Xiao Yan Xu

(许霄琰 )

IOP, CAS

7/7/2017

Self-Learning quantum Monte Carlo method

in interacting fermion systems

Serials works on SLMC

IOP: Zi Hong Liu, Zi Yang MengMIT: Junwei Liu, Yang Qi, Huitao Shen, Yuki Nagai, Liang FuUM: Kai Sun

arXiv:1610.03137arXiv:1611.09364arXiv:1612.03804 arXiv:1705.06724 arXiv:1706.10004

Similar work from Li Huang, Yi-feng Yang, Lei WangarXiv:1610.02746arXiv:1612.01871

The power of SLMC

● Reduce the time cost per sweepFor 100×100×20 latticeDQMC ~ 150000 seconds/sweepSLMC ~ 500 seconds/sweep

● Reduce the auto-correlation timeDQMC ~ may scale with system size at critical pointSLMC ~ constants ideally, model dependent

Determinantal QMC (DQMC)

DQMC(BSS algorithm)

DQMC(BSS algorithm)

~(ßN)3

~ßN3

DQMC basics

Slater determinant and its properties

Occupation number representationNe particle states

operator with form

overlap of slater determinant

DQMC basics

Hubbard-Stratonovich transformationdeal with interaction termA continuous form

Other examples

DQMC

Trotter decomposition

HS transformation

Trace out fermions (trace over all Ne particle basis, Ne=1,...,N)

DQMC

partition function

Importance sampling of configurations

Application of DQMC

Coupled Fermion-boson lattice systems ● Interacting systems after HS transformation

Hubbard like modelsMott transition, chiral Ising and chiral Heisenberg transition, bosonic SPT etc.

● build-in coupled fermion-bosonAFM in metal, SDW, nematic QCP, FM QCP, Z2 deconfined phase transtion

Issues of DQMC

● Local update is level 1 BLAS algorithmSize limited, L=20 is the typical size

● (critical) slowing down in some modelsbuild-in coupled fermion-boson problem

Slowing down for some models in DQMC

Complexity for getting an independent configuration:

Self-learning DQMC

arXiv:1612.03804arXiv:1706.10004

Self-Learning Monte Carlo

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arXiv:1612.03804

Self-Learning Determinantal Quantum Monte Carlo

Complexity

● Cumulative update:

● Detail balance:

● Sweep Green's function:

Complexity speed up

Self-Learning Determinantal Quantum Monte Carlo

Autocorrelation time

Institute of Physics, Chinese Academy of Sciences

Self-Learning Determinantal Quantum Monte Carlo

Institute of Physics, Chinese Academy of Sciences

Using SLDQMC to attack hard problems

arXiv:1706.10004

Itinerant quantum critical point with frustration

Institute of Physics, Chinese Academy of Sciences

Itinerant quantum critical point with frustration

Non-fermi liquid behavior on hot spots

Itinerant quantum critical point with frustration

dynamical exponents z=2

Itinerant quantum critical point with frustration

Linear T dependence in spin susceptibility

L=30, beta=30(30×30×600)

Itinerant quantum critical point with frustration

Hertz-Millis-Moriya theory on finite momentum QCP

Summary and outlook

SLMC can be used to attack some hard problems

How general can it be is still a question

Self-Learning Monte Carlo

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arXiv:1612.03804