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arXiv:cond-mat/9912389 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Optimization of trial wave functions for Hamiltonian lattice models

Matteo Beccaria A

Published May 1, 2000• 5 citations
arXivHTMLPDF

Abstract

We propose a new Monte Carlo algorithm for the numerical study of general lattice models in Hamiltonian form. The algorithm is based on an initial ansatz for the ground state wave function depending on a set of free parameters which are dynamically optimized. The method is discussed in detail and the results are reported from explicit simulations of a U(1) N chain, a degenerate limit of U(1) lattice gauge theory in $2+1$ dimensions.

DOI
10.1103/physrevd.61.114503

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We propose a new Monte Carlo algorithm for the numerical study of general lattice models in Hamiltonian form. The algorithm is based on an initial ansatz for the ground state wave function depending on a set of free parameters which are dynamically optimized. The method is discussed in detail and the results are reported from explicit simulations of a U(1) N chain, a degenerate limit of U(1) lattice gauge theory in 2+12+12+1 dimensions.
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Monte Carlo Hamiltonian method is a new algorithm for numerical simulation of quantum theories. The advantage is its ability to study spectrum and wave functions beyond the ground state. The previous approach needs information on free particles and is difficult to be used for lattice gauge theory. In this paper, we propose a new approach to MCH, which may overcome the difficulty. We describe the basic ideas, and use a 1-dimensional quantum mechanical model V(x) =μ2x2 + λx4(with μ2 0,λ 0) as an example.
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We study lattice QED in 2+12+12+1 dimensions coupled to Wilson fermions in the Hamiltonian formulation and develop a sign-problem-free variational Monte Carlo algorithm. Using a continuous basis to describe the gauge degrees of freedom and fermionic Gaussian states for the matter fields, we construct a variational ansatz for the ground state. The parameters of the ansatz are optimized using gradient descent and/or imaginary time evolution based on the time-dependent variational principle. For a lattice with NNN matter sites, the computational complexity for sampling a gauge configuration is O(N2log⁡N)\mathcal{O}(N^2\log N)O(N2logN), and the complexity for computing the energy per gauge configuration is O(N4)\mathcal{O}(N^4)O(N4). Comparing our method to results from exact diagonalization on small lattices, we demonstrate good agreement. In addition, we present preliminary results for the ground state for two fermion flavors, Nf=2N_f=2Nf​=2, in the presence of a non-zero chemical potential, a regime in which the conventional action-based Monte Carlo methods suffer from the sign problem.
We propose a new ansatz for the ground-state wave function of quantum many-body systems on a lattice. The key idea is to cover the lattice with plaquettes and obtain a state whose configurational weights can be optimized by means of a variational Monte Carlo algorithm. Such a scheme applies to any dimension, without any 'sign' instability. We show results for various two-dimensional spin models (including frustrated ones). A detailed comparison with available exact results, as well as with variational methods based on different ansatze, is offered. In particular, our numerical estimates are in quite good agreement with exact ones for unfrustrated systems, and compare favorably to other methods for frustrated ones.