A self-consistent constrained path auxiliary field Quantum Monte - PowerPoint PPT Presentation

A self-consistent constrained path auxiliary field Quantum Monte Carlo method Mingpu Qin ( 秦明普 ) Shanghai Jiao Tong university July 22, 2019, CAQMP 2019 ISSP, Kashiwa, Japan In collaboration with Shiwei Zhang (William & Mary, CCQ of Flatiron Institute) and Hao Shi (CCQ of Flatiron Institute) 1

Outline ● Auxiliary field Quantum Monte Carlo. ● Minus sign problem. ● Constrained path approximation and the self- consistent loop. ● Results on 2D Hubbard model. 2

Auxiliary field quantum Monte Carlo Hubbard model 3

Auxiliary field quantum Monte Carlo Hubbard model 4

Auxiliary field quantum Monte Carlo Hubbard model Hubbard-Stratonovich transformation 5

Auxiliary field quantum Monte Carlo Hubbard-Stratonovich transformation For a general Hamiltonian where 6

Open-ended projection Deal directly with ground state wave-function 7

Open-ended projection Deal directly with ground state wave-function |φ 0 0 > |φ 1 0 > |φ 2 0 > |φ 0 1 > |φ 1 1 > |φ 2 1 > Sample {x} Sample {x} ... ... ... ... |φ 1 N > |φ 2 N > |φ 0 N > | ψ 2 > | ψ 1 > | ψ 0 > 8

Physical observable Instead of using the real estimator, mixed estimator is usually used which is computational less intensive. 9

Physical observable Instead of using the real estimator, mixed estimator is usually used which is computational less intensive. For energy it is OK because: 10

Physical observable Instead of using the real estimator, mixed estimator is usually used which is computational less intensive. For energy it is OK because: For other observables, we need back-propagation: 11

Minus sign problem Sign problem results from the cancellation of positive and negative trajectories. 12

Minus sign problem Sign problem results from the cancellation of positive and negative trajectories. Projection time 13

Minus sign problem Sign problem results from the cancellation of positive and negative trajectories. Projection time 14

Minus sign problem Sign problem results from the cancellation of positive and negative trajectories. Projection time constrained path AFQMC: Keep only the positive trajectories, approximate the node structure with a trial wave-function . 15 Shiwei Zhang, J. Carlson, J.E. Gubernatis, Phys. Rev. Lett. 74, 3652 (1995)

Introduce a trial wave-function Only allow those walkers whose overlap with trial wave-function is positive to propagate. ... |φ 0 0 > |φ 1 0 > |φ 2 0 > |φ 3 0 > ... X |φ 0 1 > |φ 1 1 > |φ 2 1 > Population |φ 3 1 > control apply the constraint ... ... ... ... ... ... |φ 1 N > |φ 2 N > |φ 0 N > |φ 3 N > | ψ i > | ψ 2 > | ψ 3 > | ψ 1 > | ψ 0 > One step CP-AFQMC: sample the field, apply the projection operator 16

Results with constraint 4 x 4 system ground state energy 17 Chia-Chen Chang and Shiwei Zhang, Phys. Rev. B 78 , 165101 (2008)

The cost of introducing a trial wave-function: systematic bias Bias-variance trade-off : Common choices of trial wave-functions: 1. Free electron wf. 2. Hartree-Fock wf. 18

Add the self-consistent loop Limitation of CP-AFQMC method: how does the result depend on trial wave-function? 19

Add the self-consistent loop Limitation of CP-AFQMC method: how does the result depend on trial wave-function? Trial wave-function CP-AFQMC 20

Add the self-consistent loop Limitation of CP-AFQMC method: how does the result depend on trial wave-function? Trial wave-function CP-AFQMC Key question : How to construct a new trial wave-function from physical quantities? 21

Add the self-consistent loop Limitation of CP-AFQMC method: how does the result depend on trial wave-function? Trial wave-function CP-AFQMC Key question : How to construct a new trial wave-function from physical quantities? Residual error still exists, but at least the method is internally 22 consistent.

Couple CP-AFQMC with mean field 1. Construct a trial wave-function from the mean-field Hamiltonian, using the local density from CP-AFQMC calculation. Hubbard model 23

Couple CP-AFQMC with mean field 1. Construct a trial wave-function from the mean-field Hamiltonian, using the local density from CP-AFQMC calculation. Hubbard model Mean field 24

Couple CP-AFQMC with mean field 1. Construct a trial wave-function from the mean-field Hamiltonian, using the local density from CP-AFQMC calculation. Hubbard model Mean field Scan U eff to minimize 25

Self-consistent CP-AFQMC Choose an initial trail wf: Free electron / HF One step CP-AFQMC Physical quantities: density Optimization to match the physical quantities Trial wave-function Converged? yes no Construct a single Slater Stop determinant 26

Give the exact spin density 4 x 16, U = 8, 1/8 doped From free trial wave-function 27 Mingpu Qin , Hao Shi, Shiwei Zhang, Phys. Rev. B 94 , 235119 (2016)

Give the exact spin density 4 x 16, U = 8, 1/8 doped From UHF trial wave-function 28 Mingpu Qin , Hao Shi, Shiwei Zhang, Phys. Rev. B 94 , 235119 (2016)

Energy 4 x 16, U = 8, 1/8 doped 29 Mingpu Qin , Hao Shi, Shiwei Zhang, Phys. Rev. B 94 , 235119 (2016)

Results on Hubbard model 6 x 32, U=8, 1/8 doped, red: DMRG, Blue: QMC Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin , Reinhard M. 30 Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358 , 1155 (2017)

Ground state energy in TDL U=8 1/8 doped Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin , Reinhard M. 31 Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358 , 1155 (2017)

Ground state energy with different methods U=8 1/8 doped Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin , Reinhard M. 32 Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358 , 1155 (2017)

Ground state at 1/8 doping: stripe phase ● Arrow direction: spin direction ● Arrow size: spin density ● Symbol size: hole density ● Agreement among different methods, with discrepancy in details. Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin , Reinhard M. 33 Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358 , 1155 (2017)

Conclusion and perspective 1. We have developed a self-consistent CP-AFQMC method, which can reduce the bias from trial wave function (finite temperature: see Yuan-Yao He, Mingpu Qin , Hao Shi, Zhong-Yi Lu, Shiwei Zhang, Phys. Rev. B 99 , 045108 (2019) ). 2. We established the stripe state as the ground state of the 1/8 doped 2D Hubbard model. 3. Other ways to optimize the trial wave-function self- consistently? 34

Thanks 35