Conformal Invariance and SLE in Two-Dimensional Ising Spin Glasses

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 $\theta \approx -0.28$.

4 pages, 4 figures, changed title and discussion, additional discussion of Markov property; discussion of Markov property removed, discussion of statistics of driving function of Loewner map added, see also related work by Bernard, Le Doussal, and Middleton to appear soon