4,107 research outputs found
Sampling the ground-state magnetization of d-dimensional p-body Ising models
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.Comment: 8 pages, 6 figures, 3 table
Dynamics of the Wang-Landau algorithm and complexity of rare events for the three-dimensional bimodal Ising spin glass
We investigate the performance of flat-histogram methods based on a
multicanonical ensemble and the Wang-Landau algorithm for the three-dimensional
+/- J spin glass by measuring round-trip times in the energy range between the
zero-temperature ground state and the state of highest energy. Strong
sample-to-sample variations are found for fixed system size and the
distribution of round-trip times follows a fat-tailed Frechet extremal value
distribution. Rare events in the fat tails of these distributions corresponding
to extremely slowly equilibrating spin glass realizations dominate the
calculations of statistical averages. While the typical round-trip time scales
exponential as expected for this NP-hard problem, we find that the average
round-trip time is no longer well-defined for systems with N >= 8^3 spins. We
relate the round-trip times for multicanonical sampling to intrinsic properties
of the energy landscape and compare with the numerical effort needed by the
genetic Cluster-Exact Approximation to calculate the exact ground state
energies. For systems with N >= 8^3 spins the simulation of these rare events
becomes increasingly hard. For N >= 14^3 there are samples where the
Wang-Landau algorithm fails to find the true ground state within reasonable
simulation times. We expect similar behavior for other algorithms based on
multicanonical sampling.Comment: 9 pages, 12 figure
Genetic embedded matching approach to ground states in continuous-spin systems
Due to an extremely rugged structure of the free energy landscape, the
determination of spin-glass ground states is among the hardest known
optimization problems, found to be NP-hard in the most general case. Owing to
the specific structure of local (free) energy minima, general-purpose
optimization strategies perform relatively poorly on these problems, and a
number of specially tailored optimization techniques have been developed in
particular for the Ising spin glass and similar discrete systems. Here, an
efficient optimization heuristic for the much less discussed case of continuous
spins is introduced, based on the combination of an embedding of Ising spins
into the continuous rotators and an appropriate variant of a genetic algorithm.
Statistical techniques for insuring high reliability in finding (numerically)
exact ground states are discussed, and the method is benchmarked against the
simulated annealing approach.Comment: 17 pages, 12 figures, 1 tabl
Evolutionary Approaches to Optimization Problems in Chimera Topologies
Chimera graphs define the topology of one of the first commercially available
quantum computers. A variety of optimization problems have been mapped to this
topology to evaluate the behavior of quantum enhanced optimization heuristics
in relation to other optimizers, being able to efficiently solve problems
classically to use them as benchmarks for quantum machines. In this paper we
investigate for the first time the use of Evolutionary Algorithms (EAs) on
Ising spin glass instances defined on the Chimera topology. Three genetic
algorithms (GAs) and three estimation of distribution algorithms (EDAs) are
evaluated over hard instances of the Ising spin glass constructed from
Sidon sets. We focus on determining whether the information about the topology
of the graph can be used to improve the results of EAs and on identifying the
characteristics of the Ising instances that influence the success rate of GAs
and EDAs.Comment: 8 pages, 5 figures, 3 table
Calculation of ground states of four-dimensional +or- J Ising spin glasses
Ground states of four-dimensional (d=4) EA Ising spin glasses are calculated
for sizes up to 7x7x7x7 using a combination of a genetic algorithm and
cluster-exact approximation. The ground-state energy of the infinite system is
extrapolated as e_0=-2.095(1). The ground-state stiffness (or domain wall)
energy D is calculated. A D~L^{\Theta} behavior with \Theta=0.65(4) is found
which confirms that the d=4 model has an equilibrium spin-glass-paramagnet
transition for non-zero T_c.Comment: 5 pages, 3 figures, 31 references, revtex; update of reference
Direct sampling of complex landscapes at low temperatures: the three-dimensional +/-J Ising spin glass
A method is presented, which allows to sample directly low-temperature
configurations of glassy systems, like spin glasses. The basic idea is to
generate ground states and low lying excited configurations using a heuristic
algorithm. Then, with the help of microcanonical Monte Carlo simulations, more
configurations are found, clusters of configurations are determined and
entropies evaluated. Finally equilibrium configuration are randomly sampled
with proper Gibbs-Boltzmann weights.
The method is applied to three-dimensional Ising spin glasses with +- J
interactions and temperatures T<=0.5. The low-temperature behavior of this
model is characterized by evaluating different overlap quantities, exhibiting a
complex low-energy landscape for T>0, while the T=0 behavior appears to be less
complex.Comment: 9 pages, 7 figures, revtex (one sentence changed compared to v2
Ground-state behavior of the 3d +/-J random-bond Ising model
Large numbers of ground states of the three-dimensional random-bond
Ising model are calculated for sizes up to using a combination of a
genetic algorithm and Cluster-Exact Approximation. Several quantities are
calculated as function of the concentration of the antiferromagnetic bonds.
The critical concentration where the ferromagnetic order disappears is
determined using the Binder cumulant of the magnetization. A value of
is obtained. From the finite-size behavior of the Binder
cumulant and the magnetization critical exponents and
are calculated.Comment: 8 pages, 11 figures, revte
Ground-state landscape of 2d +-J Ising spin glasses
Large numbers of ground states of two-dimensional Ising spin glasses with
periodic boundary conditions in both directions are calculated for sizes up to
40^2. A combination of a genetic algorithm and Cluster-Exact Approximation is
used. For each quenched realization of the bonds up to 40 independent ground
states are obtained.
For the infinite system a ground-state energy of e=-1.4015(3) is
extrapolated. The ground-state landscape is investigated using a finite-size
scaling analysis of the distribution of overlaps. The mean-field picture
assuming a complex landscape describes the situation better than the
droplet-scaling model, where for the infinite system mainly two ground states
exist. Strong evidence is found that the ground states are not organized in an
ultrametric fashion in contrast to previous results for three-dimensional spin
glasses.Comment: 9 pages, revtex, 11 figures, 51 reference
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