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Bulletin of the American Physical Society

Variational Monte Carlo Study of the Two-Dimensional Hubbard Model Using Slater-Jastrow Wave Functions

Jeong-Pil Song, Leonard W. Sprague, Chia‐Chen Chang, Brenda M. Rubenstein

Published Mar 5, 2018• 0 citations
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Jastrow-Slater and correlator product state wave functions, two classes of quantum many-body wave functions, are commonly studied using Monte Carlo methods with the associated drawbacks of stochastic error. Here we show that efficient nonstochastic algorithms for these wave functions exist, both for observable evaluation and for optimization. The algorithms rely on the structure of these states as a product of local, commuting, invertible operators acting on a simple reference wave function. We describe the nonstochastic energy evaluation and optimization algorithms, and demonstrate them with applications to the Heisenberg and spinless and full Hubbard models. Our results demonstrate that the nonstochastic algorithms yield optimized wave functions and energies very close to those obtained with the variational Monte Carlo algorithm. Such algorithms provide new criteria for identifying new classes of wave functions for efficient computational simulation.
A scheme of optimization of a trial wave function in the variational Monte Carlo calculation is developed. It provides a more precise description of the ground state for strongly correlated electronic systems with highly improved trial wave function. A general type of variational wave function constructed from a linear combination of Slater determinants multiplied by Jastrow correlation factors is proposed. They can explicitly contain low-energy fluctuations and long-range correlations. Using this optimization method, we examine the properties of these wave functions by applying them to a one-dimensional Hubbard model for which the exact solution is available for comparison. It is found that these wave functions prove to be substantial improvements compared to the conventional Jastrow-type wave functions.
We propose to compute physical properties by Monte Carlo calculations using conditional expectation values. The latter are obtained on top of the usual Monte Carlo sampling by partitioning the physical space in several subspaces or fragments, and subsampling each fragment (i.e., performing side walks) while freezing the environment. No bias is introduced and a zero-variance principle holds in the limit of separability, i.e., when the fragments are independent. In practice, the usual bottleneck of Monte Carlo calculations-the scaling of the statistical fluctuations as a function of the number of particles N-is relieved for extensive observables. We illustrate the method in variational Monte Carlo on the two-dimensional Hubbard model and on metallic hydrogen chains using Jastrow-Slater wave functions. A factor O(N) is gained in numerical efficiency.