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 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 which 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.Comment: large extensions, now 12 pages, 9 figures, 27 reference

Low-energy excitations in the three-dimensional random-field Ising model

The random-field Ising model (RFIM), one of the basic models for quenched disorder, can be studied numerically with the help of efficient ground-state algorithms. In this study, we extend these algorithm by various methods in order to analyze low-energy excitations for the three-dimensional RFIM with Gaussian distributed disorder that appear in the form of clusters of connected spins. We analyze several properties of these clusters. Our results support the validity of the droplet-model description for the RFIM.Comment: 10 pages, 9 figure

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

No spin-glass transition in the "mobile-bond" model

The recently introduced ``mobile-bond'' model for two-dimensional spin glasses is studied. The model is characterized by an annealing temperature T_q. On the basis of Monte Carlo simulations of small systems it has been claimed that this model exhibits a non-trivial spin-glass transition at finite temperature for small values of T_q. Here the model is studied by means of exact ground-state calculations of large systems up to N=256^2. The scaling of domain-wall energies is investigated as a function of the system size. For small values T_q<0.95 the system behaves like a (gauge-transformed) ferromagnet having a small fraction of frustrated plaquettes. For T_q>=0.95 the system behaves like the standard two-dimensional +-J spin-glass, i.e. it does NOT exhibit a phase transition at T>0.Comment: 4 pages, 5 figures, RevTe

Lower Critical Dimension of Ising Spin Glasses

Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal (+- J) distributions of the disorder are calculated using a ``matching'' algorithm, which allows large system sizes of up to N=480^2 spins to be investigated. We study domain walls induced by two rather different types of boundary-condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy'', which we denote by DE. For Gaussian disorder, we find a power-law behavior DE ~ L^\theta, with \theta=-0.266(2) and \theta=-0.282(2) for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension d_c. For the +-J model, we obtain a different result, namely the domain-wall energy saturates at a nonzero value for L\to \infty, so \theta = 0, indicating that the lower critical dimension for the +-J model exactly d_c=2.Comment: 4 pages, 4 figures, 1 table, revte

A dedicated algorithm for calculating ground states for the triangular random bond Ising model

In the presented article we present an algorithm for the computation of ground state spin configurations for the 2d random bond Ising model on planar triangular lattice graphs. Therefore, it is explained how the respective ground state problem can be mapped to an auxiliary minimum-weight perfect matching problem, solvable in polynomial time. Consequently, the ground state properties as well as minimum-energy domain wall (MEDW) excitations for very large 2d systems, e.g. lattice graphs with up to N=384x384 spins, can be analyzed very fast. Here, we investigate the critical behavior of the corresponding T=0 ferromagnet to spin-glass transition, signaled by a breakdown of the magnetization, using finite-size scaling analyses of the magnetization and MEDW excitation energy and we contrast our numerical results with previous simulations and presumably exact results.Comment: 5 pages, 5 figure

Generating droplets in two-dimensional Ising spin glasses by using matching algorithms

We study the behavior of droplets for two dimensional Ising spin glasses with Gaussian interactions. We use an exact matching algorithm which enables study of systems with linear dimension L up to 240, which is larger than is possible with other approaches. But the method only allows certain classes of droplets to be generated. We study single-bond, cross and a category of fixed volume droplets as well as first excitations. By comparison with similar or equivalent droplets generated in previous works, the advantages but also the limitations of this approach are revealed. In particular we have studied the scaling behavior of the droplet energies and droplet sizes. In most cases, a crossover of the data can be observed such that for large sizes the behavior is compatible with the one-exponent scenario of the droplet theory. Only for the case of first excitations, no clear conclusion can be reached, probably because even with the matching approach the accessible system sizes are still too small.Comment: 11 pages, 16 figures, revte

Percolation in three-dimensional random field Ising magnets

The structure of the three-dimensional random field Ising magnet is studied by ground state calculations. We investigate the percolation of the minority spin orientation in the paramagnetic phase above the bulk phase transition, located at [Delta/J]_c ~= 2.27, where Delta is the standard deviation of the Gaussian random fields (J=1). With an external field H there is a disorder strength dependent critical field +/- H_c(Delta) for the down (or up) spin spanning. The percolation transition is in the standard percolation universality class. H_c ~ (Delta - Delta_p)^{delta}, where Delta_p = 2.43 +/- 0.01 and delta = 1.31 +/- 0.03, implying a critical line for Delta_c < Delta <= Delta_p. When, with zero external field, Delta is decreased from a large value there is a transition from the simultaneous up and down spin spanning, with probability Pi_{uparrow downarrow} = 1.00 to Pi_{uparrow downarrow} = 0. This is located at Delta = 2.32 +/- 0.01, i.e., above Delta_c. The spanning cluster has the fractal dimension of standard percolation D_f = 2.53 at H = H_c(Delta). We provide evidence that this is asymptotically true even at H=0 for Delta_c < Delta <= Delta_p beyond a crossover scale that diverges as Delta_c is approached from above. Percolation implies extra finite size effects in the ground states of the 3D RFIM.Comment: replaced with version to appear in Physical Review

Scaling and self-averaging in the three-dimensional random-field Ising model

We investigate, by means of extensive Monte Carlo simulations, the magnetic critical behavior of the three-dimensional bimodal random-field Ising model at the strong disorder regime. We present results in favor of the two-exponent scaling scenario, $\bar{\eta}=2\eta$, where $\eta$ and $\bar{\eta}$ are the critical exponents describing the power-law decay of the connected and disconnected correlation functions and we illustrate, using various finite-size measures and properly defined noise to signal ratios, the strong violation of self-averaging of the model in the ordered phase.Comment: 8 pages, 6 figures, to be published in Eur. Phys. J.