QMC Simulations within the DMFT - komet 337

Quantum Monte Carlo (QMC) Simulations 

within the Dynamical Mean-field Theory (DMFT) 

Nils Blümer 

TP6, WA Prof. van Dongen (KOMET337) 

Outline 

Introduction: Hubbard model, DMFT, self-energy 

General Monte Carlo principles 

Hirsch-Fye QMC algorithm; error bars 

Mott Insulator (1 band): QMC vs. ePT and SFT/DIA 

LDA+DMFT(QMC) for La1−xSrxTiO3; MEM 

Summary and Preview 

Theorieseminar der DFG-Forschergruppe 559 · Nov 12, 2004 · Nils Blümer (Univ. Mainz) ⊳ ←↪ △ ⊲ 1

H = 

�Ne 

i=1 

Introduction: Hubbard model, DMFT 

pi 2 

2m + 

L� 

k=1 

P k 2 

2Mk 

+ � 

k

H = 

�Ne 

i=1 


pi 2 

2m + 

L� 

k=1 

P k 2 

2Mk 

+ � 

k

H = 

�Ne 

i=1 


pi 2 

2m + 

L� 

k=1 

P k 2 

2Mk 

+ � 

k

H = 

�Ne 

i=1 


pi 2 

2m + 

L� 

k=1 

P k 2 

2Mk 

+ � 

k

H = 

�Ne 

i=1 


pi 2 

2m + 

L� 

k=1 

P k 2 

2Mk 

+ � 

k

ˆH = � 

(i,j),σ 

tij(ĉ † 

iσ ĉ jσ 

Perturbation theory, e.g. 

+ h.c.) + U � 

• U → 0: Hartree-Fock (uncorrelated) 

i 

ˆni↑ˆni↓ 

• t/U → 0: half filling (n = 1) � Heisenberg model 

• T → ∞, n → 0 

• ( Vion → 0 � jellium model � LDA) 

d = 1: Bethe ansatz, DMRG 

finite clusters: ED, QMC 


ˆH = � 

(i,j),σ 

tij(ĉ † 

iσ ĉ jσ 

Perturbation theory, e.g. 

+ h.c.) + U � 

• U → 0: Hartree-Fock (uncorrelated) 

i 

ˆni↑ˆni↓ 

• t/U → 0: half filling (n = 1) � Heisenberg model 

• T → ∞, n → 0 

• ( Vion → 0 � jellium model � LDA) 

Dynamical mean-field theory (DMFT): local self-energy Σ(k, ω) ≡ Σ(ω) 

+ non-perturbative 

� valid at MIT 

+ dynamical on-site 

correlations preserved 

+ exact for Z → ∞ 

d = 1: Bethe ansatz, DMRG 

finite clusters: ED, QMC 

+ in thermodynamic limit d=2: Z = 4 bcc: Z = 8 fcc: Z = 12 DMFT: Z = ∞ 


Excursus: Green function and self-energy 

Noninteracting (1-band) system: dispersion ɛk, density of states ρ(ɛ) = 1 

V B 

Green function (within DMFT for µ = 0): Gk(ω) = 

Σ(ω) 

G k (ω) 

4 

3 

2 

1 

0 

-1 

-1 0 1 2 3 4 5 

6 

4 

2 

0 

-2 

-4 

ω-ε k 

-6 

-1 0 1 2 3 4 5 

ω-ε k 

Σ = 0 

Σ(ω) 

G k (ω) 

4 

3 

2 

1 

0 

-1 

-1 0 1 2 3 4 5 

6 

4 

2 

0 

-2 

-4 

ω-ε k 

-6 

-1 0 1 2 3 4 5 

ω-ε k 

Σ = Σ 0 

� d 3 k δ(ɛ − ɛk) 

1 

ω − ɛk − Σ(ω) = (ω − ɛk − Σ ′ ) + iΣ ′′ 

(ω − ɛk − Σ ′ ) 2 + Σ ′′2 

G k (ω) 

4 

3 

2 

1 

0 

-1 

-1 0 1 2 3 4 5 

6 

4 

2 

0 

-2 

-4 

-6 

-1 0 1 2 3 4 5 

A(ω) = − 1 

π ImG(ω + i0+ ) Σ→0 

−→ ρ(ω); G(ω) ≡ Gii(ω) = � dɛ 

Σ(ω) 

ω-ε k 

ω-ε k 

Σ = Σ(ω) 

ρ(ɛ) 

ω−ɛ−Σ(ω) 


Example: half-filled frustrated Hubbard model, U=W=4, T=0.05 

Σ (ω) 

A ε (ω) 

2 

1 

0 

-1 

-2 

-3 

-4 

2.5 

2 

1.5 

1 

0.5 

0 

Re Σ 

Im Σ 

-4 -2 0 2 4 

∫ dω A ε (ω) 

1.15 

1.1 

1.05 

1 

interpolated 

raw 

0.95 

-2 -1 0 

ε 

1 2 

ω 

-4 -3 -2 -1 0 1 2 3 4 

ω 

ε=0.0 

ε=0.5 

ε=1.0 

ε=1.5 

ε=2.0 

G(ω) 

0.6 

0.4 

0.2 

0 

-0.2 

-0.4 

-0.6 

-0.8 

-1 

Re G 

Im G 

-4 -2 0 2 4 

Theorieseminar der DFG-Forschergruppe 559 · Nov 12, 2004 · Nils Blümer (Univ. Mainz) ⊳ ←↪ △ ⊲ 5 

ω

Iterative solution of 

DMFT equations 

Impurity solver: 

• Quanten-Monte-Carlo (QMC) 

G(τ) = −〈ΨΨ ∗ 〉 G[τ] 

✬ ✩ 

G G 

✻ 

G 

❵❵❵❵❵❵❵❵ 

� 

G(iωn) = dɛ 

• Iterative perturbation theory (IPT; not controlled) 

• Non-crossing approximation (NCA; not controlled) 

• Exact diagonalization (ED; large finite-size errors) 

• Numerical renormalization group (NRG; 1-2 bands) 

• Density matrix renormalization group (DMRG) 

• Self-energy functional theory (SFT) + ED 

Direct d = ∞ solution: PT, ePT 

G 

❵❵❵❵❵❵❵❵ 

−1 (iωn)=G −1 (iωn)+Σ(iωn) 

ρ(ɛ) 

iωn−ɛ−Σ(iωn) 

Σ ← Σ0 

✫ ✪ 


❄ 

−→

Excursus: Monte-Carlo Principles 

Example: Computation of average depth h of a lake from depth distribution h(x1, x2) 




Deterministic: lattice + trapezoid rule 

N = V/(∆x) d measurements 

∆h ∝ (∆x) 2 ∝ N −2/d 






∆h ∝ (∆x) 2 ∝ N −2/d 






∆h ∝ (∆x) 2 ∝ N −2/d 

Trapez 

Δx 

0 0.2 0.4 0.6 0.8 1 

Δx / Δx0 Theorieseminar der DFG-Forschergruppe 559 · Nov 12, 2004 · Nils Blümer (Univ. Mainz) ⊳ ←↪ △ ⊲ 7





∆h ∝ (∆x) 2 ∝ N −2/d 

Stochastic: simple Monte-Carlo 

N “configurations”, equally probable 

� 

var{h} −1/2 

∆h � ∝ N 

N 

Trapez 

Δx 

QMC 

N 

0 0.2 0.4 0.6 0.8 1 

Δx / Δx0 0 100 200 300 400 500 

N 






∆h ∝ (∆x) 2 ∝ N −2/d 

Stochastic: simple Monte-Carlo 

N “configurations”, equally probable 

� 

var{h} −1/2 

∆h � ∝ N 

N 

Stochastic: Importance sampling MC 

factorization: h(x) = p(x) o(x); 

p(x) normalized, var{o} ≪ var{h} 

� 

var{o} 

∆h � ∝ N −1/2 

Neff 

Trapez 

Δx 


N 


N 

0 0.2 0.4 0.6 0.8 1 

Δx / Δx0 0 100 200 300 400 500 

N 

0 100 200 300 400 500 

N 


Application of Monte Carlo in Statistical Physics 

〈O〉 = � 

i 

pi Oi , pi = e−Ei /(kBT ) 

Z 

≡ ˜pi 

� 

, Z = 

Z 

i 

e −Ei /(kBT ) 

Simple Monte Carlo: Estimation of both sums from a number N of equally probable configurations. 

Problem: typically � var{p} ≫ p. 


Application of Monte Carlo in Statistical Physics 

〈O〉 = � 

i 

pi Oi , pi = e−Ei /(kBT ) 

Z 

≡ ˜pi 

� 

, Z = 

Z 

i 

e −Ei /(kBT ) 

Simple Monte Carlo: Estimation of both sums from a number N of equally probable configurations. 

Problem: typically � var{p} ≫ p. 

Importance Sampling MC: Probability distribution given by Boltzmann weights pi. 

Problem: Normalization 1/Z unknown. 

Solution: generate probability distribution by random walk, e.g. using Metropolis algorithm: 

〈O〉 = 

� 

i ˜piOi 

� 

i ˜pi 

P {i → j} = min{pj/pi, 1} , pj/pi = e (Ej−Ei )/(kBT ) 

−→ 〈O〉 = 

′� 

Oi ≈ 

+ Precise computation of observable averages 〈O〉 

i 

N+N 

�0 

n=1+N 0 

− Not accessible by construction: partition function Z, free energy F 

Oin; 〈(∆O) 2 〉 ∝ var{O} 

N 


Hirsch-Fye QMC algorithm for DMFT impurity problem 

Wanted: Green-Funktion G(ω) 

Treatment in imaginary time using fermionic Grassmann variables ψ, ψ ∗ : 

Gσ(τ2 − τ1) ≡ Gσ(τ1, τ2) = 1 

� 

D[ψ] D[ψ 

Z 

∗ ] ψσ(τ1)ψ ∗ 

σ (τ2) e A A = 

, 

A0 − U � 

� β 

dτ ψ 

2 

∗ 

σ (τ)ψσ (τ)ψ∗ σ ′(τ)ψσ ′(τ) 

discretization β = Λ ∆τ, Trotter decoupling, discrete Hubbard-Stratonovich transformation 

σσ ′ 

0 

Wick theorem: 

� 

M det{M} 

G = � 

det{M} 

Metropolis MC importance sampling over auxiliary Ising field, 2 Λ configurations, 50 � Λ � 400 

+ nonperturbative, numerically exact 

− effort scales as T −3 


Hirsch-Fye QMC algorithm for DMFT impurity problem 

Wanted: Green-Funktion G(ω) 

Treatment in imaginary time using fermionic Grassmann variables ψ, ψ ∗ : 

Gσ(τ2 − τ1) ≡ Gσ(τ1, τ2) = 1 

� 

D[ψ] D[ψ 

Z 

∗ ] ψσ(τ1)ψ ∗ 

σ (τ2) e A A = 

, 

A0 − U � 

� β 

dτ ψ 

2 

∗ 

σ (τ)ψσ (τ)ψ∗ σ ′(τ)ψσ ′(τ) 

discretization β = Λ ∆τ, Trotter decoupling, discrete Hubbard-Stratonovich transformation 

σσ ′ 

0 

Wick theorem: 

� 

M det{M} 

G = � 

det{M} 

Metropolis MC importance sampling over auxiliary Ising field, 2 Λ configurations, 50 � Λ � 400 

+ nonperturbative, numerically exact 

− effort scales as T −3 

− no information for ω � ωNyquist 

G(τ) 

0.5 

Δτ = 0 

Δτ > 0 

0 

0 

τ/β 

1 


Im G(iω n ) 

ω n

Contributions to DMFT-QMC error bars: 

• statistical fluctuations + warm-up 

• convergency (of self-consistency cycle) 

• discretization (Trotter error and Fourier transform) 

Example: half-filled frustrated Hubbard model, U = 5, W = 4, T = 0.04 (Mott insulator) 

D 

0.0236 

0.0235 

0.0234 

0.0233 

0.0232 

0.0231 

0.023 

0.0229 

0.0228 

0 10 20 30 40 50 60 70 

it 

Δτ=0.12 

Δτ=0.15 

Δτ=0.18 

Δτ=0.20 

E 

-0.107 

-0.108 

-0.109 

-0.11 

-0.111 

-0.112 

0 10 20 30 40 50 60 70 

it 

Δτ=0.12 

Δτ=0.15 

Δτ=0.18 

Δτ=0.20 


Contributions to DMFT-QMC error bars: 

• statistical fluctuations + warm-up 

• convergency (of self-consistency cycle) 

• discretization (Trotter error and Fourier transform) 

Example: half-filled frustrated Hubbard model, U = 5, W = 4, T = 0.04 (Mott insulator) 

D 

D 

0.0236 

0.0235 

0.0234 

0.0233 

0.0232 

0.0231 

0.023 

0.0229 

0.0228 

0.024 

0.023 

0 10 20 30 40 50 60 70 

it 

Δτ=0.12 

Δτ=0.15 

Δτ=0.18 

Δτ=0.20 

0.022 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 

Δτ 2 

E 

E 

-0.107 

-0.108 

-0.109 

-0.11 

-0.111 

-0.112 

-0.104 

-0.106 

-0.108 

-0.11 

-0.112 

-0.114 

-0.116 

0 10 20 30 40 50 60 70 

it 

Δτ=0.12 

Δτ=0.15 

Δτ=0.18 

Δτ=0.20 

-0.118 

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 

Δτ 2 

next: prerequisite for / proof of extreme precision 


Fourier transformation schemes: 

self-energy (T = 0.1, U = 5.0) 

a) “Ulmke smoothing”, 

b) improved “Smoothing”, 

c) scheme with analytic 

high-frequency corrections 

even stronger effects at lower T 

low-frequency errors of Σ(ω) small 

in b) und c) 

Im Σ(iω n ) 

0 

-1 

-2 

-3 

-4 

-5 

-6 

-7 

-8 

U=5.0 

Δτ=0.4 

Δτ=0.25 

Δτ=0.20 

Δτ=0.156 

Δτ=0.125 

Δτ=0.1 

ω n Im Σ(iω n ) 

0 

-1 

-2 

-3 

-4 

-5 

-6 

Im Σ(iω n ) 

0 

-1 

-2 

-3 

-4 

-5 

-6 

-7 

-8 

U=5.0 

0 5 10 15 

ωn 20 25 30 

0 5 10 15 

ωn 20 25 30 

Δτ=0.4 

Δτ=0.25 

Δτ=0.20 

Δτ=0.156 

Δτ=0.125 

Δτ=0.1 


0 5 10 15 

ωn 20 25 30 

b) U=5.0 

c) 

Im Σ(iω n ) 

0 

-1 

-2 

-3 

-4 

-5 

-6 

-7 

-8 

0 

-1 

-2 

-3 

-4 

-5 

-6 

Δτ=0.4 

Δτ=0.25 

Δτ=0.20 

Δτ=0.156 

Δτ=0.125 

Δτ=0.1 

a) 

0 5 10 15 

ωn 20 25 30 


0 

-1 

-2 

-3 

-4 

-5 

-6 

0 5 10 15 

ωn 20 25 30 

0 5 10 15 

ωn 20 25 30 


E 

-0.09 

-0.1 

-0.11 

Mott insulator: energy + double occupancy I 

2 nd order 

4 th order 

10 th order PT 


5 5.5 6 

Excellent agreement at U = 6.0. 

U 

10 th 


order PT 

4 th 

order 

2 nd 

order 

5 5.5 6 

U 

0.025 

0.02 

0.015 


D

(E - E PT ) /10 -4 

0 

-1 

-2 

Mott insulator: energy + double occupancy II 

12 th order 

20 th order 

40 th order 

QMC + ePT 

5 5.5 6 

U 

QMC + ePT 

40 th order 

20 th order 

12 th order 

5 5.5 6 

Mott insulator: Uc1, critical exponents, low-T parameter for Uc(T ) 

high-precision results for E, D at all U (parametrizations available) � benchmark 

[Blümer, Kalinowski, cond-mat/0404568 (2004), cond-mat/0407442 (2004)] 


U 

8 

7 

6 

5 

4 

3 

2 

1 

0 

(D - D PT ) /10 -4

La1−xSrxTiO3 

• perovskite structure 

• 1−x t2g electrons per site 

• AF for x � 0.05 

• strongly correlated metal 

Realistic LDA+DMFT(QMC) calculations 

• density functional theory (LDA) fails 

Ti 3d 

e 

g 

t 2g 

• Ti 

• La, Sr 

• O 


multi-band issues 

2M(2M − 1) 

2 

Hubbard-Stratonovich fields 

approximation: Hund terms J αα ′ s α · s α′ 

−→ Jαα ′ s α 

z sα′ 

z 

to avoid sign problem in QMC 



2M(2M − 1) 

2 



Analytic continuation 

Aim: (ill-conditioned) inversion of G(τ) = 

−→ Jαα ′ s α 

z sα′ 

z 

� 


dω exp[−τω] 

1 + exp[−βω] A(ω) 

Maximum entropy method (MEM): introduce entropy function to find smoothest solution A(ω) 

compatible with G(τ) 

G(τ) MEM 

−→ ImG(ω) Kra-Kro 

−→ ReG(ω) 

MEM: not fully controlled; no ∆τ extrapolation possible 



2M(2M − 1) 

2 



Analytic continuation 

Aim: (ill-conditioned) inversion of G(τ) = 

−→ Jαα ′ s α 

z sα′ 

z 

� 


dω exp[−τω] 

1 + exp[−βω] A(ω) 

Maximum entropy method (MEM): introduce entropy function to find smoothest solution A(ω) 

compatible with G(τ) 

G(τ) MEM 

−→ ImG(ω) Kra-Kro 

−→ ReG(ω) 

MEM: not fully controlled; no ∆τ extrapolation possible 

for ARPES, σ(ω): use 2-dimensional Newton scheme to invert Dyson equation for G(ω) −→ Σ(ω) 


Photoemission spectra for La1−xSrxTiO3 (x=0.06) 

Intensity (arb. Units) 

LDA (U=0) 

QMC: U=5.0 

Exp. (T=80K) 

Exp. (T=150K) 

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 

Energy (eV) 

[Nekrasov, Held, Blümer, Poteryaev, Anisimov, Vollhardt, EPJB 18, 55 (2000)] 


Computers 

speed (accepted Megaunitsweeps ~ MFlops) on 2 CPUs 

350 

300 

250 

200 

150 

100 

50 

0 

250 

200 

150 

100 

50 

0 

1 band 

3 bands 

0 50 100 150 200 250 300 

# of time slices Λ (linear Matrix size) 

Opteron 244 (32 bit) 

HP Superdome 

P4 2.6 GHz, serial 

Athlon MP 2200+ 

Athlon MP 1.2 GHz 

Group cluster (2nd stage) 

JUMP cluster (NIC Jülich) 


Summary 

“derivation” of (1-band/multi-band) Hubbard model 

characterization of DMFT 

role of self-energy 

Monte Carlo principles 

QMC: principles and caveats (convergency, ∆τ extrapolation) 

extreme-precision QMC results for 1-band model (vs. ePT, SFT/DIA) 


Summary 







multi-band issue: spin-flip terms (not in DIA) 

challenge: analytic continuation using MEM 

LDA+DMFT: significant improvements for PES of La1−xSrxTiO3 

technical prerequisite: computer power 


Summary 







multi-band issue: spin-flip terms (not in DIA) 

challenge: analytic continuation using MEM 

LDA+DMFT: significant improvements for PES of La1−xSrxTiO3 

technical prerequisite: computer power 

Further talks: 

DMFT for double perowskite models 

high-frequency corrected QMC (multi-band); observables 

MEM and LDA+DMFT

QMC Simulations within the DMFT - komet 337

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