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arXiv:1811.07290 · Physical review. B./Physical review. B

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

Yuan-Yao He, Mingpu Qin, Hao Shi, Zhong-Yi Lu, Shiwei Zhang

Published Jan 3, 2019• 66 citations
arXivHTMLPDF

Abstract

$A\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ computations of finite-temperature properties are important in fermion systems such as correlated models, ultracold Fermi atomic gases, and optical lattices, and realistic materials. Here, the authors present a highly accurate approach that eliminates the sign problem in standard determinantal quantum Monte Carlo (DQMC) using a gauge constraint in auxiliary-field space. The constraint can be systematically improved by a self-consistent procedure coupling the many-body computation with an independent-particle calculation. A benchmark is performed in the Hubbard model across the entire temperature range.

DOI
10.1103/physrevb.99.045108

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A\phantom{\rule{0}{0ex}}b i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o computations of finite-temperature properties are important in fermion systems such as correlated models, ultracold Fermi atomic gases, and optical lattices, and realistic materials. Here, the authors present a highly accurate approach that eliminates the sign problem in standard determinantal quantum Monte Carlo (DQMC) using a gauge constraint in auxiliary-field space. The constraint can be systematically improved by a self-consistent procedure coupling the many-body computation with an independent-particle calculation. A benchmark is performed in the Hubbard model across the entire temperature range.
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Quantum Monte Carlo (QMC) methods are one of the most important tools for studying interacting quantum many-body systems. The vast majority of QMC calculations in interacting fermion systems require a constraint to control the sign problem. The constraint involves an input trial wave function which restricts the random walks. We introduce a systematically improvable constraint which relies on the fundamental role of the density or one-body density matrix. An independent-particle calculation is coupled to an auxiliary-field QMC calculation. The independent-particle solution is used as the constraint in QMC, which then produces the input density or density matrix for the next iteration. The constraint is optimized by the self-consistency between the many-body and the independent-particle calculations. The approach is demonstrated in the two-dimensional Hubbard model by accurately determining the ground state when collective modes separated by tiny energy scales are present in the magnetic and charge correlations. Our approach also provides an ab initio way to predict effective interaction parameters for independent-particle calculations.
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We present a quantum Monte Carlo method which allows calculations on many-fermion systems at finite temperatures without any sign decay. This enables simulations of the grand-canonical ensemble at large system sizes and low temperatures. Both diagonal and off-diagonal expectations can be computed straightforwardly. The sign decay is eliminated by a constraint on the fermion determinant. The algorithm is approximate. Tests on the Hubbard model show that accurate results on the energy and correlation functions can be obtained.
We introduce a method that combines the power of both the lattice Green' function Monte Carlo (LGFMC) with the auxiliary field quantum Monte Carlo (AFQMC) techniques, and allows us to compute exact ground-state properties of the Hubbard model for $U\ensuremath{\lesssim}4t$ on finite clusters. Thanks to LGFMC, one obtains unbiased zero temperature results, not affected by the so-called Trotter approximation of the imaginary time propagator ${e}^{\ensuremath{-}H\ensuremath{\tau}}$. At the same time, the AFQMC formalism yields a remarkably fast convergence in $\ensuremath{\tau}$ before the fermion sign problem becomes prohibitive. As an application we report ground-state energies of the Hubbard model at U/t=4U/t=4U/t=4 with up to 100 sites.