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Tensor learning approach to sparse QMC sampling of two-particle Greens 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 Matsubara


  1. 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

  2. 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

  3. <latexit sha1_base64="cFHez4i4vao19ALqOcNKzU7MQb0=">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</latexit> <latexit sha1_base64="cFHez4i4vao19ALqOcNKzU7MQb0=">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</latexit> <latexit sha1_base64="cFHez4i4vao19ALqOcNKzU7MQb0=">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</latexit> <latexit sha1_base64="GnqyXkjB+3+pEhwGAq9ZwAXtkrk=">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</latexit> Intermediate representation (IR) 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) G α ( i ω n ) = ∫ ω max ρ ( ω ) d ω K α ( i ω n , ω ) ρ α ( ω ) − ω max α = F (fermion), B (boson) 0 ∞ Model independent ∑ K α ( i ω n , ω ) = − S α l U α l ( i ω n ) V α l ( ω ) orthonormal basis sets l =0 Λ ≡ βω max Λ ≡βω max G α l = − S α l ρ α l Λ =10 2 Λ =10 4 Size ~ O (log Λ )

  4. 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

  5. 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) ������������������ ������������������� ������������ ���������������������������� � ������� �� � ������������������������� �������� ������������ ������������������ ������� � ��������������������������� �� �� ��������������������� ������� ��� �������������� ��������� �� ������� ��������������� � � ��������� �������������� ������� ��� ���������������������� ������������ ���������������������� ��������������� ���������������������������� �������� ������������������������������ ���������������������������������������� ����������������������������������������

  6. 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

  7. � ��������������������� ���� �������� ����������� ������������������������� ���� � �������� ���� ��������������� 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 � ��� � ���

  8. 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

  9. 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

  10. 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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