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 ( ) Shanghai Jiao Tong university July 22, 2019, CAQMP 2019 ISSP, Kashiwa, Japan In collaboration with Shiwei Zhang (William & Mary, CCQ of
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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)
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Outline
consistent loop.
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Auxiliary field quantum Monte Carlo
Hubbard model
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Auxiliary field quantum Monte Carlo
Hubbard model
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Auxiliary field quantum Monte Carlo
Hubbard-Stratonovich transformation Hubbard model
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Auxiliary field quantum Monte Carlo
Hubbard-Stratonovich transformation For a general Hamiltonian where
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Open-ended projection
Deal directly with ground state wave-function
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Open-ended projection
Deal directly with ground state wave-function
|φ0
0>|ψ0>
|φ0
1>|φ0
N>|φ1
0>|φ1
1>|φ1
N>... ...
|φ2
0>|φ2
1>|φ2
N>... |ψ1> |ψ2>
Sample {x} Sample {x}
...
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Physical observable
Instead of using the real estimator, mixed estimator is usually used which is computational less intensive.
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Physical observable
Instead of using the real estimator, mixed estimator is usually used which is computational less intensive. For energy it is OK because:
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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:
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Minus sign problem
Sign problem results from the cancellation of positive and negative trajectories.
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Minus sign problem
Projection time
Sign problem results from the cancellation of positive and negative trajectories.
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Minus sign problem
Projection time
Sign problem results from the cancellation of positive and negative trajectories.
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Minus sign problem
Projection time
Sign problem results from the cancellation of positive and negative trajectories. constrained path AFQMC:
Keep only the positive trajectories, approximate the node structure
with a trial wave-function.
Shiwei Zhang, J. Carlson, J.E. Gubernatis, Phys. Rev. Lett. 74, 3652 (1995)
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Introduce a trial wave-function
Only allow those walkers whose overlap with trial wave-function is positive to propagate.
|φ0
0>|ψ0>
|φ0
1>|φ0
N>|φ1
0>|φ1
1>|φ1
N>|φ3
0>|φ3
1>|φ3
N>... ... ...
|φ2
0>|φ2
1>|φ2
N>...
X
apply the constraintPopulation control
... ... ... ... |ψ1> |ψ2> |ψ3> |ψi>
One step CP-AFQMC: sample the field, apply the projection operator
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Results with constraint
4 x 4 system ground state energy
Chia-Chen Chang and Shiwei Zhang, Phys. Rev. B 78, 165101 (2008)
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The cost of introducing a trial wave-function: systematic bias Common choices of trial wave-functions:
Bias-variance trade-off:
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Add the self-consistent loop
Limitation of CP-AFQMC method: how does the result depend on trial wave-function?
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Add the self-consistent loop
CP-AFQMC
Limitation of CP-AFQMC method: how does the result depend on trial wave-function?
Trial wave-function
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Add the self-consistent loop
CP-AFQMC
Key question: How to construct a new trial wave-function from physical quantities? Limitation of CP-AFQMC method: how does the result depend on trial wave-function?
Trial wave-function
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Add the self-consistent loop
CP-AFQMC
Key question: How to construct a new trial wave-function from physical quantities? Limitation of CP-AFQMC method: how does the result depend on trial wave-function?
Trial wave-function
Residual error still exists, but at least the method is internally consistent.
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Couple CP-AFQMC with mean field
using the local density from CP-AFQMC calculation. Hubbard model
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Couple CP-AFQMC with mean field
using the local density from CP-AFQMC calculation. Hubbard model Mean field
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Couple CP-AFQMC with mean field
using the local density from CP-AFQMC calculation. Scan Ueff to minimize Hubbard model Mean field
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Self-consistent CP-AFQMC
Choose an initial trail wf: Free electron / HF One step CP-AFQMC Optimization to match the physical quantities Physical quantities: density Construct a single Slater determinant Trial wave-function Stop Converged? no yes
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Give the exact spin density
Mingpu Qin, Hao Shi, Shiwei Zhang, Phys. Rev. B 94, 235119 (2016)
4 x 16, U = 8, 1/8 doped From free trial wave-function
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Give the exact spin density
Mingpu Qin, Hao Shi, Shiwei Zhang, Phys. Rev. B 94, 235119 (2016)
4 x 16, U = 8, 1/8 doped
From UHF trial wave-function
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Energy
Mingpu Qin, Hao Shi, Shiwei Zhang, Phys. Rev. B 94, 235119 (2016)
4 x 16, U = 8, 1/8 doped
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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. Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358, 1155 (2017)
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Ground state energy in TDL
Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin, Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358, 1155 (2017)
U=8 1/8 doped
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Ground state energy with different methods
Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin, Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358, 1155 (2017)
U=8 1/8 doped
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Ground state at 1/8 doping: stripe phase
methods, with discrepancy in details.
Ref: Bo-Xiao Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Ming-Pu Qin, Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, Garnet Kin-Lic Chan, Science 358, 1155 (2017)
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Conclusion and perspective
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) ).
consistently?
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