Ground state of many-body lattice systems: an analytical probabilistic approach

TL;DR

This paper introduces an analytical probabilistic method to determine the ground state properties of many-body lattice systems, providing explicit solutions in certain cases and a semi-analytical approach for general interactions.

Contribution

It develops a Feynman-Kac--type formula for lattice Hamiltonians to derive analytical expressions for ground state energies and correlations, applicable at long times.

Findings

Ground-state energy and correlations are determined by solving a scalar equation.

Explicit solutions are obtained for noninteracting systems.

The approach is valid at long times when a central limit theorem applies.

Abstract

On the grounds of a Feynman-Kac--type formula for Hamiltonian lattice systems we derive analytical expressions for the matrix elements of the evolution operator. These expressions are valid at long times when a central limit theorem applies. As a remarkable result we find that the ground-state energy as well as all the correlation functions in the ground state are determined semi-analytically by solving a simple scalar equation. Furthermore, explicit solutions of this equation are obtained in the noninteracting case.