A Monte Carlo method for critical systems in infinite volume: the planar Ising model

TL;DR

This paper introduces a Monte Carlo method that effectively generates finite-domain marginals of critical statistical models in infinite volume, overcoming boundary effects by using scale invariance and renormalization ideas.

Contribution

It presents a novel Monte Carlo algorithm that encodes infinite volume effects via holographic boundary conditions, improving the simulation of critical systems.

Findings

Accurately reproduces multi-point functions of the Ising model.

Defines a lattice stress-energy tensor and verifies conformal Ward identities.

Numerically determines the Ising model's central charge.

Abstract

In this paper we propose a Monte Carlo method for generating finite-domain marginals of critical distributions of statistical models in infinite volume. The algorithm corrects the problem of the long-range effects of boundaries associated to generating critical distributions on finite lattices. It uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of "holographic" boundary condition that encodes the presence of an infinite volume beyond it. We check the quality of the distribution obtained in the case of the planar Ising model by comparing various observables with their infinite-plane prediction. We accurately reproduce planar two-, three- and four-point functions of spin and energy operators. We also define a lattice stress-energy tensor, and numerically obtain the associated conformal Ward identities and the Ising central charge.