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. Kune š
Matsubara Green’s functions χ Γ χ = + Many perturbative theories, dynamical mean-field theory, quantum Monte Carlo Challenges at low T and complex systems Storage Intermediate representation (IR) Manipulation Sparse sampling and tensor learning Main focus of my talk: two-particle quantities
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Simple example ρ ( ω ) G ( τ ) β = 100 G α l = − S α l ρ α l − 1 +1 0 Metal 10 − 1 S l 10 − 5 | ρ l | | G l | 10 − 9 0 10 20 30 l Insulator Super exponential decay
Scaling at low T 0 β τ Legendre Matsubara frequency with tail O ( β ) 10 4 Chebyshev Legendre and Chebyshev L. Boehnke et al . (2011) Number of coefficients IR E. Gull et al . (2018) O ( β ) i ω n 10 3 N IR O (log β ) 10 2 10 1 10 2 10 4 10 6 β ω max = 10 eV, T = 10 K → ω max β = 10 4 • Python/C++ implementation of IR basis Single pole https://github.com/SpM-lab/irbasis ω | δ G( τ =0)| < 10 -8 • Application to DMFT with ED solver -1 0 � ������� ������ Y. Nagai and H. Shinaoka, JPSJ 88 , 064004 (2019) ������������������ ������������������� ������������ ���������������������������� � ������� �� � ������������������������� �������� ������������ ������������������ ������� � ��������������������������� �� �� ��������������������� ������� ��� �������������� ��������� �� ������� ��������������� � � ��������� �������������� ������� ��� ���������������������� ������������ ���������������������� ��������������� ���������������������������� �������� ������������������������������ ���������������������������������������� ����������������������������������������
How to do math ? Simplest example: Dyson equation 1 G ( i ω n ) = Σ ( i ω n ) → G ( i ω n ) i ω n − H − Σ ( i ω n ) Solve at all Matsubara frequencies Sparse sampling O (log β N 3 O ( N i ω N 3 orbital ) ≃ O ( β N 3 orbital ) orbital ) J. Li, M. Wallerberger, N. Chikano, E. Gull, HS, in preparation
� ��������������������� ���� �������� ����������� ������������������������� ���� � �������� ���� ��������������� Sparse sampling ���������������������������������������� ���������������������������������������� We know how many coefficients are required. ( ≦ 100) ���������������������������� ��������� J. Li, M. Wallerberger, N. Chikano, ���������������������������������������� U F l ( i ω n ) E. Gull, HS, in preparation ��� Λ =10 4 10 − 2 G l n | G l − G inv | l 10 − 6 G l 10 − 10 10 − 14 10 − 18 0 20 40 60 80 N l − 1 l ∑ G F ( i ω n ) = U F l ( i ω n ) G l l Unknown � ��������������� Known � ���� ����� •Model-independent sampling points (# of sampling points) = (# of IR coefficients to be fitted) •Stable fitting � ��� � ���
Example: quantum chemistry calculations GF2 G ( i ω n ) G l G ( τ i ) Fitting G ( i ω n ) = ( i ω n − H − Σ ( i ω n )) − 1 Σ ( i ω n ) Σ ( τ i ) Σ l O ( N τ N 5 orb ) Fitting ω n ∈ sampling points τ i ∈ sampling points Hydrogen atom chain GF(2) new Dyson, H 10 ( R = 1), β = 10 3 , STO-6g, E tot = − 3.8101298016 10 0 The same idea applies to Eliashberg T ~ 260 K IR basis ( Λ = 10 4 ) 10 − 1 Chebyshev basis and GW-type equations! Chem. accuracy 10 − 2 10 − 3 | ∆ E tot | [H] 10 − 4 10 − 5 10 − 6 10 − 7 J. Li, M. Wallerberger, N. Chikano, 10 − 8 E. Gull, HS, in preparation 10 − 9 0 50 100 150 200 250 300 350 400 number of coefficients
Why two-particle objects? Vertex corrections My motivation: Dynamical mean-field theory (DMFT) • Dynamic susceptibility (inelastic scattering experiments) • Non-local correlations beyond DMFT χ Γ χ = + Bethe-Salpeter equation
Rich frequency structure fermionic frequency Matsubara representation ⟨ T τ c ( τ 1 ) c † ( τ 2 ) c ( τ 3 ) c † ( τ 4 ) ⟩ Discontinuities cf . G ( τ ) 0 β τ M. Wallerberger, PhD thesis Power-law decay fermionic frequency • Large size: O ( β 3 ) • More indices for spin, orbital, wave vector… • IR approach frequency dependence • Dimensionality reduction
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