Gauge-invariant method for the +/-J spin-glass model

Exact algorithm for sampling the two-dimensional Ising spin glass

A sampling algorithm is presented that generates spin-glass configurations of the two-dimensional Edwards-Anderson Ising spin glass at finite temperature with probabilities proportional to their Boltzmann weights. Such an algorithm overcomes the slow dynamics of direct simulation and can be used to study long-range correlation functions and coarse-grained dynamics. The algorithm uses a correspondence between spin configurations on a regular lattice and dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson, Proceedings of the Eighth Symposium on Discrete Algorithms (SIAM, Philadelphia, 1997), p 258] for sampling dimer coverings on a planar lattice is adapted to generate samplings for the dimer problem corresponding to both planar and toroidal spin-glass samples. This algorithm is recursive: it computes probabilities for spins along a "separator" that divides the sample in half. Given the spins on the separator, sample configurations for the two separated halves are generated by further division and assignment. The algorithm is simplified by using Pfaffian elimination rather than Gaussian elimination for sampling dimer configurations. For n spins and given floating point precision, the algorithm has an asymptotic run-time of O(n(3/2)); it is found that the required precision scales as inverse temperature and grows only slowly with system size. Sample applications and benchmarking results are presented for samples of size up to n=128(2), with fixed and periodic boundary conditions.

Ground-state clusters of two-, three-, and four-dimensional +/-J Ising spin glasses

A huge number of independent true ground-state configurations is calculated for two-, three- and four-dimensional +/-J spin-glass models. Using the genetic cluster-exact approximation method, system sizes up to N=20(2),8(3),6(4) spins are treated. A "ballistic-search" algorithm is applied, which allows even for large system sizes to identify clusters of ground states that are connected by chains of zero-energy flips of spins. The number of clusters n(C) diverges with N going to infinity. For all dimensions considered here, an exponential increase of n(C) appears to be more likely than a growth with a power of N. The number of different ground states is found to grow clearly exponentially with N. A zero-temperature entropy per spin of s(0)=0.078(5)k(B) (2D), s(0)=0.051(3)k(B) (3D), respectively, s(0)=0.027(5)k(B) (4D) is obtained.