Droplets in the two-dimensional ±J Ising spin glass

Abstract

By using problem-dependent mappings to computer-science problems and by applying sophisticated algorithms, one can numerically study some important problems much better compared to applying standard approaches, such as Monte Carlo simulations. Here, by using calculations of ground states of suitable perturbed systems, droplets are obtained in two-dimensional $\ifmmode\pm\else\textpm\fi{}J$ spin glasses, which are in the focus of a currently very lively debate. By using a sophisticated matching algorithm, exact ground states of large systems up to ${L}^{2}={256}^{2}$ spins can be generated. Furthermore, no equilibration or extrapolation to $T=0$ is necessary. Three different $\ifmmode\pm\else\textpm\fi{}J$ models are studied here: (a) with free boundary conditions, (b) with fixed boundary conditions, and (c) a diluted system wherein a fraction $p=0.125$ of all bonds is zero. For large systems, the droplet energy shows for all three models a power-law behavior ${E}_{\text{dp}}\ensuremath{\sim}{L}^{{\ensuremath{\theta}}_{\text{dp}}^{\ensuremath{'}}}$ with ${\ensuremath{\theta}}_{\text{dp}}^{\ensuremath{'}}<0$. This is different from the previous studies of domain walls, wherein a convergence to a constant nonzero value $({\ensuremath{\theta}}_{\text{DW}}=0)$ was found. After correcting for the noncompactness of the droplets, the results are likely to be compatible with ${\ensuremath{\theta}}_{\text{dp}}\ensuremath{\approx}\ensuremath{-}0.29$ for all three models. This is in accordance with the Gaussian system wherein ${\ensuremath{\theta}}_{\text{dp}}=\ensuremath{-}0.287(4)$ ($\ensuremath{\nu}\ensuremath{\approx}3.5$ via $\ensuremath{\nu}=\ensuremath{-}1/{\ensuremath{\theta}}_{\text{dp}}$). Nevertheless, the disorder-averaged spin-spin correlation exponent $\ensuremath{\eta}$ is determined here via the probability to have a nonzero-energy droplet, and $\ensuremath{\eta}\ensuremath{\approx}0.22$ is found for all three models, in contrast to the model with the Gaussian interactions, where $\ensuremath{\eta}=0$ is exact.