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arXiv:cond-mat/0510614 · Physical Review Letters

Ground States and Defect Energies of the Two-Dimensional XY Spin Glass from a Quasiexact Algorithm

Martin Weigel, Michel J. P. Gingras

Published Mar 9, 2006• 16 citations
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Abstract

We employ a novel algorithm using a quasiexact embedded-cluster matching technique as minimization method within a genetic algorithm to reliably obtain numerically exact ground states of the Edwards-Anderson XY spin-glass model with bimodal coupling distribution for square lattices of up to 28 x 28 spins. Contrary to previous conjectures, the ground state of each disorder replica is nondegenerate up to a global O(2) rotation. The scaling of spin and chiral defect energies induced by applying several different sets of boundary conditions exhibits strong crossover effects. This suggests that previous calculations have yielded results far from the asymptotic regime. The novel algorithm and the aspect-ratio scaling technique consistently give theta(s)=-0.308(30) and theta(c)=-0.114(16) for the spin and chiral stiffness exponents, respectively.

DOI
10.1103/physrevlett.96.097206

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We employ a novel algorithm using a quasiexact embedded-cluster matching technique as minimization method within a genetic algorithm to reliably obtain numerically exact ground states of the Edwards-Anderson XY spin-glass model with bimodal coupling distribution for square lattices of up to 28 x 28 spins. Contrary to previous conjectures, the ground state of each disorder replica is nondegenerate up to a global O(2) rotation. The scaling of spin and chiral defect energies induced by applying several different sets of boundary conditions exhibits strong crossover effects. This suggests that previous calculations have yielded results far from the asymptotic regime. The novel algorithm and the aspect-ratio scaling technique consistently give theta(s)=-0.308(30) and theta(c)=-0.114(16) for the spin and chiral stiffness exponents, respectively.
For many real spin-glass materials, the Edwards-Anderson model with continuous-symmetry spins is more realistic than the rather better understood Ising variant. In principle, the nature of an occurring spin-glass phase in such systems might be inferred from an analysis of the zero-temperature properties. Unfortunately, with few exceptions, the problem of finding ground-state configurations is a nonpolynomial problem computationally, such that efficient approximation algorithms are called for. Here, we employ the recently developed genetic embedded matching (GEM) heuristic to investigate the nature of the zero-temperature phase of the bimodal XYXYXY spin glass in two dimensions. We analyze bulk properties such as the asymptotic ground-state energy and the phase diagram of disorder strength vs disorder concentration. For the case of a symmetric distribution of ferromagnetic and antiferromagnetic bonds, we find that the ground state of the model is unique up to a global O(2)\mathrm{O}(2)O(2) rotation of the spins. In particular, there are no extensive degeneracies in this model. The main focus of this work is on an investigation of the excitation spectrum as probed by changing the boundary conditions. Using appropriate finite-size scaling techniques, we consistently determine the stiffness of spin and chiral domain walls and the corresponding fractal dimensions. Most noteworthy, we find that the spin and chiral channels are characterized by two distinct stiffness exponents and, consequently, the system displays spin-chirality decoupling at large length scales. Results for the overlap distribution do not support the possibility of a multitude of thermodynamic pure states.
For many real spin-glass materials, the Edwards-Anderson model with continuous-symmetry spins is more realistic than the rather better understood Ising variant. In principle, the nature of an occurring spin-glass phase in such systems might be inferred from an analysis of the zero-temperature properties. Unfortunately, with few exceptions, the problem of finding ground-state configurations is a non-polynomial problem computationally, such that efficient approximation algorithms are called for. Here, we employ the recently developed genetic embedded matching (GEM) heuristic to investigate the nature of the zero-temperature phase of the bimodal XY spin glass in two dimensions. We analyze bulk properties such as the asymptotic ground-state energy and the phase diagram of disorder strength vs. disorder concentration. The main focus of this work is on an investigation of the excitation spectrum as probed by changing the boundary conditions. Using appropriate finite-size scaling techniques, we consistently determine the stiffness of spin and chiral domain walls and the corresponding fractal dimensions. Results for the overlap distribution do not support the possibility of a multitude of thermodynamic pure states.
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.
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​.
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.
The XY\mathrm{XY}XY model with quenched random phase shifts is studied by a T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 finite size defect energy scaling method in 2D and 3D. The defect energy is defined by a change in the boundary conditions from those compatible with the true ground state configuration for a given realization of disorder. A numerical technique, which is exact in principle, is used to evaluate this energy and to estimate the stiffness exponent $\ensuremath{\theta}$. This method gives $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}0.36\ifmmode\pm\else\textpm\fi{}0.01$ in 2D and $\ensuremath{\theta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}+0.31\ifmmode\pm\else\textpm\fi{}0.015$ in 3D, which are considerably larger than previous estimates, strongly suggesting that the lower critical dimension is less than 3. Some arguments in favor of these new estimates are given.