Numerically exact correlations and sampling in the two-dimensional Ising spin glass

Abstract

A powerful existing technique for evaluating statistical mechanical quantities in two-dimensional Ising models is based on constructing a matrix representing the nearest neighbor spin couplings and then evaluating the Pfaffian of the matrix. Utilizing this technique and other more recent developments in evaluating elements of inverse matrices and exact sampling, a method and computer code for studying two-dimensional Ising models is developed. The formulation of this method is convenient and fast for computing the partition function and spin correlations. It is also useful for exact sampling, where configurations are directly generated with probability given by the Boltzmann distribution. These methods apply to Ising model samples with arbitrary nearest-neighbor couplings and can also be applied to general dimer models. Example results of computations are described, including comparisons with analytic results for the ferromagnetic Ising model, and timing information is provided.

Universal critical behavior of the two-dimensional Ising spin glass

“…Disentangling the effects of the three sources of corrections to scaling will require a strong analytical guidance. Probably, simulating much larger systems, which is possible using special methods [46], will be useful.…”

“…For Gaussian couplings we also obtain a precise estimate of ν [specifically, 1/ν = 0.283(6)]. Notice that there are methods more powerful than Monte Carlo for simulating the model we study (namely, the Edwards-Anderson model on the square lattice with nearest-neighbor couplings [33,46]). However, our focus here is on the finite-size scaling analysis regardless of how data were obtained.…”

Topologically invariant tensor renormalization group method for the Edwards-Anderson spin glasses model

“…If all the external random fields are zero, the partition function and the pair-spin correlations can be calculated exactly in polynomial time [23,31,38]. However, for general external fields {h i }, the problem is proved to be in the NP-hard class [23].…”

“…In the previous study, the mean-field approximation [24][25][26][27][28] and Monte Carlo sampling [29], transfer matrix method [30], and numerical exact algorithm for 2D without the external field [23,31] are used to calculate local properties for individual finite size instances. These methods are combined with finite size scaling to investigate the thermodynamic limit properties.…”

Extracting thermodynamic behavior of spin glasses from the overlap function

“…In the 2D ±J model, a large number of degenerate configurations exist at all scales, but entropic effects suppress the effects of this degeneracy at large scales, just as large scale excitations are suppressed by large free-energy costs within the droplet model for 3D spin glasses at low enough temperatures T < T 3D g . The advantage of the two-dimensional model is that configurations can be exactly sampled 14 for relatively large linear sizes L 256. The sample average of P (q) near q = 0 varies slowly with L for L ≈ 10, resembling the results 15,16 for three-and four-dimensional Ising spin glasses that would appear to support a many-states picture.…”