Exact Algorithm for Sampling the 2D Ising Spin Glass

A sampling algorithm is presented that generates spin glass configurations of the 2D 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, Proc. 8th Symp. Discrete Algorithms 258, (1997)] 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.Comment: 18 pages, 10 figures, 1 table; minor clarification

Simplest model to study reentrance in physical systems

We numerically investigate the necessary ingredients for reentrant behavior in the phase diagram of physical systems. Studies on the possibly simplest model that exhibits reentrance, the two-dimensional random bond Ising model, show that reentrant behavior is generic whenever frustration is present in the model. For both discrete and continuous disorder distributions, the phase diagram in the disorder-temperature plane is found to be reentrant, where for some disorder strengths a paramagnetic phase exists at both high and low temperatures, but an ordered ferromagnetic phase exists for intermediate temperatures.Comment: 4 pages, 5 figure

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

Boolean decision problems with competing interactions on scale-free networks: Critical thermodynamics

We study the critical behavior of Boolean variables on scale-free networks with competing interactions (Ising spin glasses). Our analytical results for the disorder-network-decay-exponent phase diagram are verified using Monte Carlo simulations. When the probability of positive (ferromagnetic) and negative (antiferromagnetic) interactions is the same, the system undergoes a finite-temperature spin-glass transition if the exponent that describes the decay of the interaction degree in the scale-free graph is strictly larger than 3. However, when the exponent is equal to or less than 3, a spin-glass phase is stable for all temperatures. The robustness of both the ferromagnetic and spin-glass phases suggests that Boolean decision problems on scale-free networks are quite stable to local perturbations. Finally, we show that for a given decay exponent spin glasses on scale-free networks seem to obey universality. Furthermore, when the decay exponent of the interaction degree is larger than 4 in the spin-glass sector, the universality class is the same as for the mean-field Sherrington-Kirkpatrick Ising spin glass.Comment: 14 pages, lots of figures and 2 table

Irrational mode locking in quasiperiodic systems

A model for ac-driven systems, based on the Tang-Wiesenfeld-Bak-Coppersmith-Littlewood automaton for an elastic medium, exhibits mode-locked steps with frequencies that are irrational multiples of the drive frequency, when the pinning is spatially quasiperiodic. Detailed numerical evidence is presented for the large-system-size convergence of such a mode-locked step. The irrational mode locking is stable to small thermal noise and weak disorder. Continuous time models with irrational mode locking and possible experimental realizations are discussed.Comment: 4 pages, 3 figures, 1 table; revision: 2 figures modified, reference added, minor clarification

Novel disordering mechanism in ferromagnetic systems with competing interactions

Ferromagnetic Ising systems with competing interactions are considered in the presence of a random field. We find that in three space dimensions the ferromagnetic phase is disordered by a random field which is considerably smaller than the typical interaction strength between the spins. This is the result of a novel disordering mechanism triggered by an underlying spin-glass phase. Calculations for the specific case of the long-range dipolar LiHo_xY_{1-x}F_4 compound suggest that the above mechanism is responsible for the peculiar dependence of the critical temperature on the strength of the random field and the broadening of the susceptibility peaks as temperature is decreased, as found in recent experiments by Silevitch et al. [Nature (London) 448, 567 (2007)]. Our results thus emphasize the need to go beyond the standard Imry-Ma argument when studying general random-field systems.Comment: 4+2 pages, 3 figure

Shapes of pored membranes

We study the shapes of pored membranes within the framework of the Helfrich theory under the constraints of fixed area and pore size. We show that the mean curvature term leads to a budding- like structure, while the Gaussian curvature term tends to flatten the membrane near the pore; this is corroborated by simulation. We propose a scheme to deduce the ratio of the Gaussian rigidity to the bending rigidity simply by observing the shape of the pored membrane. This ratio is usually difficult to measure experimentally. In addition, we briefly discuss the stability of a pore by relaxing the constraint of a fixed pore size and adding the line tension. Finally, the flattening effect due to the Gaussian curvature as found in studying pored membranes is extended to two-component membranes. We find that sufficiently high contrast between the components' Gaussian rigidities leads to budding which is distinct from that due to the line tension.Comment: 8 pages, 9 figure