Cluster statistics of critical Ising and Ashkin-Teller models

TL;DR

This paper investigates the scaling behavior of the number of boundary-crossing clusters in 2D classical models at criticality, revealing a conformal invariance-driven logarithmic scaling similar to quantum entanglement entropy.

Contribution

It demonstrates that the boundary cluster count exhibits universal logarithmic scaling at criticality, connecting conformal invariance with cluster statistics in classical models.

Findings

Logarithmic boundary cluster scaling observed at criticality

Scaling coefficient depends on system geometry and boundary conditions

Numerical Monte Carlo simulations confirm theoretical predictions

Abstract

Motivated by recent progress on the scaling behavior of entanglement entropy, we study the scaling behavior of the number of clusters crossing the boundary between two subsystems for several classical statistical models in two dimension. This number exhibits a subleading logarithmic dependence of the linear dimension of the boundary when the model is at critical, in analogy to the entanglement entropy of a quantum system. It is shown that the logarithmic scaling of the cluster number originates from the conformal invariance of the critical system. We check this numerically for Ising and Ashkin-Teller models by using Monte Carlo simulations, and show that whether a universal coefficient of the logarithmic term can be observed numerically may strongly depend on the geometry and boundary condictions of the system.