Tensor learning approach to sparse QMC sampling of two-particle - - PowerPoint PPT Presentation

Tensor learning approach to sparse QMC sampling of two-particle Green’s function in DMFT Hiroshi SHINAOKA

Collaborators

• N. Chikano, J. Otsuki, M. Ohzeki, K. Yoshimi, K. Haule,
• M. Wallerberger, J. Li, E. Gull, D. Geffroy, J. Kuneš

Matsubara Green’s functions

χ χ Γ

+ =

Challenges at low T and complex systems Storage Manipulation

Many perturbative theories, dynamical mean-field theory, quantum Monte Carlo Intermediate representation (IR) Sparse sampling and tensor learning Main focus of my talk: two-particle quantities

Intermediate representation (IR)

Gα

l = −Sα l ρα l

Model independent

• rthonormal basis sets

α = F (fermion), B (boson)

HS, J. Otsuki, M. Ohzeki, K. Yoshimi, PRB 96, 035147 (2017)

• J. Otsuki, M. Ohzeki, HS, K. Yoshimi, PRE 95, 061302(R) (2017)
• N. Chikano, J. Otsuki, HS, PRB 98, 035104 (2018)

Gα(iωn) = ∫

ωmax −ωmax

dω Kα(iωn, ω)ρα(ω) Kα(iωn, ω) = −

∞

∑

l=0

Sα

l Uα l (iωn)Vα l (ω)

ρ(ω)

Size ~ O(log Λ) Λ=102 Λ=104

Λ≡βωmax

Λ ≡ βωmax

Metal

10 20 30 l 10−9 10−5 10−1

Sl |ρl| |Gl|

Insulator

Simple example

−1 +1

β = 100

Gα

l = −Sα l ρα l

G(τ) ρ(ω)

Super exponential decay

Scaling at low T

102 104 106 β 101 102 103 104 N

Legendre Chebyshev IR iωn

• Number of coefficients

O( β) O(β)

O(log β)

• L. Boehnke et al. (2011)
• E. Gull et al. (2018)

Matsubara frequency with tail Legendre and Chebyshev

IR

Single pole |δG(τ=0)| < 10-8

ω

• 1

β τ

ωmax = 10 eV, T = 10 K → ωmaxβ = 104

• Python/C++ implementation of IR basis


https://github.com/SpM-lab/irbasis

• Application to DMFT with ED solver
• Y. Nagai and H. Shinaoka, JPSJ 88, 064004 (2019)

How to do math?

G(iωn) = 1 iωn − H − Σ(iωn)

Simplest example: Dyson equation

O(NiωN3

• rbital) ≃ O(βN3
• rbital)

O(log β N3

• rbital)

Solve at all Matsubara frequencies Sparse sampling

• J. Li, M. Wallerberger, N. Chikano, E. Gull, HS,

in preparation

Σ(iωn) → G(iωn)

• Model-independent sampling points

(# of sampling points) = (# of IR coefficients to be fitted)

• Stable fitting

Sparse sampling

Λ=104

• 20

40 60 80 l 10−18 10−14 10−10 10−6 10−2 Gl Gl |Gl − Ginv

l

|

• Known

GF(iωn) =

Nl−1

∑

l

UF

l (iωn)Gl

Unknown

We know how many coefficients are required. (≦ 100)

UF

l (iωn)

n

• J. Li, M. Wallerberger, N. Chikano,
• E. Gull, HS, in preparation

Example: quantum chemistry calculations

50 100 150 200 250 300 350 400 number of coefficients 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 |∆Etot| [H]

GF(2) new Dyson, H10(R = 1), β = 103, STO-6g, Etot = −3.8101298016 IR basis (Λ = 104) Chebyshev basis

• Chem. accuracy

T ~ 260 K

Σ(τi) G(τi) Gl Σl Σ(iωn) G(iωn)

τi ∈ sampling points

G(iωn) = (iωn − H − Σ(iωn))−1

ωn ∈ sampling points

The same idea applies to Eliashberg and GW-type equations!

O(NτN5

• rb)

GF2

Hydrogen atom chain

• J. Li, M. Wallerberger, N. Chikano,
• E. Gull, HS, in preparation

Fitting Fitting

Why two-particle objects?

Vertex corrections My motivation: Dynamical mean-field theory (DMFT)

• Dynamic susceptibility (inelastic scattering experiments)
• Non-local correlations beyond DMFT

χ χ Γ

+ =

Bethe-Salpeter equation

Rich frequency structure

Power-law decay

Matsubara representation

• Large size: O(β3)
• More indices for spin, orbital, wave vector…

Discontinuities

• M. Wallerberger, PhD thesis

β

τ

fermionic frequency fermionic frequency

cf . G(τ)

• IR approach frequency dependence
• Dimensionality reduction

⟨Tτc(τ1)c†(τ2)c(τ3)c†(τ4)⟩

IR approach

(a)

Discontinities

(c) (c) (d)

Discontinuities

= + +

HS, J. Otsuki, M. Ohzeki, K. Yoshimi, K. Haule, M. Wallerberger, E. Gull, PRB 97, 205111 (2018)

0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 10-4 10-3 10-2 10-1 100 101 102 l1 l2 g(1)

l1 l2

0 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 10-4 10-3 10-2 10-1 100 101 102 l1 l2 g(2)

l1 l2

0 2 4 6 8 10 12 14 16 2 4 6 8 10 10-4 10-3 10-2 10-1 100 101 102 l1 g(3)

l1 l2

• Exponential decay
• Small size: O((log β)3)
• QMC measurement difficult.

Dynamic susceptibility in DMFT

計算物性物理

ハイパフォーマンスコンピューティング モデル計算 第一原理計算

現実のシュレディンガー方程式を忠実に 解き、物質の「個性」を定量的に再現 物理現象の本質をとらえた簡 潔な模型を解析 独自アルゴリズムによって計算物理学のフロンティアを開拓

相互作用する多数の電子が起こす非自明な集団運動

磁性、超伝導、金属絶縁体転移、トポロジカル相 ハバード模型、スピン模型 密度汎関数理論、バンド計算 、並列コンピューティング 量子モンテカルロ法、動的平均場近似法

共同研究

東大、スイス連邦工科大学チューリッヒ校、フリブール大学 スイス 、 エコール・ポリテクニーク フランス など

• Single-particle level: A(k, ω)
• Two-particle level: χ(q, ω)

Review: G. Kotliar et al., Rev. Mod. Phys. 78, 865 (2006)

Dynamical effective bath

e-

Correlated atom

ARPES

Inelastic neutron scattering

Xloc

abcd(iωn, iωn′; iνm)

χabcd(q, iνm) boson fermion Huge data spin/orbital

Local two-particle Green’s functions

QMC

Bethe-Salpeter equation

Three-orbital t2g model (atom)

U = 5 eV, J = 0.1 eV, β = 10 Exact diagonalization

Gaσ1,bσ2,cσ3,dσ4(iωn, iωn′, iνm)

νm = 0

n′ = 2

Gxy↑,xy↑,xy↑,xy↑ n′

Λ = 1000

a,b,c,d = dxy, dyz, dzx

νm = ν2

Gxy↑,xy↑,yz↑,yz↑

n′ = 2

n n′ n

Sparse sampling 3 GB → 180 KB (D=8)

Perfect fit with small D

Static susceptibility in DMFT

Single-site Hubbard model on square lattice

U = 12, band width = 8, β=2.5 (slightly above AF Tc) Hubbard-I solver Bethe-Salpeter equation in Matsubara frequency D=5

5 10 15 D 10−5 10−4 10−3 10−2 10−1 |∆Xloc|max/|Xloc|max

Relative fitting error

Exponential convergence Indistinguishable from exact values AF spin order

interpolation Sparse sampling

Tensor regression

Implementation: DCore v2β + BSE-tool

χ(q)

More realistic systems (work in progress)

• D. Geffroy, J. Kaufmann, A. Hariki, P. Gunacker, A. Hausoel and
• J. Kuneš, PRL 122, 127601 (2019)

DMFT calculations of dynamic susceptibilities

훥 T

exciton condensate

Two-band model with crystal-field splitting

• Warm sampling in Legendre

basis by ALPS/CT-HYB

• D. Geffroy@Brno
• J. Kuneš@Wien

40 hours with 840 processes

Ga↑a↑a↑a↑(iωn, iωn′, iν0)

n′ n

Ga↑a↑b↓b↓(iωn, iωn′, iν10)

n

Sparse QMC sampling with ALPS/CT-HYB

D=60 is enough.

β=60

Box (-1500 < n, n’, m < 1500) 110 TB Sparse QMC sampling 600 MB Fitting parameters (D=60) 330 KB

≫ ≫

Summary

Sparse sampling Intermediate representation + Compact representation of Green’s functions Diagrammatic equations at single-particle level Sparse QMC measurement Diagrammatic equations at two-particle level + Tensor decomposition

• HS, J. Otsuki, M. Ohzeki, K. Yoshimi, PRB 96, 035147 (2017)
• N. Chikano, J. Otsuki, HS, PRB 98, 035104 (2018)
• HS, J. Otsuki, M. Ohzeki, K. Yoshimi, K. Haule, M. Wallerberger, E. Gull, PRB 97, 205111 (2018)
• J. Li, M. Wallerberger, N. Chikano, E. Gull, HS, in preparation
• HS, D. Geffroy, M. Wallerberger, J. Otsuki, K. Yoshimi, E. Gull, J. Kuneš, in preparation

Open source implementation (under development) DCore v2β + BSE-tool Unconventional superconductivity etc.