Page 1

Effect of Electron-Phonon Interaction Range for a Half-Filled Band in One Dimension

Martin Hohenadler,1Fakher F. Assaad,1and Holger Fehske2

1Institut f¨ ur Theoretische Physik und Astrophysik, Universit¨ at W¨ urzburg, 97074 W¨ urzburg, Germany

2Institut f¨ ur Physik, Ernst-Moritz-Arndt-Universit¨ at Greifswald, 17487 Greifswald, Germany

(Dated: September 19, 2012)

We demonstrate that fermion-boson models with nonlocal interactions can be simulated at finite

band filling with the continuous-time quantum Monte Carlo method. We apply this method to

explore the influence of the electron-phonon interaction range for a half-filled band in one dimension,

covering the full range from the Holstein to the Fr¨ ohlich regime.

metallic, Peierls, and phase-separated regions. Nonlocal interactions suppress the Peierls instability,

and thereby lead to almost degenerate power-law exponents for charge and pairing correlations.

The phase diagram contains

PACS numbers: 71.10.Hf, 71.10.Pm, 71.30.+h, 71.45.Lr

Introduction.—Electron-phonon interaction has an es-

sential influence on the properties of many materials [1].

It plays a key role for pairing and superconductivity, mass

renormalization, and charge ordering phenomena. Tak-

ing into account quantum lattice fluctuations leads to a

complex, many-body problem. Consequently, theoreti-

cal studies usually rely on simplified microscopic models.

A frequently invoked approximation, in particular for nu-

merical studies, is to consider a completely local electron-

phonon coupling as in Holstein’s molecular-crystal model

[2]. However, nonlocal interactions are expected to play

an important role in materials with incomplete screen-

ing such as quasi-one-dimensional (quasi-1D) organics

[3]. Long-range interactions, as described by the Fr¨ ohlich

model [4], have been investigated in the context of high-

temperature superconducting cuprates [5–9], and were

found to support light polarons and bipolarons [10–12].

Exact numerical methods have played an important

role for the understanding of coupled electron-phonon

systems. Whereas efficient algorithms exist for Holstein-

type models at arbitrary band filling [13–18], extended

interactions could so far be addressed only in the empty-

band limit, see, for example, [10, 19–22], and [8] for a re-

view. Consequently, key phenomena such as the Peierls

instability [23] were out of reach. The latter occurs in

quasi-1D systems with commensurate fillings, for exam-

ple in TTF-TCNQ [24], and drives a transition to a

Peierls insulator with charge-density-wave order. For the

Holstein model, it is known that quantum lattice fluc-

tuations can destroy the charge order [25], leading to

a metal-insulator quantum phase transition at a finite

value of the electron-phonon coupling strength.

Unbiased investigations of the effect of nonlocal inter-

actions at finite band-filling represent a long-standing,

open problem. In this Letter, we use the continuous-

time quantum Monte Carlo (CTQMC) method [26] to

study a model that interpolates between and includes

the paradigmatic Holstein and Fr¨ ohlich limits.

Model.—We consider a HamiltonianˆH = ˆH0+ˆH1,

where

ˆH0

= −t?

?ij?σ(c†

iσcjσ+ H.c.) describes 1D

fermions with nearest-neighbor hopping t, and

?

The first term describes lattice fluctuations in the har-

monic approximation, with the phonon frequency ω0and

the stiffness constant K = ω2

resents the electron-phonon interaction, in the form of a

nonlocal density-displacement coupling, with the density

operator ˆ ni =

elements are chosen as [9, 10, 27]

ˆH1=

i

?

1

2MˆP2

i+K

2ˆQ2

i

?

−γ

?

i,r

f(r)ˆQi+r(ˆ ni− 1). (1)

0M. The second term rep-

?

σˆ niσ and ˆ niσ = c†

iσciσ. The matrix

f(r) =

1

(r2+ 1)3/2e−r/ξ,0 ≤ r < L/2,(2)

where the lattice constant a = 1. For ξ → ∞, this cou-

pling represents a lattice version of the Fr¨ ohlich interac-

tion [27]. More generally,ˆH1 may be viewed as an ex-

tended Holstein interaction with screening length ξ; the

original Holstein model [2] is recovered in the limit ξ → 0.

For ω0→ ∞, the model maps onto an attractive, general-

ized Hubbard model. Our method can be applied to any

coupling which preserves translational invariance. The

restriction of r is due to periodic boundary conditions.

Method.—For electron-phonon problems [18],

starting point is the partition function at inverse tem-

perature β = 1/kBT,

?

where ψ,ψ are Grassmann fields, and q denotes phonon

coordinates. The phonons can be integrated out exactly

[28], leading to a purely fermionic action with a nonlocal

(in space and time) interaction

the

Z =

D(ψ,ψ)e−S0[ψ,ψ]

?

D(q)e−S1[ψ,ψ,q], (3)

Sf

1= −

??β

0

dτdτ??

ij

[ni(τ) − 1]Dτ,τ?

i,j[nj(τ?) − 1]. (4)

The phonon propagator takes the form Dτ,τ?

j)D(τ − τ?) with F(i − j) =?

i,j

= F(i −

rf(r + i − j)f(r) and

arXiv:1205.0612v2 [cond-mat.str-el] 18 Sep 2012

Page 2

2

the Holstein propagator D(τ − τ?).

range in space (time) is determined by ξ (ω0).

CTQMC method used here is based on an exact expan-

sion around γ = 0 [26]. A hybridization expansion algo-

rithm for electron-phonon impurity problems also exists

[29]. Monte Carlo updates consist of adding or removing

single vertices, and flipping auxiliary Ising spins [18, 26].

The numerical effort scales with the cube of the aver-

age expansion order. Because of the underlying weak-

coupling expansion, the CTQMC method is particularly

efficient for problems with small expansion orders, such

as the Peierls transition in the adiabatic regime ω0/t ? 1

[30]. Importantly, the method enables us to study the

many-electron problem [31]. We have verified that it re-

produces exact results in the Holstein limit ξ → 0, and

in the anti-adiabatic limit ω0→ ∞ where the model (1)

maps onto an extended attractive Hubbard model.

Results.—We choose a phonon frequency ω0/t = 0.5.

In this regime, representative of many materials, neither

static nor instantaneous approximations are valid, and

numerical simulations are essential. The dimensionless

ratio λ = ?p/2t, where ?pis the polaron binding energy

in the atomic limit (t = 0) [27] and 2t is half of the free

bandwidth, allows us to compare different ξ at the same

effective coupling strength. We use t as the energy unit,

and set ? = M = 1. All results are for a half-filled band.

The phase diagram as a function of ξ and λ, obtained

from CTQMC simulations with up to L = 42 sites, is

shown in Fig. 1. Because the Holstein model is recov-

ered for ξ = 0, its previously studied metallic and Peierls

insulating phases [15, 32, 33] smoothly extend to ξ > 0.

However, we observe a significant ξ-dependence for small

ξ, and saturation for larger values. Additionally, for suf-

ficiently large ξ and λ, we find a region of phase sep-

aration or charge segregation. The metallic phase and

the metal-insulator transition extend all the way to the

Fr¨ ohlich limit ξ = ∞ [λPI

the range of λ shown in Fig. 1, phase separation is absent

for ξ ≤ 2, and becomes more favorable with increasing ξ.

The different phases can be characterized by the den-

sity correlator Sρ(r) = ?(ˆ nr−1)(ˆ n0−1)? and the density

structure factor Sρ(q). In the metallic phase [Fig. 2(a)],

Sρ(r) shows a power-law decay of 2kFcorrelations (with

exponent Kρ) as expected from bosonization, and is lin-

ear for q → 0. Together with exponentially suppressed

spin correlations (not shown), these findings are consis-

tent with a bipolaronic Luther-Emery phase [34, 35].

The Peierls state exhibits quasi-long-range 2kFdensity

correlations [Fig. 2(a)], corresponding to charge-density-

wave order at T = 0 with two electrons of opposite spin

forming bipolarons on every other site. The phase bound-

ary for the Peierls transition can be determined from the

staggered charge susceptibility [32]

The interaction

The

c = 0.48(2)]; see also Fig. 3. For

χρ(π) =1

L

?

ij

(−1)i−j

?β

0

dτ?ˆ ni(τ)ˆ nj(0)?.(5)

FIG. 1.

action range ξ and electron-phonon coupling strength λ. The

regions correspond to a metal, a Peierls insulator, and phase

separation. The metallic and Peierls phases (and presumably

also phase separation) extend to the Fr¨ ohlich limit ξ = ∞.

Lines are guides to the eye. Here ω0/t = 0.5.

(Color online) Phase diagram as a function of inter-

For fixed β/L, χρ(π)/L is universal at the critical point,

and the crossing of curves for different L gives λPI

example, we have λPI

c

= 0.33(1) for ξ = 2 in Fig. 2(c).

As for the Holstein model [32, 36], the Peierls transition

is expected to be of the Kosterlitz-Thouless type also for

ξ > 0. Extended interactions (ξ > 0) promote metallic

behavior by dissociating the onsite bipolarons predomi-

nant in the Holstein regime, and we see a Peierls insulator

to metal transition as a function of ξ [Fig. 2(a)]. By the

same mechanism, the critical coupling for the transition

to the Peierls insulator, λPI

c, increases with increasing ξ,

see Fig. 1. The onset of charge order is also reflected in

the divergent q = 2kFpeak in Sρ(q), see Fig. 2(b).

Phase separation as a result of the phonon-induced at-

traction manifests itself as a peak at small q in Sρ(q),

as shown in Fig. 2(b). In the phase-separated region of

Fig. 1, the quantity πSρ(q1)/q1(with q1= 2π/L)—whose

thermodynamic limit is related to Kρ in a Luttinger

liquid—diverges with system size. In the Peierls phase

Kρ= 0, as verified on very large systems [32]. Formally,

Kρ= ∞, reflecting phase separation, implies a divergent

compressibility [34]. The divergence of πSρ(q1)/q1in the

phase-separated region is shown for ξ = 6 in Fig. 2(d),

and we deduce a critical value of λPS

are two possible scenarios for the transition from the

Peierls to the phase separated region. A continuous tran-

sition would imply a melting of the charge-ordered Peierls

state before the formation of multipolaron droplets, al-

lowing for an intervening (narrow) metallic region with

finite Kρ. Alternatively, the insulator-insulator transi-

tion could be of first order. Evidence for the latter pos-

sibility comes from the occurrence of metastable config-

urations and hysteresis at low temperatures in our sim-

ulations. The occurrence of phase separation in models

with long-range electron-phonon coupling had been sug-

c; for

c

= 0.55(2). There

Page 3

3

−0.3

0

0.3

0.6

0246810

Sρ(r)

r

(a) λ = 0.325

0

2

4

0

π

q

2π

Sρ(q)

(b) ξ = 6

0.3

0.4

0.5

0.6

0.310.33

λ

0.35

χρ(π)/L

(c) ξ = 2

100

101

102

00.050.10

LSρ(2π/L)/2

1/L

(d) ξ = 6

ξ = 0.5(PI)

ξ = 5(M)

λ = 0.20(M)

λ = 0.45(PI)

λ = 0.55(PS)

L = 14

L = 18

L = 22

L = 26

λ = 0.525

λ = 0.540

λ = 0.545

λ = 0.550

λ = 0.560

FIG. 2.

(PI, ξ = 0.5) and the metallic phase (M, ξ = 5). (b) Structure factor Sρ(q) at ξ = 6 in the three phases of Fig. 1. We used

L = 22, and β = L (L/2) for λ = 0.2, 0.45 (0.55). (c) Scaling of the charge susceptibility χρ(π) [Eq. (5)] at ξ = 2, defining the

critical point λPI

c

= 0.33(1) of the Peierls transition. Here β = L. (d) Finite-size scaling of πSρ(q1)/q1 = LSρ(2π/L)/2 with

β = L/2. The divergence for λ ≥ 0.55 indicates phase separation (PS). All results are for ω0/t = 0.5.

(Color online) (a) Density-density correlation function Sρ(r), for λ = 0.325 and βt = L = 22, in the Peierls insulator

gested before [20] and observed in analytical work [6, 7];

for short-range interactions, it is suppressed by the ab-

sence of bound triplet states [20], but may occur in the

vicinity of a Mott transition [37]. Phase separation is

expected to be suppressed in the presence of additional

long-range electron-electron interaction [38].

In combination with analytical continuation [39], we

can calculate the single-particle spectral function

?

A(k,ω) =1

Z

ij

|?i|ckσ|j?|2(e−βEi+e−βEj)δ(∆ji−ω),

(6)

where |i? is an eigenstate with energy Ei, and ∆ji =

Ej−Ei. Results in the metallic phase (λ = 0.2) are shown

in Fig. 3 for the extreme Holstein (ξ = 0.1) and Fr¨ ohlich

limits (ξ = ∞). The locus of spectral weight follows the

free band dispersion, −2tcosk. The exponentially small

Luther-Emery spin gap is not resolved for the parame-

ters chosen, and the spectrum is particle-hole symmet-

ric. Excitations are sharp inside the coherent interval

[−ω0,ω0], whereas they are substantially broadened as a

result of multiphonon processes at higher energies [40].

As expected for our 1D model, the spectrum agrees well

with the exact bosonization result [41], which predicts

a hybridization of the spin, charge and phonon modes,

although spin-charge separation is not visible due to the

weak coupling and small ω0[42]. Comparing Figs. 3(a)

and (b) we see that in contrast to previous work on one

and two electrons [8, 10, 19–22], the impact of the inter-

action range is remarkably small. This important char-

acteristic of the many-electron case can be related to the

absence of significant polaron and bipolaron effects in the

metallic phase of Fig. 1.

Figures 4(a),(b) show A(k,ω) in the Peierls phase

(λ = 0.4). In the Holstein regime (ξ = 0.1), the spec-

trum consists of two sets of features. The cosine band

seen in Fig. 3 has acquired a gap at the Fermi level and

reveals additional, backfolded shadow bands as a result of

dimerization and the corresponding doubling of the unit

cell [43]. These signatures are labeled (1) and (1’) in

Fig. 4(a), respectively. In addition, we find lower-energy

excitations labeled (2) corresponding to bound soliton-

antisoliton pairs or, equivalently, polarons [30, 44], which

are absent in a homogeneous mean-field solution that

captures only (1) and (1’) [30]. The soliton dispersion

indicates a mass larger than the electron mass. Because

their energy at the Fermi level is lower than the Peierls

gap, doping would lead to the formation of solitons [45].

Increasing the interaction range from ξ = 0.1 to ξ = 4

Page 4

4

−4

−2

0

2

4

0 π/2

k

π

ω/t

0π/2

k

π

0.010.010.10.11110 10

(a) A(k,ω), ξ = 0.1

(b) A(k,ω), ξ = ∞

FIG. 3.

A(k,ω) [Eq. (6)] in the metallic phase at λ = 0.2 for (a)

ξ = 0.1, (b) ξ = ∞. Here ω0/t = 0.5, βt = L = 30. The

dashed lines indicate the Fermi level (ω = 0) and ω = ±ω0.

(Color online) Single-particle spectral function

−4

−2

0

2

4

0 π/2

k

π

ω/t

0π/2

k

π

0

2

4

0π/2

q

π

ω/t

0.010.010.010.10.10.1111101010

(a) A(k,ω), ξ = 0.1

(1)(1’)

(2)

(b) A(k,ω), ξ = 4

(c) Sρ(q,ω), ξ = 4

FIG. 4.

tion A(k,ω) in the Peierls phase at λ = 0.4 for (a) ξ = 0.1,

(b) ξ = 4. (c) Dynamical charge structure factor [Eq. (7)] for

the same parameters as in (b). Here ω0/t = 0.5, βt = L = 22.

The dashed lines indicate (a),(b) the Fermi level and (c) ω0.

The labels (1), (1’), and (2) in (a) are explained in the text.

(Color online) (a),(b) Single-particle spectral func-

drives the system into the vicinity of the metal-insulator

transition, see Fig. 1. This is reflected by a much smaller

gap at the Fermi level, reduced spectral weight of the po-

laron excitations, and the suppression of shadow bands.

Consequently, the spectral function becomes quite simi-

lar (but not identical) to that shown in Fig. 3, and illus-

trates the continuous evolution of A(k,ω) across λPI

c.

A hallmark feature of the Peierls state is phonon soft-

ening at q = 2kF, which is visible [30, 46] in the dynami-

10−4

10−3

10−2

10−1

8163264

r

(a) ξ = 0.1

10−4

10−3

10−2

10−1

8163264

r

(b) ξ = ∞

Sρ(r)

P(r)

FIG. 5.

distance r = L/2 for (a) ξ = 0.1, (b) ξ = ∞. Here ω0/t = 0.5

and λ = 0.2. Lines are fits to a power law f(r) = Ar−η.

(Color online) Charge and pairing correlations at

cal charge structure factor [ˆ ρq=?

Sρ(q,ω) =1

Z

reiqr(ˆ nr−?ˆ nr?)/√L]

?

ij

|?i| ˆ ρq|j?|2e−βEjδ(∆ji− ω).(7)

The results in Fig. 4(c) for λ = 0.4, ξ = 4 [the same

parameters as in Fig. 4(b)] reveal a clear signature of

the renormalized phonon dispersion. The spectrum is

dominated by the soft phonon mode at ω = 0, q = 2kF.

Furthermore, we observe a continuum of particle-hole ex-

citations, and a charge gap at long wavelengths (q → 0).

Finally, we consider the interaction-range effect on the

competition of charge and pairing correlations. Figure 5

shows results for Sρ(r) and the s-wave pair correlator

P(r) = ?∆†

phase. For ξ = 0.1, Fig. 5(a) reflects the dominance of

charge correlations previously observed for the Holstein

model [47]. However, Fig. 5(b) reveals that long-range

electron-phonon interaction (here ξ = ∞) suppresses

charge correlations and thereby results in almost identical

power-law exponents in both channels. Additional short-

range electron-electron repulsion is expected to further

suppress onsite bipolaron formation—promoting nonlo-

cal pairing—and to lead to a more general phase diagram

with Mott, metallic, Peierls and phase-separated ground

states. It will be interesting to explore this issue further,

both at and away from half filling, also in the light of a

recently reported phase of correlated singlets [48].

Conclusions.—We used exact Monte Carlo simulations

to study many-electron systems with nonlocal and even

long-range electron-phonon interaction.

the Holstein model, extended interactions suppress the

Peierls instability—making pairing more favorable—and

can lead to phase separation.

materials modeling are that interactions of finite but

small range are well described by Holstein-type models,

whereas long-range interactions can have substantial ef-

fects on the balance of pairing and charge correlations.

We are grateful to F. Essler and A. Muramatsu for

helpful discussions, and acknowledge support from the

DFG Grant No. Ho 4489/2-1 as well as computer time at

the LRZ Munich and the J¨ ulich Supercomputing Centre.

r∆0? (with ∆†

r= c†

r↑c†

r↓) in the metallic

Compared to

Key implications for

Page 5

5

[1] Polarons in Advanced Materials, edited by A. S. Alexan-

drov (Canopus Publishing and Springer Verlag GmbH,

Bristol (UK), 2007).

[2] T. Holstein, Ann. Phys. (N.Y.) 8, 325; 8, 343 (1959).

[3] W. Barford, Electronic and Optical Properties of Conju-

gated Polymers (Oxford University Press, Oxford, 2005).

[4] H. Fr¨ ohlich, Adv. Phys. 3, 325 (1954).

[5] R. Zeyher, Z. Phys. B 80, 187 (1990).

[6] A. S. Alexandrov and P. E. Kornilovitch, J. Phys.: Con-

dens. Matter 14, 5337 (2002).

[7] A. S. Alexandrov and P. E. Kornilovitch, Phys. Lett. A

299, 650 (2002).

[8] J. T. Devreese and A. S. Alexandrov, Rep. Prog. Phys.

72, 066501 (2009).

[9] T. M. Hardy, J. P. Hague, J. H. Samson, and A. S.

Alexandrov, Phys. Rev. B 79, 212501 (2009).

[10] H. Fehske, J. Loos, and G. Wellein, Phys. Rev. B 61,

8016 (2000).

[11] J. Bonˇ ca, T. Katraˇ snik, and S. A. Trugman, Phys. Rev.

Lett. 84, 3153 (2000).

[12] J. P. Hague, P. E. Kornilovitch, J. H. Samson, and A. S.

Alexandrov, Phys. Rev. Lett. 98, 037002 (2007).

[13] R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, Phys.

Rev. D 24, 2278 (1981).

[14] J. E. Hirsch and E. Fradkin, Phys. Rev. Lett. 49, 402

(1982).

[15] H. Fehske, G. Hager, and E. Jeckelmann, Europhys. Lett.

84, 57001 (2008).

[16] H. Fehske, G. Wellein, G. Hager, A. Weiße, and A. R.

Bishop, Phys. Rev. B 69, 165115 (2004).

[17] R. T. Clay and R. P. Hardikar, Phys. Rev. Lett. 95,

096401 (2005).

[18] F. F. Assaad and T. C. Lang, Phys. Rev. B 76, 035116

(2007).

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

[20] J. Bonˇ ca and S. A. Trugman, Phys. Rev. B 64, 094507

(2001).

[21] A. S. Mishchenko, N. V. Prokof’ev, A. Sakamoto, and

B. V. Svistunov, Phys. Rev. B 62, 6317 (2000).

[22] A. S. Mishchenko, N. Nagaosa, N. V. Prokof’ev, A.

Sakamoto, and B. V. Svistunov, Phys. Rev. Lett. 91,

236401 (2003).

[23] R. Peierls, Surprises in Theoretical Physics (Princeton

University Press, New Jersey, 1979).

[24] D. J´ erome, Chem. Rev. 104, 5565 (2004).

[25] E. Jeckelmann, C. Zhang, and S. R. White, Phys. Rev.

B 60, 7950 (1999).

[26] A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein,

Phys. Rev. B 72, 035122 (2005).

[27] A. S. Alexandrov and P. E. Kornilovitch, Phys. Rev. Lett.

82, 807 (1999).

[28] R. P. Feynman, Phys. Rev. 97, 660 (1955).

[29] P. Werner and A. J. Millis, Phys. Rev. Lett. 99, 146404

(2007).

[30] M. Hohenadler, H. Fehske, and F. F. Assaad, Phys. Rev.

B 83, 115105 (2011).

[31] M. Hohenadler, D. Neuber, W. von der Linden, G.

Wellein, J. Loos, and H. Fehske, Phys. Rev. B 71, 245111

(2005).

[32] R. P. Hardikar and R. T. Clay, Phys. Rev. B 75, 245103

(2007).

[33] M. Tezuka, R. Arita, and H. Aoki, Phys. Rev. B 76,

155114 (2007).

[34] J. Voit, Rep. Prog. Phys. 58, 977 (1995).

[35] S. Ejima and H. Fehske, J. Phys. Conf. Ser. 200, 012031

(2010).

[36] J. E. Hirsch and E. Fradkin, Phys. Rev. B 27, 4302

(1983).

[37] M. Capone, G. Sangiovanni, C. Castellani, C. Di Castro,

and M. Grilli, Phys. Rev. Lett. 92, 106401 (2004).

[38] A. S. Alexandrov and V. V. Kabanov, Pis’ma Zh. Eksp.

Teor. Fiz. 72, 825 (2000), [JETP Lett. 72, 569 (2000)].

[39] K. S. D. Beach, cond-mat/0403055.

[40] J. Loos, M. Hohenadler, and H. Fehske, J. Phys.: Con-

dens. Matter 18, 2453 (2006).

[41] V. Meden, K. Sch¨ onhammer, and O. Gunnarsson, Phys.

Rev. B 50, 11179 (1994).

[42] W. Q. Ning, H. Zhao, C. Q. Wu, and H. Q. Lin, Phys.

Rev. Lett. 96, 156402 (2006).

[43] J. Voit, L. Perfetti, F. Zwick, H. Berger, G. Margari-

tondo, G. Gr¨ uner, H. H¨ ochst, and M. Grioni, Science

290, 501 (2000).

[44] A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su,

Rev. Mod. Phys. 60, 781 (1988).

[45] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev.

Lett. 42, 1698 (1979).

[46] F. F. Assaad, Phys. Rev. B 78, 155124 (2008).

[47] K.-M. Tam, S.-W. Tsai, and D. K. Campbell, Phys. Rev.

B 84, 165123 (2011).

[48] S. Reja, S. Yarlagadda, and P. B. Littlewood, Phys. Rev.

B 86, 045110 (2012).