Geometric percolation of spins and spin-dipoles in Ashkin-Teller model

TL;DR

This paper studies the geometric percolation of spin and spin-dipole clusters in the Ashkin-Teller model, revealing a superuniversality class with invariant Binder cumulant along the critical line.

Contribution

It introduces a detailed analysis of percolation thresholds, fractal dimensions, and superuniversality in the Ashkin-Teller model's geometric clusters.

Findings

Largest clusters become macroscopic at critical threshold

Fractal dimension relates to critical exponents and varies with parameters

Binder cumulant remains invariant along the critical line, indicating superuniversality

Abstract

Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength $J$, and the spin-dipoles (product of spins) interact with neighbors with strength $\lambda.$ The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the $\lambda$-$J$ plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted respectively as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent $\beta_{m,e}$ as $D_{m,e}=d-\frac{5}{12}\frac{\beta_{m,e}}\nu,$ where $d=2$ is the spatial dimension and $\nu$ is the correlation length exponent. This relation determines all other percolation exponents and their variation wrt $\lambda.$ We show that for magnetic Percolation, the Binder cumulant, as a function of $\xi_2/L$ with $\xi_2$ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.