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arXiv:hep-lat/0105024 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Adaptive optimization of wave functions for fermion lattice models

Matteo Beccaria A, Antonio Moro

Published Sep 5, 2001• 2 citations
arXivHTMLPDF

Abstract

We present a simulation algorithm for Hamiltonian fermion lattice models. A guiding trial wave function is adaptively optimized during Monte Carlo evolution. We apply the method to the two dimensional Gross-Neveu model and analyze systematic errors in the study of ground state properties. We show that accurate measurements can be achieved by a proper extrapolation in the algorithm-free parameters.

DOI
10.1103/physrevd.64.077502

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We present a simulation algorithm for Hamiltonian fermion lattice models. A guiding trial wave function is adaptively optimized during Monte Carlo evolution. We apply the method to the two dimensional Gross-Neveu model and analyze systematic errors in the study of ground state properties. We show that accurate measurements can be achieved by a proper extrapolation in the algorithm-free parameters.
The accuracy of Green Function Monte Carlo (GFMC) simulations can be greatly improved by a clever choice of the approximate ground state wave function that controls configuration sampling. This trial wave function typically depends on many free parameters whose fixing is a non trivial task. Here, we discuss a general purpose adaptive algorithm for their non-linear optimization. As a non trivial application we test the method on the two dimensional Wess-Zumino model, a relativistically invariant supersymmetric field theory with interacting bosonic and fermionic degrees of freedom.
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.
The purpose of this paper is to present some simple variants of a recently developed iterative algorithm for improvement of any given trial wave function. These modifications improve the convergence rate of the iterative process and allow us to obtain a good variational state for larger lattices. We apply the method to the one-dimensional Gross-Neveu model with one flavor in a finite-size lattice. The ground-state energy, the density-density correlation function, and the single-particle Green's function are calculated. They are in good agreement with existing results.
We have developed a Quantum Monte Carlo optimization (QMCO) method for many fermion systems. The ground state wave function is written as a linear combination of the basis states generated using the auxiliary field method The bases are chosen with weights that are determined by diagonalizing the Hamiltonian to lower the ground state energy. The QMCO method is free from a difficulty originating from the negative sign problem. We apply the QMCO method to the two‐leg ladder and two‐dimensional Hubbard model. We can reproduce the exact results for the Hubbard model on small clusters. The pair correlation function is indeed enhanced compared to that for U=0 as the strength of the repulsive Coulomb interaction increases.
We describe and discuss a recently proposed quantum Monte Carlo algorithm to compute the ground-state properties of various systems of interacting fermions. In this method, the ground state is projected from an initial wave function by a branching random walk in an overcomplete basis of Slater determinants. By constraining the determinants according to a trial wave function |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉 is exact. We illustrate the method by describing in detail its implementation for the two-dimensional one-band Hubbard model. We show results for lattice sizes up to 16\ifmmode\times\else\texttimes\fi{}16 and for various electron fillings and interaction strengths. With simple single-determinant wave functions as |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, the method yields accurate (often to within a few percent) estimates of the ground-state energy as well as correlation functions, such as those for electron pairing. We conclude by discussing possible extensions of the algorithm.
We present a simple, robust, and highly efficient method for optimizing all parameters of many-body wave functions in quantum Monte Carlo calculations, applicable to continuum systems and lattice models. Based on a strong zero-variance principle, diagonalization of the Hamiltonian matrix in the space spanned by the wave function and its derivatives determines the optimal parameters. It systematically reduces the fixed-node error, as demonstrated by the calculation of the binding energy of the small but challenging C(2) molecule to the experimental accuracy of 0.02 eV.