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arXiv:0705.0046

Comment on "Conformal invariance and stochastic Loewner evolution processes in two-dimensional Ising spin glasses"

Ronald Fisch

Published May 1, 2007• 2 citations
arXivHTMLPDF

Abstract

By combining the scaling relation of Amoruso {\it et al.}, PRL {\bf 97}, 267202 (2006) with standard droplet model assumptions, a value $θ= (\sqrt{6} - 3) / 2$ is obtained. This conjecture is reasonably consistent with the best existing numerical calculations, and may be exact.

DOI
10.48550/arxiv.0705.0046

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By combining the scaling relation of Amoruso {\it et al.}, PRL {\bf 97}, 267202 (2006) with standard droplet model assumptions, a value θ=(6−3)/2θ= (\sqrt{6} - 3) / 2θ=(6​−3)/2 is obtained. This conjecture is reasonably consistent with the best existing numerical calculations, and may be exact.
Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional ±J\pm J±J Ising model'
Contrary to the suggestion of Kitatani and Sinada (2000 J. Phys. A: Math. Gen. {\bf 33} 3545-3553), the scaling analysis of the order parameter distributions suggests the existence of a finite-temperature spin-glass phase transition Tc≠0T_c\ne 0Tc​=0, even if the corrections to scaling are taken into account.
Reply to Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional ±J\pm J±J Ising model'
We show how the recent computer simulation data reported by Caracciolo et al. for the three-dimensional ±\pm± JJJ Ising spin glass in a uniform magnetic field are, in contrast to their claims, quite consistent with the simple scaling or droplet theory developed by McMillan, Fisher and Huse, and Bray and Moore, and do notnotnot provide evidence for a de Almeida-Thouless transition.
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 approximately 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) approximately 1.27 and theta approximately -0.28.