Properties of the multicritical point of +/- J Ising spin glasses on the square lattice

TL;DR

This study uses numerical transfer-matrix methods to analyze the multicritical point of +/- J Ising spin glasses on a square lattice, estimating key critical exponents and comparing them to known universality classes.

Contribution

It provides new numerical estimates of critical exponents and conformal anomaly at the multicritical point, testing universality class predictions for the spin glass model.

Findings

Estimated conformal anomaly c=0.46(1)

Decay-of-correlations exponent η≈0.194

Susceptibility exponent ratios γ/ν≈1.797 and γ^{nl}/ν≈5.59

Abstract

We use numerical transfer-matrix methods to investigate properties of the multicriticalpoint of binary Ising spin glasses on a square lattice, whose location we assume to be given exactly by a conjecture advanced by Nishimori and Nemoto. We calculate the two largest Lyapunov exponents, as well as linear and non-linear zero-field uniform susceptibilities, on strip of widths $4 \leq L \leq 16$ sites, from which we estimate the conformal anomaly $c$, the decay-of-correlations exponent $\eta$, and the linear and non-linear susceptibility exponents $\gamma/\nu$ and $\gamma^{nl}/\nu$, with the help of finite-size scaling and conformal invariance concepts. Our results are: $c=0.46(1)$; $0.187 \lesssim \eta \lesssim 0.196$; $\gamma/\nu=1.797(5)$; $\gamma^{nl}/\nu=5.59(2)$. A direct evaluation of correlation functions on the strip geometry, and of the statistics of the zeroth moment of the associated probability distribution, gives $\eta=0.194(1)$, consistent with the calculation via Lyapunov exponents. Overall, these values tend to be inconsistent with the universality class of percolation, though by small amounts. The scaling relation $\gamma^{nl}/\nu=2 \gamma/\nu+d$ (with space dimensionality $d=2$) is obeyed to rather good accuracy, thus showing no evidence of multiscaling behavior of the susceptibilities.