Coupling quantum Monte Carlo and independent-particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix

Qin, Mingpu; Shi, Hao; Zhang, Shiwei

doi:10.1103/physrevb.94.235119

Cited by 56 publications

(87 citation statements)

References 48 publications

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“…It generalizes the method discussed in Ref. 40 under the ZT-AFQMC framework to finite temperatures. We will optimize the trial Hamiltonian (or trial propagator) step by step iteratively through a sequence of FT-AFQMC calculation so as to reduce the error from the constraint.…”

Section: B Ft-afqmc Calculations With Self-consistent Proceduresmentioning

confidence: 93%

“…In practice it is implemented approximately with a trial wave function or trial density matrix, which introduces a possible systematic bias in the numerical results but in turn removes the exponentially growing computational cost and recovers the algebraic complexity. Through many tests and developments in the last two decades [29][30][31][32][33][34][35][36][37][38][39][40][41] , the zero-temperature (ZT) CP-AFQMC method has been proved to be a highly accurate, general numerical approach for studying groundstate properties of various interacting fermion models as well as molecules and realistic materials by its generalization, the phaseless AFQMC method 28,42,43 .…”

Section: Introductionmentioning

confidence: 99%

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Finite-temperature auxiliary-field quantum Monte Carlo: Self-consistent constraint and systematic approach to low temperatures

Qin

Shi

et al. 2019

Phys. Rev. B

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We describe an approach for many-body calculations with a finite-temperature, grand canonical ensemble formalism using auxiliary-field quantum Monte Carlo (AFQMC) with a self-consistent constraint to control the sign problem. The usual AFQMC formalism of Blankenbecler, Scalapino, and Sugar suffers from the sign problem with most physical Hamiltonians, as is well known. Building on earlier ideas to constrain the paths in auxiliary-field space [Phys. Rev. Lett. 83, 2777(1999] and incorporating recent developments in zero-temperature, canonical-ensemble methods, we discuss how a self-consistent constraint can be introduced in the finite-temperature, grand-canonical-ensemble framework. This together with several other algorithmic improvements discussed here leads to a more accurate, more efficient, and numerically more stable approach for finite-temperature calculations. We carry out a systematic benchmark study in the two-dimensional repulsive Hubbard model at 1/8 doping. Temperatures as low as T = 1/80 (in units of hopping) are reached. The finite-temperature method is exact at very high temperatures, and approaches the result of the zero-temperature constrained-path AFQMC as temperature is lowered. The benchmark shows that systematically accurate results are obtained for thermodynamic properties.

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Section: B Ft-afqmc Calculations With Self-consistent Proceduresmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

Finite-temperature auxiliary-field quantum Monte Carlo: Self-consistent constraint and systematic approach to low temperatures

Qin

Shi

et al. 2019

Phys. Rev. B

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“…The smaller differences in this regime tend to be more pronounced at shorter imaginary-times. A very interesting perspective will be the study of the possibility to use effective parameters [37] in the dynamical BCS theory to improve the agreement with exact results.…”

Section: Discussionmentioning

confidence: 99%

Response Functions for the Two-Dimensional Ultracold Fermi Gas: Dynamical BCS Theory and Beyond

et al. 2017

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Response functions are central objects in physics. They provide crucial information about the behavior of physical systems, and they can be directly compared with scattering experiments involving particles like neutrons, or photons. Calculations of such functions starting from the many-body Hamiltonian of a physical system are challenging, and extremely valuable. In this paper we focus on the two-dimensional (2D) ultracold Fermi atomic gas which has been realized experimentally. We present an application of the dynamical BCS theory to obtain response functions for different regimes of interaction strengths in the 2D gas with zero-range attractive interaction. We also discuss auxiliary-field quantum Monte Carlo (AFQMC) methods for the calculation of imaginary-time correlations in these dilute Fermi gas systems. Illustrative results are given and comparisons are made between AFQMC and dynamical BCS theory results to assess the accuracy of the latter.

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“…For general Hamiltonians, for instance the doped Hubbard model or realistic electronic Hamiltonians in solids or molecules, a constraint can be applied to control the sign or phase problem. This framework, even with simple trial wave functions to impose an approximate constraint, has been shown to be very accurate and has been applied widely in ground-state calculations [23][24][25][26][27][28][29][30][31]. Finitetemperature generalization of the approach has also been developed [32][33][34][35].These finite temperature calculations all have computational complexity of O(N 3 s ) [36], as they are formulated in the grand-canonical ensemble to analytically evaluate the fermion trace along each path in auxiliary-field space, leading to determinants with dimension N s .…”

mentioning

confidence: 99%

Reaching the Continuum Limit in Finite-Temperature Ab Initio Field-Theory Computations in Many-Fermion Systems

Shi

Zhang

2019

Phys. Rev. Lett.

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Finite-temperature, grand-canonical computations based on field theory are widely applied in areas including condensed matter physics, ultracold atomic gas systems, and lattice gauge theory. However, these calculations have computational costs scaling as N 3 s with the size of the lattice or basis set, Ns. We report a new approach based on systematically controllable low-rank factorization which reduces the scaling of such computations to NsN 2 e , where Ne is the average number of fermions in the system. In any realistic calculations aiming to describe the continuum limit, Ns/Ne is large and needs to be extrapolated effectively to infinity for convergence. The method thus fundamentally changes the prospect for finite-temperature many-body computations in correlated fermion systems. Its application, in combination with frameworks to control the sign or phase problem as needed, will provide a powerful tool in ab initio quantum chemistry and correlated electron materials. We demonstrate the method by computing exact properties of the two-dimensional Fermi gas with zero-range attractive interaction, as a function of temperature in both the normal and superfluid states.Computations are playing an increasingly important role in addressing the fundamental challenges of understanding strong correlations in interacting quantum systems. Understandably, a major part of the development and application efforts in both physics and chemistry have focused on ground-state properties. However, experimental conditions are always at finite temperatures, where often rich and new properties can be revealed [1-4]. One example is the rapid development in the area of experiments with ultracold atoms, where temperature plays a crucial role and very precise measurements of properties are often possible with exquisite control over interaction strengths, environments, etc [5,6]. Accurate computations of thermodynamic properties allow direct comparisons with experiments, but are challenging because of the presence of strong coupling and thermal fluctuations. A second example is in strongly correlated materials, including high-temperature superconductors [7,8], where some of the outstanding and most interesting physics questions concern finite-temperature properties [9].A common finite-temperature formalism is based on field theory in which the thermodynamic properties are computed as path-integrals in field space. Approximations can be used to perform the path integrals, including the simplest, namely mean-field calculations. A more powerful approach is to evaluate the manydimensional integration by Monte Carlo methods [10]. This has become a key technique for many-body finitetemperature computations, widely applied in several fields of physics [11][12][13][14][15][16][17][18][19][20][21][22]. For example, many of the sign-problem-free computations in lattice models in condensed matter, and in Fermi gas and optical lattices of ultracold atoms, have been performed this way. For general Hamiltonians, for instance the doped Hubbard model or rea...

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Coupling quantum Monte Carlo and independent-particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix

Cited by 56 publications

References 48 publications

Finite-temperature auxiliary-field quantum Monte Carlo: Self-consistent constraint and systematic approach to low temperatures

Finite-temperature auxiliary-field quantum Monte Carlo: Self-consistent constraint and systematic approach to low temperatures

Response Functions for the Two-Dimensional Ultracold Fermi Gas: Dynamical BCS Theory and Beyond

Reaching the Continuum Limit in Finite-Temperature Ab Initio Field-Theory Computations in Many-Fermion Systems

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