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arXiv:cond-mat/0205659 · Physical review. B, Condensed matter

Large-scale low-energy excitations in the two-dimensional Ising spin glass

Alexander K. Hartmann, A. P. Young

Published Sep 16, 2002• 45 citations
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Abstract

We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ${d}_{s},$ and stiffness exponent, ${\ensuremath{\theta}}^{\ensuremath{'}},$ for ``droplet'' excitations, are in reasonable agreement with estimates from ``domain wall'' calculations, and so support the predictions of the ``droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the ``TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.

DOI
10.1103/physrevb.66.094419

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We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ds,{d}_{s},ds​, and stiffness exponent, \ensuremathθ\ensuremath′,{\ensuremath{\theta}}^{\ensuremath{'}},\ensuremathθ\ensuremath′, for droplet'' excitations, are in reasonable agreement with estimates from domain wall'' calculations, and so support the predictions of the droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.
We investigate the nature of the low-energy, large-scale excitations in the three-dimensional Edwards-Anderson Ising spin glass with Gaussian couplings and free boundary conditions, by studying the response of the ground state to a coupling-dependent perturbation introduced previously. The ground states are determined exactly for system sizes up to 123{12}^{3}123 spins using a branch-and-cut algorithm. The data are consistent with a picture where the surface of the excitations is not space filling, such as the droplet or the trivial-non-trivial (TNT) picture, with only minimal corrections to scaling. When allowing for very large corrections to scaling, the data are also consistent with a picture with space-filling surfaces, such as replica symmetry breaking. The energy of the excitations scales with their size with a small exponent ${\ensuremath{\theta}}^{\ensuremath{'}},$ which is compatible with zero if we allow moderate corrections to scaling. We compare the results with data for periodic boundary conditions obtained with a genetic algorithm, and discuss the effects of different boundary conditions on corrections to scaling. Finally, we analyze the performance of our branch-and-cut algorithm, finding that it is correlated with the existence of large-scale, low-energy excitations.
We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for nonfree boundary conditions is governed by the stiffness exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282$ describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.
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. 69,69,69, 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.
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 L2=2562{L}^{2}={256}^{2}L2=2562 spins can be generated. Furthermore, no equilibration or extrapolation to T=0T=0T=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=0.125p=0.125p=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.
Results are presented for the geometry of low-energy excitations in the one-dimensional Ising spin chain with power-law interactions, in which the model parameters are chosen to yield a finite spin-glass transition temperature. Both finite-temperature and ground-state studies are carried out. For the range of sizes studied the data cannot be fitted to any of the standard spin-glass scenarios without including corrections to scaling. Incorporating such corrections we find that the fractal dimension of the surface of the excitations, is either equal to the space dimension, consistent with replica symmetry breaking, or very slightly less than it. The latter case is consistent with the droplet and ``trivial-nontrivial'' pictures.
Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal $(\ifmmode\pm\else\textpm\fi{}J)$ distributions of the disorder are calculated using a ``matching''algorithm, which allows large system sizes of up to N=4802{N=480}^{2}N=4802 spins to be investigated. We study domain walls induced by two rather different types of boundary condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy,'' which we denote by $\ensuremath{\Delta}E.$ For Gaussian disorder, we find a power-law behavior $\ensuremath{\Delta}E\ensuremath{\sim}{L}^{\ensuremath{\theta}},$ with $\ensuremath{\theta}=\ensuremath{-}0.266(2)$ and $\ensuremath{\theta}=\ensuremath{-}0.282(2)$ for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension dc.{d}_{c}.dc​. For the $\ifmmode\pm\else\textpm\fi{}J$ model, we obtain a different result, namely that the domain-wall energy saturates at a nonzero value for $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty},$ so $\ensuremath{\theta}=0,$ indicating that the lower critical dimension for the $\ifmmode\pm\else\textpm\fi{}J$ model is exactly dc=2.{d}_{c}=2.dc​=2.