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arXiv:1108.1953 · Physical Review B

Sampling the ground-state magnetization of d -dimensional p -body Ising models

Creighton K. Thomas, Helmut G. Katzgraber

Published Nov 4, 2011• 6 citations
arXivHTMLPDF

Abstract

We demonstrate that a recently introduced heuristic optimization algorithm [Phys. Rev. E 83, 046709 (2011)] that combines a local search with triadic crossover genetic updates is capable of sampling nearly uniformly among ground-state configurations in spin-glass-like Hamiltonians with $p$-spin interactions in $d$ space dimensions that have highly degenerate ground states. Using this algorithm we probe the zero-temperature ferromagnet to spin-glass transition point ${q}_{c}$ of two example models, the disordered version of the two-dimensional three-spin Baxter-Wu model [${q}_{c}=0.1072(1)$] and the three-dimensional Edwards-Anderson model [${q}_{c}=0.2253(7)$], by computing the Binder ratio of the ground-state magnetization.

DOI
10.1103/physrevb.84.174404

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We demonstrate that a recently introduced heuristic optimization algorithm [Phys. Rev. E 83, 046709 (2011)] that combines a local search with triadic crossover genetic updates is capable of sampling nearly uniformly among ground-state configurations in spin-glass-like Hamiltonians with ppp-spin interactions in ddd space dimensions that have highly degenerate ground states. Using this algorithm we probe the zero-temperature ferromagnet to spin-glass transition point qc{q}_{c}qc​ of two example models, the disordered version of the two-dimensional three-spin Baxter-Wu model [qc=0.1072(1){q}_{c}=0.1072(1)qc​=0.1072(1)] and the three-dimensional Edwards-Anderson model [qc=0.2253(7){q}_{c}=0.2253(7)qc​=0.2253(7)], by computing the Binder ratio of the ground-state magnetization.
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.
Large numbers of ground states of the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model are calculated for sizes up to 143{14}^{3}143 using a combination of a genetic algorithm and cluster-exact approximation. Several quantities are calculated as functions of the concentration p of the antiferromagnetic bonds. The critical concentration where the ferromagnetic order disappears is determined using the Binder cumulant of the magnetization. A value of ${p}_{c}=0.222\ifmmode\pm\else\textpm\fi{}0.005$ is obtained. From the finite-size behavior of the Binder cumulant and the magnetization critical exponents \ensuremath{\nu}=1.1\ifmmode\pm\else\textpm\fi{}0.3 and \ensuremath{\beta}=0.2\ifmmode\pm\else\textpm\fi{}0.1 are calculated. The behavior of the distribution of overlaps P(q)P(q)P(q) is used to investigate how the spin-glass phase evolves with increasing concentration p.p.p. The spin-glass order is characterized by a broad distribution of overlaps which extends down to q=0.q=0.q=0.
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.
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​.
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.
Abstract Systems with quenched disorder possess complex energy landscapes that are challenging to explore under conventional Monte Carlo methods. In this work, we implement an efficient entropy sampling scheme for accurate computation of the entropy function in low-energy regions. The method is applied to the two-dimensional ± J random-bond Ising model, where frustration is controlled by the fraction p of ferromagnetic bonds. We investigate the low-temperature paramagnetic–ferromagnetic phase boundary below the multicritical point at T N = 0.9530(4), P N = 0.89078(8), as well as the zero-temperature ferromagnetic–spin-glass transition. Finite-size scaling analysis reveals that the phase boundary for T < T N exhibits reentrant behavior. By analyzing the evolution of the magnetization-resolved density of states g ( E , M ) and ground-state spin configurations against increasing frustration, we provide strong evidence that the zero-temperature transition is a mixed-order. Finite-size scaling conducted on the spin-glass side supports the validity of β = 0, where β is the magnetization exponent, with a correlation length exponent ν = 1.50(8). Our results provide new insights into the nature of the ferromagnetic-to-spin-glass phase transition in an extensively degenerate ground state.