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International Journal of Modern Physics C

GROUND STATES IN THREE-DIMENSIONAL ±J EDWARDS–ANDERSON SPIN GLASSES WITH FREE BOUNDARIES

Frauke Liers, Michael Jünger

Published May 1, 2000• 0 citations
Publisher

Abstract

By an exact calculation of the ground states for the ±J Edwards–Anderson spin glass, one can extrapolate the ground state energy to infinite system sizes. We calculate the exact ground states for the three-dimensional spin glass with free boundaries for system sizes up to 10 and fit different finite-size functions. We cannot decide, only from the quality of the fit, which fitting function to choose. Relying on the literature values for the extrapolated energy, we find the finite-size corrections to vary as 1/L.

DOI
10.1142/s0129183100000493

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By an exact calculation of the ground states for the ±J Edwards–Anderson spin glass, one can extrapolate the ground state energy to infinite system sizes. We calculate the exact ground states for the three-dimensional spin glass with free boundaries for system sizes up to 10 and fit different finite-size functions. We cannot decide, only from the quality of the fit, which fitting function to choose. Relying on the literature values for the extrapolated energy, we find the finite-size corrections to vary as 1/L.
Extensive computations of ground-state energies of the Edwards-Anderson spin glass on bond-diluted, hypercubic lattices are conducted in dimensions d =3, ..., 7. Results are presented for bond densities exactly at the percolation threshold, p = p c , and deep within the glassy regime, p > p c , where finding ground states is one of the hardest combinatorial optimization problems. Finite-size corrections of the form 1/ N ω are shown to be consistent throughout with the prediction ω=1− y / d , where y refers to the "stiffness" exponent that controls the formation of domain wall excitations at low temperatures. At p = p c , an extrapolation for d → ∞ appears to match our mean-field results for these corrections. In the glassy phase, however, ω does not approach its anticipated mean-field value of 2/3, obtained from simulations of the Sherrington-Kirkpatrick spin glass on an N -clique graph. Instead, the value of ω reached at the upper critical dimension matches another type of mean-field spin glass models, namely those on sparse random networks of regular degree called Bethe lattices.
Large numbers of ground states of three-dimensional (3D) Edwards-Anderson Ising spin glasses are calculated for sizes up to 103{10}^{3}103 using a combination of a genetic algorithm and cluster-exact approximation. A detailed analysis shows that true ground states are obtained. The ground-state stiffness (or domain wall) energy \ensuremath{\Delta} is calculated. A $|\ensuremath{\Delta}|\ensuremath{\sim}{L}^{{\ensuremath{\Theta}}_{S}}$ behavior with ${\ensuremath{\Theta}}_{S}=0.19(2)$ is found that strongly indicates that the 3D model has an equilibrium spin-glass-paramagnet transition for nonzero Tc.{T}_{c}.Tc​.
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 study zero-temperature properties of the three-dimensional Edwards-Anderson Ising spin glass on finite lattices up to size 123{12}^{3}123. Using multicanonical sampling we generate large numbers of ground-state configurations in thermal equilibrium. Finite-size scaling fits of the data are carried out for two hypothetical scenarios: Parisi mean-field theory versus a droplet scaling ansatz. With a zero-temperature scaling exponent y=0.72\ifmmode\pm\else\textpm\fi{}0.12 the data are well described by the droplet scaling ansatz. Alternatively, a description in terms of the Parisi mean-field behavior is still possible. The two scenarios give significantly different predictions on lattices of size \ensuremath{\ge}123{12}^{3}123.
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.
We present a collection of simulations of the Edwards–Anderson lattice spin glass at T=0 to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample the ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a variety of accurate exponents for domain wall stiffness and finite-size corrections, which allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattice Ising spin glasses.