Conformal Invariance and SLE in Two-Dimensional Ising Spin Glasses

TL;DR

This paper provides numerical evidence that conformal field theory techniques apply to 2D Ising spin glasses, showing domain walls behave as SLE processes with specific fractal dimensions and energy exponents.

Contribution

It demonstrates the applicability of conformal invariance and SLE to 2D Ising spin glasses, linking domain wall properties to conformal transformations and fractal dimensions.

Findings

Domain walls follow SLE with κ ≈ 2.1

Conformal transformations relate domain wall distributions in different geometries

Fractal dimension d_f ≈ 1.27 consistent with interface energy exponent θ ≈ -0.28

Abstract

We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain wall distributions in one geometry can be related to that in a second geometry by a conformal transformation. We also present direct evidence that the domain walls are stochastic Loewner (SLE) processes with $\kappa \approx 2.1$. An argument is given that their fractal dimension $d_f$ is related to their interface energy exponent $\theta$ by $d_f-1=3/[4(3+\theta)]$, which is consistent with the commonly quoted values $d_f \approx 1.27$ and $\theta \approx -0.28$.