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Nature of the zero-temperature phase of the planar spin glass in two dimensions

Martin Weigel, Michel J. P. Gingras

Published Jun 1, 2007• 0 citations
OpenAlex

Abstract

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.

Related papers

Zero-temperature phase of the XY spin glass in two dimensions: Genetic embedded matching heuristic

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Number of thermodynamic states in the three-dimensional Edwards-Anderson spin glass

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Finite-size scaling analysis of exact ground states for ± J spin glass models in two dimensions

Naoki Kawashima, Heiko Rieger

With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.

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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.
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.
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.
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.
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, ${\ensuremath{\theta}}^{\ensuremath{'}},$ 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.
The question of the number of thermodynamic states present in the low-temperature phase of the three-dimensional Edwards-Anderson Ising spin glass is addressed by studying spin- and link-overlap distributions using population annealing Monte Carlo simulations. We consider overlaps between systems with the same boundary condition, which are the usual quantities measured, and also overlaps between systems with different boundary conditions, both for the full systems and also within a smaller window within the system. Our results appear to be fully compatible with a single pair of pure states such as in the droplet-scaling picture. However, our results for whether or not domain walls induced by changing boundary conditions are space filling or not are also compatible with scenarios having many thermodynamic states, such as the chaotic-pairs picture and the replica-symmetry-breaking picture. The differing results for spin overlaps in the same and different boundary conditions suggest that finite-size effects are very large for the system sizes currently accessible in low-temperature simulations.
With the help of exact ground states obtained by a polynomial algorithm we compute the domain wall energy Δ at zero temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic and the spin glass order ceases to exist at a unique concentration p c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependence of the first and second moment of the domain wall energy are, for both models, described by a common finite-size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative, θ S = − 0.056(6), indicating the absence of any intermediate spin glass phase at non-zero temperature.