Numerical tests of conjectures of conformal field theory for three-dimensional systems

TL;DR

This paper numerically investigates three-dimensional conformal field theory conjectures by analyzing the finite-size scaling of classical spin models on hyper-cylindrical geometries, revealing analogies with two-dimensional results especially under antiperiodic boundary conditions.

Contribution

It provides the first numerical evidence for conformal field theory scaling relations in three-dimensional systems with specific boundary conditions.

Findings

Strong evidence for 3D conformal scaling relations

Antiperiodic boundary conditions reveal similar behavior to 2D cases

Monte Carlo simulations confirm theoretical conjectures

Abstract

The concept of conformal field theory provides a general classification of statistical systems on two-dimensional geometries at the point of a continuous phase transition. Considering the finite-size scaling of certain special observables, one thus obtains not only the critical exponents but even the corresponding amplitudes of the divergences analytically. A first numerical analysis brought up the question whether analogous results can be obtained for those systems on three-dimensional manifolds. Using Monte Carlo simulations based on the Wolff single-cluster update algorithm we investigate the scaling properties of O(n) symmetric classical spin models on a three-dimensional, hyper-cylindrical geometry with a toroidal cross-section considering both periodic and antiperiodic boundary conditions. Studying the correlation lengths of the Ising, the XY, and the Heisenberg model, we find strong evidence for a scaling relation analogous to the two-dimensional case, but in contrast here for the systems with antiperiodic boundary conditions.