Random walks near Rokhsar-Kivelson points

TL;DR

This paper explores how classical Monte Carlo methods can be used to study the static and dynamic properties of Rokhsar-Kivelson Hamiltonians, leveraging their known ground states and Markovian imaginary-time evolution.

Contribution

It introduces a Diffusion Monte Carlo method for RK-Hamiltonians that accurately captures their static and dynamic properties without time discretization errors.

Findings

Classical Monte Carlo can simulate quantum dynamics for sign-problem-free RK-Hamiltonians.

The relation between quantum dynamics and classical Monte Carlo follows from Markovian imaginary-time evolution.

The method avoids time discretization errors in studying perturbed RK-Hamiltonians.

Abstract

There is a class of quantum Hamiltonians known as Rokhsar-Kivelson(RK)-Hamiltonians for which static ground state properties can be obtained by evaluating thermal expectation values for classical models. The ground state of an RK-Hamiltonian is known explicitly, and its dynamical properties can be obtained by performing a classical Monte Carlo simulation. We discuss the details of a Diffusion Monte Carlo method that is a good tool for studying statics and dynamics of perturbed RK-Hamiltonians without time discretization errors. As a general result we point out that the relation between the quantum dynamics and classical Monte Carlo simulations for RK-Hamiltonians follows from the known fact that the imaginary-time evolution operator that describes optimal importance sampling, in which the exact ground state is used as guiding function, is Markovian. Thus quantum dynamics can be studied by a classical Monte Carlo simulation for any Hamiltonian that is free of the sign problem provided its ground state is known explicitly.