G. Wellein1, A. R. Bishop2, M. Hohenadler1,3, W. von der Linden3, D. Neuber3, J. Loos4, A. Alvermann5, G. Schubert5, and H. Fehske5

University Erlangen1 (Germany), TU Graz3 (Austria), University Greifswald5 (Germany)Los Alamos National Laboratory2 (USA), Czech Academy of Sciences Prague4 (Czech Rep.)

Motivation

• Optical measurements have proven the importanceof electron-phonon (EP) coupling and even polaroneffects in important classes of materials, includingquasi-1d MX chains, quasi-2d cuprate superconduc-tors, and 3d colossal magnetoresistance manganites.

• In all these systems, a noticeable density of (pola-ronic) charge carriers is observed, which puts the ap-plicability of single-polaron theories into question,particularly in the often realized case of intermediateEP couplings and phonon frequencies.

• Questions: What happens to polarons if their densityis large enough so that individual quasiparticles (QP)would overlap? Is there a density-driven crossoverfrom (large) polarons to weakly dressed electrons?

• Problem: Even for the most simplified models no re-liable analytical results exist for finite carrier densi-ties in the intermediate EP coupling regime!

Model

1D spinless fermion Holstein model:

H = −t∑

〈i,j〉c†icj − g ω0

∑

i

(b†i + bi )ni + ω0

∑

i

b†ibi

Parameter ratios:

g2 = εp/ω0 ; λ = εp/2tα = ω0/t ; n = Ne/N

Known results:

• For a single particle a transition from a large polaronto a small polaron takes place with increasing EPcoupling strength (at least in 1D).

• In the intermediate coupling regime λ ' 1, g2 ' 1,the size of the polaron is strongly dependent on thephonon frequency:

α� 1: rather extended distortionα� 1: localised distortion.

• Focusing on the intermediate coupling adiabaticregime we expect strong density-effects due to a pos-sible overlap of phonon clouds (of course, in theU → ∞ limit, bipolaron formation is suppressed)!

Methods

Quantum Monte Carlo (QMC) [1,2]

• Grand-canonical approach, free of autocorrelations.

• Finite-temperature Trotter discretization (∆τ = 0.1).

• Lang-Firsov transformed model ; moderate signproblem [which is most pronounced for intermediateλ and small α at low temperatures T (β = 1/kBT )].

• Maximum entropy method to get dynamic quantities.

Exact diagonalization (ED) [3]

• Lanczos technique HD → TL (Krylov subspaces):Fast convergence for extremal eigenvalues E0 , |ψ0〉(D . 1011, L = 100 ; ∆E0 . 10−9)!

• Phonons? D = ∞! ; Controlled Hilbert space trun-cation combined with a density-matrix based phononbasis optimisation procedure.

Dynamical properties at T=0?

AO(ω) = − 1

πlimη→0

⟨

ψ0

∣

∣

∣

∣

O† 1

ω −H + E0 + iηO∣

∣

∣

∣

ψ0

⟩

=∑

n

|〈ψn|O|ψ0〉|2δ[ω − (En − E0)]

↪→ complete spectrum necessary!?

Kernel polynomial method (KPM) [4]

• Expansion of δ[. . .] using Chebyshev polynomials

AO(x) =1

π√

1 − x2

(

µO0

+ 2∞∑

m=1

µOmTm(x)

)

• Determination of moments

µOm =1∫

−1dxTm(x)AO(x) = 〈ψ0|O†Tm(X)O|ψ0〉

by iterative MVM (X = (H− b)/a→ En ∈ [−1, 1])

• Problem: M < ∞! ; Gibbs oscillations ; trunca-tion errors! Solution: Damping factors (e.g., Jacksonor Lorentz kernels)!

• (FFT) Reconstruction of AO(x) from the M calcu-lated moments via linear approximation (KPM) ornonlinear optimisation procedure (MEM).

Advantages of KPM:

• High-resolution applications – polarons!

• Uniform reconstruction of spectra – gap features)!

• CPU-time (∝ MD); trace – average over random|r〉.

Cluster Perturbation Theory (CPT) [5]

Green functionG(k, ω) on an infinite lattice (Nc = ∞)?

G (k,ω)mnc

tt t t t

GCPT t tt t t

(ω)mnc

G (ω)mnc

G (ω)

• We have: Green function Gcmn(ω) on finite cluster(s)

of Nc sites (OBC) !

• 1st order perturbation in V =∑

[

t−−−]

G(1)ij (ω) = Gcij(ω) +

∑

rs

Gcir(ω)VrsG(1)sj (ω)

G(1)mn(K,ω) =

(

Gc(ω)

1 − V (K)Gc(ω)

)

mn, (K = Nc k)

• Fourier transform:

GCPT(k, ω) =1

Nc

Nc∑

m,n=1

G(1)mn(K,ω)e−ik(m−n)

Observables

• Single-particle spectral function:

A±k (ω) =

∑

m

|〈m(Nel±1)|c±k |0(Nel)〉|2

×δ[ω ∓ (E(Nel±1)m − E

(Nel)0 )]

with c+k = c†k, c−k = ck.

A−k (ω) [A+

k (ω)] is related to the [inverse (I)] photo-emission (PE) of an electron.

Note that Ak(ω − µ) = −1πImG(k, ω − µ), where

the imaginary-time Green functionG(k, τ ) can be di-rectly measured by QMC.

• Partial densities of states (DOS):

ρ±(ω) =∑

k

A±k (ω) .

• Optical conductivity, Reσ(ω) = Dδ(ω) + σreg(ω),the regular part of which is given by

σreg(ω) =π

N

∑

n>0

|〈n||0〉|2E0 − En

δ(ω − (E0 − En)) ,

where =iet∑

i(c†ici+1−c

†i+1ci ) (current operator).

Numerical results

Weak coupling

λ = 0.1, α = 0.4 – QMC results for N = 32, βt = 8

single-particle spectral function (QMC)

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4 6 8

0 2 4 6 8

10A(k,ω−µ)

(a) n = 0.1

k / π

(ω−µ) / t

A(k,ω−µ)

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4 6 8

0 2 4 6 8

10A(k,ω−µ)

(b) n = 0.2

k / π

(ω−µ) / t

A(k,ω−µ)

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4 6 8

0 2 4 6 8

10A(k,ω−µ)

(c) n = 0.3

k / π

(ω−µ) / t

A(k,ω−µ)

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4 6 8

0 2 4 6 8

10A(k,ω−µ)

(d) n = 0.4(d) n = 0.4(d) n = 0.4

k / π

(ω−µ) / t

A(k,ω−µ)

density of states

ρ(ω

− µ

)

-5 -4 -3 -2 -1 0 1 2 3 4 5(ω − µ) / t

n = 0.2

n = 0.1

n = 0.4

n = 0.5

n = 0.3

• pronounced QP peak ∀n• gap feature develops for n→ 0.5 ⇔ CDW

↪→ dressed “electronic” QPs!

Strong coupling

λ = 2.0, α = 0.4 – QMC results for N = 32, βt = 8

single-particle spectral function (QMC)

0.2

0.4

0.6

0.8

1 -12 -8 -4 0 4 8 12

0

0.1

0.2

0.3

0.4A(k,ω−µ) (a) n = 0.1

k / π

(ω−µ) / t

A(k,ω−µ)

0.2

0.4

0.6

0.8

1 -12 -8 -4 0 4 8 12

0

0.1

0.2

0.3

0.4A(k,ω−µ) (b) n = 0.2

k / π

(ω−µ) / t

A(k,ω−µ)

0.2

0.4

0.6

0.8

1 -12 -8 -4 0 4 8 12

0

0.1

0.2

0.3

0.4A(k,ω−µ) (c) n = 0.3

k / π

(ω−µ) / t

A(k,ω−µ)

0.2

0.4

0.6

0.8

1 -12 -8 -4 0 4 8 12

0

0.1

0.2

0.3

0.4A(k,ω−µ) (d) n = 0.4

k / π

(ω−µ) / t

A(k,ω−µ)

density of states

0.2

0.2

0.2

ρ(ω

− µ

)

0.2

-12 -8 -4 0 4 8 12(ω − µ) / t

0

0.2

n = 0.2

n = 0.1

n = 0.4

n = 0.5

n = 0.3

• exponentially small spectral weight at µ ∀n• QP band - “gap” - broad incoherent feature

↪→ small “polaronic” QPs!

Intermediate coupling

λ = 1.0, α = 0.4, N = 10 (ED) and N = 10, βt = 8 (QMC)

single-particle spectral function (ED)

-6 -4 -2 0 2 4ω / t

00.5

11.5

00.5

11.5

00.51.01.5

A+ (

k,ω

), A

− (k,

ω)

00.5

11.5

00.5

11.5

00.5

11.5

k=0

k=π

k=4π/5

k=3π/5

k=2π/5

k=π/5

(a) n = 0.1

-6 -4 -2 0 2 4ω / t

00.5

11.5

00.5

11.5

00.5

11.5

A+ (

k,ω

), A

− (k,

ω)

00.5

11.5

00.5

11.5

00.5

11.5

k=0

k=π

k=4π/5

k=3π/5

k=2π/5

k=π/5

(b) n = 0.2

-6 -4 -2 0 2 4ω / t

00.5

11.5

00.5

11.5

00.5

11.5

A+ (

k,ω

), A

− (k,

ω)

00.5

11.5

00.5

11.5

00.5

11.5

k=0

k=π

k=4π/5

k=3π/5

k=2π/5

k=π/5

(c) n = 0.3

-6 -4 -2 0 2 4ω / t

00.5

11.5

00.5

11.5

00.5

11.5

A+ (

k,ω

), A

− (k,

ω)

00.5

11.5

00.5

11.5

00.5

11.5

k=0

k=π

k=4π/5

k=3π/5

k=2π/5

k=π/5

(d) n = 0.4

integrated spectral weight (QMC vs. ED)

-6 -4 -2 0 2 4 6ω / t

0

0.5

1

Int.

spec

tral

wei

ght

EDQMC

k=0 k=2π/5 k=π

spectral functions (CPT)

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4

0

0.5

1

1.5

A-(k,ω), A+(k,ω)(a) n = 0.1

k / π

ω / t

A-(k,ω),

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4

0

0.5

1

1.5

A-(k,ω), A+(k,ω)(a) n = 0.1

k / π

ω / t

A-(k,ω),

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4

0

0.5

1

1.5

A-(k,ω), A+(k,ω)(b) n = 0.2

k / π

ω / t

A-(k,ω),

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4

0

0.5

1

1.5

A-(k,ω), A+(k,ω)(b) n = 0.2

k / π

ω / t

A-(k,ω),

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4

0

0.5

1

1.5

A-(k,ω), A+(k,ω)(c) n = 0.3

k / π

ω / t

A-(k,ω),

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4

0

0.5

1

1.5

A-(k,ω), A+(k,ω)(c) n = 0.3

k / π

ω / t

A-(k,ω),

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4

0

0.5

1

1.5

A-(k,ω), A+(k,ω)(d) n = 0.4

k / π

ω / t

A-(k,ω),

0.2

0.4

0.6

0.8

1 -6 -4 -2 0 2 4

0

0.5

1

1.5

A-(k,ω), A+(k,ω)(d) n = 0.4

k / π

ω / t

A-(k,ω),

• n = 0.1: Fermi level lies within the polaron band;band flattening at large wave-vectors k.

• Incoherent part of the spectrum closely follows thefree electron (cosine) dispersion.

• Polaron band merges with incoherent excitations atabout n = 0.3 ; no clear separation of the QP peak!

↪→ density-driven change of the “character” of thecharge carriers!

density of states (CPT)

0

0.2

0.4

0

0.2

0.4

0

0.2

0.4

ρ+(ω

), ρ

- (ω)

0

0.2

0.4

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6(ω - E

F) / t

0

0.2

0.4

n = 0.2

n = 0.1

n = 0.4

n = 0.5

n = 0.3

• n ↗: small (polaronic) DOS at EF (∝ e−g2

) →“metallic” DOS at EF (“polaron dissociation”) →pseudo-gap - precursor of CDW (∃ for λ > λc(ω0))

↪→ crossover from polaronic to metallic behaviour!

partial DOS & optical response (ED)

λ = 1.0, α = 0.4, i.e., g2 = 5, N = 10

0.0

0.2

0.4

0.6

0.8

1.0

ρ+(ω

), ρ

- (ω)

0.0

0.2

0.4

0.6

0.8

ρ+(ω

), ρ

- (ω)

-5.0 0.0 5.0(ω - E

F) / t

0.0

0.2

0.4

0.6

0.8

ρ+(ω

), ρ

- (ω)

0.0

1.0

2.0

σreg (ω

)

0.0

1.0

2.0

σreg (ω

)

0.0 2.0 4.0 6.0 8.0ω / t

0.0

1.0

2.0

σreg (ω

)

n=0.1

n=0.3

n=0.5

For comparison, the analytical strong coupling re-sult [6]

σreg(ω) =σ0n

ω0gωexp

[

(ω − 2g2ω0)2

4g2ω20

]

,

is included (σ0 = 8) – noticable deviations!.

Anti-adiabatic (strong-coupling) regime

λ = 2.0, α = 4.0, N = 10 (ED)

-12 -8 -4 0 4 8 12ω / t

0

1

0

1

0

1

A+ (

k,ω

), A

− (k,

ω)

0

1

0

1

0

1 k=0

k=π

k=4π/5

k=3π/5

k=2π/5

k=π/5

• Phonon absorption bands with electronic satellites.

• Spectra are mainly unchanged going from n = 0.1[green/blue] to n = 0.3 [red/black] (i.e., ther are al-most no finite-density effects).

↪→ Carriers are small polarons also at large fillings!

Conclusions

In the physically most important adiabatic intermediateelectron-phonon coupling regime, for which no analyt-ical results are available, we observe a dissociation ofpolarons with increasing band filling, leading to nor-mal metallic behaviour, while for parameters favour-ing small polarons, no such density-driven changes oc-cur. The present work points towards the inadequacy ofsingle-polaron theories for a number of polaronic ma-terials such as the manganites.

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[2] Hohenadler et al., Phys. Rev. B 71, 245111 (2005).

[3] Bauml, Wellein, Fehske, Phys. Rev. B 58, 3663 (1998);Weiße et al., Phys. Rev. B 62, R747 (2000).

[4] Weiße et al., cond-mat/0504627 (review).

[5] Senechal, Perez, Plouffe, Phys. Rev. B 66, 075129 (2002);Hohenadler, Aichhorn, von der Linden, Phys. Rev. B 68, 184304 (2003).

[6] D. Emin, Phys. Rev. B 48, 13691 (1993).