Eq. 3.31). A fundamental difference is observed betweenthe one-dimensional and the two-dimensional case. Unlike the 1D case, in the transitionregion the 2D polaron exhibits a phonon distribution with two maxima, one at zero andthe other corresponding to a large phonon number. We observed the same feature evenat higher phonon frequency (at ω0 = 1, not shown), but less pronounced. In the anti-adiabatic limit (large ω0) the two-peak structure disappears. Similar characteristics werealso noticed in other polaron models [30]. This peculiar distribution of phonons in thepolaronic cloud suggests that the transition region polaron is a mixture of a state wherethe free electron configuration has the most significant contribution and another one witha large number of phonons Gaussian distributed. In V. Cataudella et al.’s calculation [14]the variational space consists of two kinds of states, states with phonon distributioncharacteristic of the large polaron and states with phonon distribution characteristic ofthe small polaron. The agreement of their one-dimensional and two-dimensional resultswith the exact ones is remarkable, proving that the polaron at the intermediate couplingis a mixture of large and small polarons. This is true even in the one-dimensional caseand the absence of the zero-maximum in the phonon distribution we believe is due tothe fact that the one-dimensional large polaron state has a small Z0 at the intermediateelectron-phonon coupling, according to the results shown in Fig. 3.7. However the abovestatement does not say anything about the excited states which cannot be described onlyas a superposition of large and small polarons presumably because they are made-up alsowith states from the continuum.

Preliminary calculations show that the number of the stable excited states is stronglydependent of the adiabaticity ratio and lattice dimension. We did not study this problemsystematically. In Fig. 3.11 we show only one example of such calculation, for the one-dimensional case, at the adiabaticity ratio ω0/t = 0.5. The quantity plotted is the spectralfunction at k = 0 defined as:

S(k, ω) =∑

ν

|〈ν|c†k|0〉|2δ(ω − Eν(k)) =∑

ν

Z0νδ(ω − Eν(k)) (3.41)

46 Chapter 3. Holstein Polaron

Two polaron states exist below the continuum, both of them having a pretty large quasi-particle weight (Z00,1 > 0.1).

3.4.2 Momentum Dependent Properties

Even for weak electron-phonon interaction the polaron problem is far from being trivial.The difficulty occurs at large momenta where the perturbation theory fails. The best an-alytical results at large momenta are obtained by summing all the non-crossing diagramsin the calculation of the self-energy. However this approximation is still poor underesti-mating the energy correction. Therefore, the exact solution obtained with DiagrammaticQMC is of extreme importance.

In Fig. 3.12 we present the energy dispersion for the one-dimensional polaron at asmall electron-phonon coupling. When the polaron momentum is small, the Rayleigh-Schrödinger perturbation theory works well, even in first order. Close to the momentumvalue where the bare electron energy reaches the phonon energy (ε(k) − ε(0) = ω0), theperturbation theory fails, producing an unphysical peak in the polaron dispersion (seethe dashed line). The QMC results show that, in fact, the polaron energy continues toincrease and the closer it gets to the threshold value E(0) + ω0, corresponding to thecontinuum, the flatter the dispersion becomes. The dispersion is weakly renormalized atsmall momenta until the polaron energy gets close to the continuum threshold from wherethe dispersion becomes suddenly flat. The polaron at large k is a weakly bound state.In the intermediate coupling region the same flattening of the polaron dispersion at largek is seen but now the binding energy is large everywhere in the Brillouin Zone (see theinset).

Not only the dispersion but also the quasiparticle weight dependence on k, presentedin Fig 3.13-a, shows that the polaron at large momentum is fundamentally different fromthe one at small momentum. At small coupling (solid line) and small k the quasiparticleweight is large, the polaron being a lightly dressed electron. From Fig 3.13-b it can beseen that at large k the number of phonons in the polaronic cloud is larger by almost one.Here the quasiparticle weight is vanishing small. The reason is that the polaron becomesmostly a one electron plus one phonon state which is characterized by a zero Z0(k) and aflat dispersion, now the momentum being entirely carried by the dispersionless phonon.However for larger α (dashed line), the number of phonons at large k is larger by morethan one than the corresponding number at k = 0, showing that states with two and morephonons also participate in the electron dressing process.

The two-dimensional polaron behavior is similar. The situation is presented in Fig. 3.14.At the zone boundary the polaron is weakly bound and the dispersion is flat. The quasi-particle weight is large at the zone center and is decreasing rapidly with increasing k.Because we are not in the very small coupling regime as in the previous one-dimensionalcase, the number of phonons at large k is larger by more than one than the correspondingk = 0 number.

In both the one- and the two-dimensional cases we noticed that the number of phononsand the dispersion at large momenta are related. In the extremely weak coupling regimethe dispersion is close to the perfectly flat shape and the number of phonons equalsthe number of phonons at k = 0 plus one. When the coupling is increased the large k

3.4. Results of the QMC Calculation 47

0 0.2 0.4 0.6 0.8 1k/

-2.25

-2

-1.75

-1.5

E(k

)

-2.6

-2.4

1D, =0.5 t

=0.5

ω

α

π

-2 t cos(k)

RS pe

rturb

ation

QMC

E +

0

0ω

=2.0α

π

E(k

)

k/

Figure 3.12: The one-dimensional polaron dispersion at small electron-phonon coupling.The intermediate coupling case is shown in the inset.

0 0.2 0.4 0.6 0.8k/

0

0.2

0.4

0.6

0.8

Z (

k)

=0.5 =2

0 0.2 0.4 0.6 0.8 1k/

0

2

4

6

8

N

(k)

=3.92

π π

0 ph

αα

1D, =0.5 tω

a) b)

α

Figure 3.13: One-dimensional case. The quasiparticle weight (a) and the number ofphonons (b) versus momentum at small (solid line) and intermediate (dashed line)electron-phonon coupling. The phonon number versus k at large coupling is shown withthe dotted line in b).

48 Chapter 3. Holstein Polaron

-4.5

-4.2

-3.9E

(k)

0

0.3

0.6

Z (

k)

0

1

2

3

N

(k)

0ph

(0,0) (π,π) (0,0)(π,0)

E (0) + ω00 a)

b)

c)

2D, =1.69=0.5 t αω

Figure 3.14: a) The two-dimensional polaron dispersion along the principal symmetryaxes of the Brillouin Zone. b) The two-dimensional polaron quasiparticle weight versusmomentum. c) The phonon number of the two-dimensional polaron versus momentum.

0 2 4 6 8n

0

0.2

0.4

0.6

0.8 k=0k=

0 2 4 6 8 10n

0

0.2

0.4

0.6

0.8k=(0,0)k=(0.2 , 0.2 )k=( , )

πππ π

π

1D, 2D,=0.5 t =0.5 tω ω

αα

=0.5=1.69

a) b)

Figure 3.15: Phonon distribution for different values of k. a) One-dimensional case. b)Two dimensional case.

3.4. Results of the QMC Calculation 49

polaron dispersion departs from the perfectly flat shape and the number of phonons keepincreasing with k.

The phononic cloud evolution with k is shown in Fig. 3.15 for one and for two-dimensional case. Notice that the phonon distribution at large k has a strong maximumat one, showing that the one-electron-and-one-phonon states are the most important con-figurations. For the very weak coupling case, which characterizes the one-dimensionalexample, the distribution at large k can be obtained by adding one more phonon to thezero momentum one. It shows that the large k polaron is a bound state between a zeromomentum polaron and a phonon which carries the momentum. In the two-dimensionalexample the phonon distribution still has a maximum at one but now the configurationswith more than one phonon participate also significantly to the large k polaron state for-mation. The qualitative differences noticed between the one-dimensional case (Fig. 3.15-a)and two-dimensional case (Fig. 3.15-b) results from the electron-phonon coupling strengthand not from the dimensionality. We should mention that for the one (two) -dimensionalcase the phonon distribution at other large k values not shown in the figure is practicallythe same with the shown distribution at k = π (k = (π, π)).

In the strong coupling regime, the momentum dependence of the polaron is a con-sequence of the exponentially reduced hopping parameter (Eq. 3.19). Therefore we didnot expect significant changes as in the weak and intermediate coupling cases. The dot-ted line plot in Fig 3.13-b, where the phonon number versus momentum at large α isshown, confirms that. Unlike the weak-coupling regime, the number of phonons is weaklyk dependent here.

For the Fröhlich polaron model, analytical calculations [20, 21] confirmed by numericalresults [7] show that, unlike in our case, there is a critical momentum kc above which thepolaron state does not exist. We found that for all k in the Brillouin Zone there is astable polaron state. However this statement can not be proved numerically for the entireparameter space (α, ω0) because, even if true, at very small electron-phonon coupling thepolaron binding energy at large k will be so small that our method can not resolve it. TheFröhlich model is special because the electron band, k2/2m, is infinite. Other differencefrom our model is the electron-phonon coupling matrix form, V (q) ∝ α/q, which is smallwhen large momentum phonon scatter (and they enter with predilection in the formationof large momentum polarons), but presumably this is not so important for the existenceof the kc end point. We believe a relevant fact is that in the extreme case of an infinitebare electron bandwidth the derivative ∂E(k)/∂k is zero at kc. The Fröhlich polaron nearkc is a weakly bound state of the zero momentum polaron with one more phonon [8] .It can be said that for k larger than kc the polaron band is perfectly flat and has zerobinding energy (which indicates a zero momentum polaron and a free phonon). Based onour results, we believe that when the infinite band condition is released, the zero bindingenergy at large momenta will become finite and the perfectly flat dispersion will transforminto a very slowly increasing one. Of course, for a real proof an analytical approach shouldbe taken.

50 Chapter 3. Holstein Polaron

3.5 Conclusions

In this chapter we have calculated the Holstein polaron properties using the DiagrammaticQuantum Monte Carlo. The algorithm which we have discussed in detail here is anexample of the general technique introduced in Chapter 2. It has been proved to bevery efficient in calculating both the ground state (k = 0) and the momentum dependentpolaron properties.

We have studied the one-dimensional and the two-dimensional polaron transition fromthe weak electron-phonon coupling regime to the strong electron-phonon coupling one.The transition is always continuous, but sharp when the phonon frequency ω0 is small.It takes place around the critical electron-phonon coupling αc = 2 at which the latticedeformation energy becomes equal to the bare electron bandwidth. The weak couplingpolaron is a light state carrying a small number of phonons which results in a weak latticepolarization extended over a large distance around the electron. The strong couplingpolaron is a heavy state with an exponentially large effective mass and a large numberof phonons. The electron polarizes the lattice strongly over a small distance aroundits position. The dimensionality has a strong influence on the weak coupling polaronproperties. In the one-dimensional case, the electron gets dressed very rapidly as thecoupling increases, resulting in a small quasiparticle weight before the transition. In thetwo-dimensional case where the polaron “before the transition” is characterized by a largequasiparticle weight which drops abruptly when the transition takes place. The evolutionof the phononic cloud with α shows that the transition region polaron is a mixture of largepolaron like states with small polaron like ones. The involvement of more than two statesis required to explain the number and the properties of the excited stable states whichappear in the transition region. However for a complete understanding of the transitionregion physics a systematic study of the excited states properties is needed, and a possibleand promising way is the spectral analysis of Diagrammatic Quantum Monte Carlo data.

In the weak-coupling region the polaron at large momentum is fundamentally differentfrom the one at small momentum. At small α the polaron changes from the almost freeelectron state at k = 0 to the weakly bound zero momentum polaron with one phononstate at large k. Consequently the large momentum polaron will be characterized by aflat dispersion, an almost vanishing quasiparticle weight and a number of phonons largerby one than the corresponding k = 0 polaron. At larger α (intermediate coupling regime)the large momentum polaron contains a significant fraction of states with two and morephonons, therefore the number of phonons will be larger by more than one than thenumber of phonons at k = 0 and the the dispersion will deviate from the perfectly flatshape, but the phonon distribution at large k still has a strong maximum at one. Thedimensionality does not play an important role on the momentum dependent polaronproperties. In the strong-coupling regime the polaron properties are weakly momentumdependent as a consequences of the exponentially small effective hopping integral.

Unlike the Fröhlich polaron where a critical momentum kc was found, above whichthe polaron state cannot exist, in the Holstein model we found stable polaron states atall k in the Brillouin Zone. Even though numerically it is impossible to prove the abovestatement at extremely small α and ω0, our calculation suggests that this is indeed trueand the kc end point in Fröhlich model is a consequence of the infinite electron bandwidth.

3.6. Appendix 51

τ τ’0 τ

k

τ1

τ1

0

k

21

Figure 3.16: Changing the diagram length

3.6 Appendix

3.6.1 Diagrams Updates

The weight of every diagram can be determined using a simple set of rules. These rulescan be found by inspecting directly Eq. 3.37. This calculation is easy but tedious, andwill result in the following: (i) every electron-phonon vertex corresponds to a term g dτ ,(ii) every phonon propagator corresponds to a term dq, (iii) every electron propagator of

length τ and moment k corresponds to a term e−ε(k)τ and (iv) every phonon propagatorof length τ corresponds to a term e−ω0τ . An example is given in Fig. 3.3.

In order to cover all the possible configurations, we need at least three types of updateprocedures: updates which change the diagram length (time), updates which add/removeinternal phonons and updates which add/remove external phonons.

Changing the diagram length These updates do not change the integration multi-plicity. We propose the following procedure which changes the length of the last (the oneat the right end) interval. An interval is a part of the diagram between two successiveinteraction vertices or between the last (first) vertex and the final (initial) end. The situ-ation is illustrated in Fig. 3.16. If we propose the end time of last interval to change fromτ to τ ′ the ratio between the final and the initial diagram weight will be

D2D1 = e

−(ε(k) + nω0 − µ)(τ ′ − τ) (3.42)

where n is the number of external phonons. We have more options. One is to chooseτ ′ uniformly random between τ1 and τmax and accept the change with the probabilityp = D2/D1, but a more efficient one is to choose τ ′ with the probability

P(τ ′) = e−(ε(k) + nω0 − µ)(τ ′ − τ)dτ ′ (3.43)

between τ1 and τmax which results in an acceptance ratio of 1.

Adding/Removing internal phonons These updates change the integration mul-tiplicity. The situation is illustrated in Fig. 3.17. According to the general discussionfrom Chapter 2 two distinct related subroutines are necessary: one for adding phononpropagators and the other for removing them.

Adding one phonon propagator (1 −→ 2). 1) Choose an initial τ1 and afinal τ2 time for the phonon propagator, with a probability P(τ1, τ2)dτ1dτ2. 2) Choose

52 Chapter 3. Holstein Polaron

a momentum q for the phonon propagator, with a probability P(q)dq. 3) The diagramsweight ratio will be

D2/D1 = g2dτ1dτ2dq∑

i

e−(ε(ki − q) + ω0 − ε(ki))∆τi (3.44)

where a sum over all intervals i of length ∆τi between τ1 and τ2 is performed. Accordingto Eq. 2.20 and Eq. 2.22 the change will be accepted with probability

p =prempadd

g2∑

i

e−(ε(ki − q) + ω0 − ε(ki))∆τi niph + 1P(τ1, τ2)P(q) (3.45)

where niph is the number of internal phonon propagators (which will become niph + 1 ifthe update is accepted) and padd (prem) is the calling probability for the internal phononaddition (removal) subroutine.

Removing one phonon propagator (2 −→ 1). 1) Choose a phonon propagatorwith a probability 1/niph. 2) The diagrams weight ratio is

D1/D2 = 1g2dτ1dτ2dq

∑i

e−(ε(ki + q)− ω0 − ε(ki))∆τi (3.46)

where τ1 (τ2) is the initial (final) phonon with momentum q propagator time. Accordingto Eq. 2.21 and Eq. 2.22 the change will be accepted with probability

p =paddprem

1

g2

∑i

e−(ε(ki + q)− ω0 − ε(ki))∆τiP(τ1, τ2)P(q)niph

(3.47)

The probability functions P(τ1, τ2) and P(q) are arbitrary and can be tuned to max-imize the code efficiency. We made the following choices. First we choose τ1 randomlybetween 0 and τ and afterwards we choose τ2 > τ1 with a probability ∝ e−ω0(τ2−τ1). Withrespect to the phonon momentum q, we chose it uniformly random in the Brillouin Zone.

Adding/Removing external phonon The procedures of adding and removing ex-ternal phonon (see Fig. 3.18) are similar to the ones regarding the internal phonon. Thedifference is that now τ1 (τ2) represents the connection time of the left (right) end externalphonon propagator and the summation over i in Eq. 3.44 and in the similar subsequentequations means summation over all intervals from left end (zero time) to τ1 and from τ2to τ . The number of internal phonons niph should also be replaced with the number ofexternal phonons n.

1 2

p k p kq

τ τ1 2τ τ

Figure 3.17: Adding/Removing an internal phonon

3.6. Appendix 53

1τ 2τ

1 2

k k−p

p q

k

ττk−p k−q

q

k

p

k−q

Figure 3.18: Adding/Removing an external phonon

The updates introduced above constitute a minimal set for ensuring the ergodicitycondition. Other updates can be imagined. For example we used three more subroutines,one which changes the time of the interaction vertices, one which interchanges two verticesand another which stretches the diagram. We did not present them here. However,introducing them turns out to be important, the convergence being improved by oneorder of magnitude.

3.6.2 Estimators

Our algorithm generates all the possible diagrams with the length between 0 and τmaxand with the external phonon number between 0 and Nmaxph . Suppose we want to measurea certain quantity A. Every diagram (i) generated according to its weight P ni(k, τi)contributes to A with a certain amount given by the estimator a(i), such as in the end

A =1

M

M∑i=1

P ni(k, τi)a(i) (3.48)

We used estimators to compute the Green’s function, the polaron energy, the phonondistribution in the polaronic cloud, the polaron phonon number and the polaron effectivemass.

Green’s function estimator The n0-phonon Green’s function at time τ0 is

P n0(k, τ0) =1

M

1

Γ

M∑i=1

P ni(k, τi)δ(τi − τ0)δni,n0 (3.49)

where Γ is

Γ =1

M

M∑i=1

δ(τi − τ0)δni,n0 (3.50)

Therefore it seems that we should define the estimator for P n0(k, τ0) as (1/Γ) δ(τi −τ0)δni,n0 . However the diagram time can take continuous values, which means a infinitelysmall probability for the diagram time, τi, to be exactly τ0. Consequently the statistics(and also Γ) will be infinitely small and the measurements are impossible. In order toovercome this difficulty we write the Green’s function in a different way

P n0(k, τ0) =1

M

1

Γ

M∑i=1

P ni(k, τi)P n0(k, τ0)

P ni(k, τi)Θ(a− |τi − τ0|)δni,n0 (3.51)

54 Chapter 3. Holstein Polaron

where

Γ =1

M

M∑i=1

Θ(a− |τi − τ0|) = 2a (3.52)

is a constant equal to the area of the (arbitrary chosen) radius a time window in whichthe statistics are collected. Θ(x) is the Heaviside step function. The Green’s functionestimator will be

pn0(k, τ0) =1

2a

P n0(k, τ0)

P ni(k, τi)Θ(a− |τi − τ0|)δni,n0 (3.53)

This estimator allows us to collect statistics at all the times in the window (τi− a, τi + a)from a particular diagram with time τi, making the algorithm very efficient.

Energy estimator In order to determine an estimator for the energy, let’s notice thatat large τ

P (k, (1 + λ)τ)

P (k, τ)−→ e−λE(k)τ (3.54)

The estimator for the left hand side of Eq. 3.54 is

Q(i, λ) =P ni(k, (1 + λ)τi)

P ni(k, τi)= (1 + λ)Nv

∏

l

e−λεl(k)∆τl∏

j

e−λω0∆τj (3.55)

where l (j) counts the electron (phonon) propagators of time ∆τl (∆τj), and Nv = 2(n+niph) is the total number of vertices. Up to the first order in λ, Eq. 3.55 becomes

Q(i, λ) = 1 + λ(Nv −∑

l

εl(kl)∆τl −∑

j

ω0∆τj) (3.56)

The right hand side of Eq. 3.54 is, up to first order in λ, equal to 1−λE(k)τ which makesthe polaron energy estimator to be

e(i) =1

τi(−Nv +

∑

l

εl(kl)∆τl +∑

j

ω0∆τj) (3.57)

The measurements should be taken only on the large time diagrams where the asymptoticregime is reached.

Inverse of the effective mass estimator Analogously, let’s notice that at large timeand for small λ

P (λ, τ)

P (0, τ)−→ e−(E(λ)− E(0))τ = e−λ2τ/2m∗ (3.58)

The estimator for the left hand side of Eq. 3.58 is

Q(i, λ) =∏

l

e−(εl(kl + λ)− εl(kl))∆τl (3.59)

3.6. Appendix 55

Expanding Eq. 3.55 up to second order in λ we get

Q(i, λ) = 1− λ2(z∑

δ=1

∑

l

(t cos(klδ)− 2t2 sin2(klδ))∆τl) . (3.60)

The inverse of the effective mass estimator will result in

mem∗

(i) =1

τi

z∑

δ=1

∑

l

(2t cos(klδ)− 4t2 sin2(klδ)) , (3.61)

where me is the free electron mass.

n-phonon probability estimator The estimator for Zn0 (k) is

zn(i) = δni,n (3.62)

For n = 0 Eq. 3.62 gives the quasiparticle weight estimator.

Total number of phonon estimator is

nph(i) = nδni,n (3.63)

In order to determine the total number of phonons accurately, Nmaxph should be takenlarge enough such that the contribution of diagrams with a larger than Nmaxph number ofphonons is exponentially small and therefore negligible.

56 Chapter 3. Holstein Polaron

References

[1] L. Landau, Z. Phys 3, 664 (1933).

[2] E. de Mello and J. Ranninger, Phys. Rev. B 55, 14872 (1997).

[3] M. Capone et al., Phys. Rev. B 56, 448 (1997).

[4] A. S. Alexandrov, Phys. Rev. B 49, 9915 (1994).

[5] G. Wellein and H. Fehske, Phys. Rev. B 56, 4513 (1997).

[6] G. Wellein and H. Fehske, Phys. Rev. B 58, 6208 (1998).

[7] N. Prokof’ev and B. Svistunov, Phys. Rev. Lett. 81, 2514 (1998).

[8] A. Mishchenko, N. Prokof’ev, A.Sakamoto, and B. Svistunov, Phys. Rev. B 62, 6317(2000).

[9] H. De Raedt and A. Langedijl, Phys. Rev. B 27, 6097 (1982).

[10] P. E. Kornilovitch, Phys. Rev. Lett. 81, 5382 (1998).

[11] P. E. Kornilovitch, Phys. Rev. B 60, 323 (1999).

[12] P. E. Kornilovitch, Phys. Rev. B 56, 4494 (1997).

[13] A. H. Romero, et al., Phys. Rev. B 59, 13728 (1999).

[14] V. Cataudella, G. D. Filippis, and G. Iadonisi, Phys. Rev. B 62, 1496 (2000).

[15] V. Cataudella, G. D. Filippis, and G. Iadonisi, Phys. Rev. B 60, 15163 (1999).

[16] G. Iadonisi, V. Cataudella, G. D. Filippis, and D. Ninno, Europhys. Lett 41, 309(1998).

[17] E. Jeckelmann, et al., Phys. Rev. B 57, 6367 (1998).

[18] J. Bonca, S. A. Trugman, and I. Batistic, Phys. Rev. B 60, 1633 (1999).

[19] J. W. Negel and H. Orland, Quantum Many-Particle Systems (Addison-Wesley,1988).

[20] G. D. Whitfield and R. Puff, Phys. Rev. 139, A338 (1965).

[21] D. M. Larsen, Phys. Rev. 14, 697 (1966).

[22] I. G. Lang and Y. A. Firsov, Sov. Phys. JETP 16, 1301 (1963).

[23] C. Kittel, Introduction to solid state physics (New-York, 1996).

[24] F. Marsiglio, Physica C 224, 21 (1995).

3.6. Appendix 57

[25] H. Lowen, Phys. Rev. B 37, 8661 (1988).

[26] B. Gerlach and H. Lowen, Rev. Mod. Phys. 63, 63 (1991).

[27] D. M. Eagles, Phys. Rev. 145, 645 (1966).

[28] D. M. Eagles, J. Phys. C 17, 637 (1984).

[29] D. M. Eagles, Phys. Rev. 181, 1278 (1969).

[30] A. Mishchenko, N. Nagaosa, N. Prokof’ev, A.Sakamoto, and B. Svistunov, Phys.Rev. B 66, 020301 (2002).

[31] A. Mishchenko, private communications.

58