A Perturbative/Variational Approach to Quantum Lattice Hamiltonians

TL;DR

This paper introduces a combined perturbative and variational method to approximate ground states of local lattice Hamiltonians, transforming the quantum problem into a solvable statistical mechanical system, with applications demonstrated on the transverse field Ising model.

Contribution

It presents a novel exponential ansatz-based approach that integrates perturbation theory and variational techniques for quantum lattice Hamiltonians.

Findings

Exact upper bounds to the energy in some cases

Transforming quantum problems into solvable statistical models

Application to the transverse field Ising model

Abstract

We propose a method to construct the ground state $\psi(\lambda)$ of local lattice hamiltonians with the generic form $H_0 + \lambda H_1$, where $\lambda$ is a coupling constant and $H_0$ is a hamiltonian with a non degenerate ground state $\psi_0$. The method is based on the choice of an exponential ansatz $\psi(\lambda) = {\rm exp}(U(\lambda)) \psi_0$, which is a sort of generalized lattice version of a Jastrow wave function. We combine perturbative and variational techniques to get succesive approximations of the operator $U(\lambda)$. Perturbation theory is used to set up a variational method which in turn produces non perturbative results. The computation with this kind of ansatzs leads to associate to the original quantum mechanical problem a statistical mechanical system defined in the same spatial dimension. In some cases these statistical mechanical systems turn out to be integrable, which allow us to obtain exact upper bounds to the energy. The general ideas of our method are illustrated in the example of the Ising model in a transverse field.