Statistics of lowest droplets in two-dimensional Gaussian Ising spin glasses

Abstract

An approach to determine the value of the zero-temperature thermal exponent $\ensuremath{\theta}$ in spin glasses is presented. It consists in describing the energy level spectrum in spin glasses only in terms of the properties of the lowest energy droplets and the lowest droplet exponents (LDE's) ${\ensuremath{\lambda}}_{l},{\ensuremath{\theta}}_{l}$ that describe the statistics of their sizes and gaps. We show how these LDE's yield the standard thermal exponent of droplet theory $\ensuremath{\theta}$ through the relation $\ensuremath{\theta}={\ensuremath{\theta}}_{l}+d{\ensuremath{\lambda}}_{l}.$ The present approach provides a new way to measure the thermal exponent $\ensuremath{\theta}$ without any assumption about the correct procedure to generate typical low-lying excitations as is commonly done in many perturbation methods including domain wall calculations. To illustrate the usefulness of the method we present a detailed investigation of the properties of the lowest energy droplets in two-dimensional Gaussian Ising spin glasses. By independent measurements of both LDE's and an aspect-ratio analysis, we find $\ensuremath{\theta}(2d)\ensuremath{\simeq}\ensuremath{-}0.46(1)<{\ensuremath{\theta}}_{\mathrm{DW}}(2d)\ensuremath{\simeq}\ensuremath{-}0.287$ where ${\ensuremath{\theta}}_{\mathrm{DW}}$ is the thermal exponent obtained in domain-wall theory. We also discuss the origin of finite-volume corrections in the behavior of the LDE ${\ensuremath{\theta}}_{l}$ and relate them to the finite-volume corrections in the statistics of extreme values. Finally, we analyze some geometrical properties of the lowest energy droplets, finding results in agreement with those recently reported by Kawashima and Aoki [J. Phys. Soc. Jpn. $69,$ 169 (2000)]. All in all, we show that typical large-scale droplets are not probed by most of the present perturbation methods, since they probably do not have a compact structure as has been recently suggested. We speculate that a multifractal scenario could be at the roots of the reported discrepancies on the value of the thermal exponent $\ensuremath{\theta}$ in the two-dimensional Gaussian Ising spin glass.