Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass (original) (raw)
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Finite-Size Scaling of the Domain Wall Entropy Distributions for the 2D ± J Ising Spin Glass
Journal of Statistical Physics, 2006
The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for L × L square lattices with L ≤ 48, and p = 0.5, where p is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. When L is even, almost all domain walls have energy E dw = 0 or 4. When L is odd, most domain walls have E dw = 2. The probability distribution of the entropy, S dw , is found to depend strongly on E dw . When E dw = 0, the probability distribution of |S dw | is approximately exponential. The variance of this distribution is proportional to L, in agreement with the results of Saul and Kardar. For E dw = k > 0 the distribution of S dw is not symmetric about zero. In these cases the variance still appears to be linear in L, but the average of S dw grows faster than √ L. This suggests a one-parameter scaling form for the L-dependence of the distributions of S dw for k > 0.
Finite-size scaling of the Domain Wall Entropy for the 2D \pm J Ising Spin Glass
2005
The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and -1 bonds are studied for LtimesLL \times LLtimesL square lattices with Lle20L \le 20Lle20, and xxx = 0.25 and 0.5, where xxx is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. Under these conditions, almost all domain walls have an energy EdwE_{dw}Edw equal to 0 or 4. The probability distribution of the entropy, SdwS_{dw}Sdw, is found to depend strongly on EdwE_{dw}Edw. The results for SdwS_{dw}Sdw when Edw=4E_{dw} = 4Edw=4 agree with the prediction of the droplet model. Our results for SdwS_{dw}Sdw when Edw=0E_{dw} = 0Edw=0 agree with those of Saul and Kardar. In addition, we find that the distributions do not appear to be Gaussian in that case. The special role of Edw=0E_{dw} = 0Edw=0 domain walls is discussed, and the discrepancy between the prediction of Amoruso, Hartmann, Hastings and Moore and the result of Saul and Kardar is explained.
Scalings of Domain Wall Energies in Two Dimensional Ising Spin Glasses
Physical Review Letters, 2003
We study domain wall energies of two dimensional spin glasses. The scaling of these energies depends on the model's distribution of quenched random couplings, falling into three different classes. The first class is associated with the exponent θ ≈ −0.28, the other two classes have θ = 0, as can be justified theoretically. In contrast to previous claims we find that θ = 0 does not indicate d = d c l but rather d ≤ d c l , where d c l is the lower critical dimension. PACS numbers: 75.10.Nr, 75.40.Mg, 02.60.Pn
Subextensive Singularity in the 2D ± J Ising Spin Glass
Journal of Statistical Physics, 2007
The statistics of low energy states of the 2D Ising spin glass with +1 and -1 bonds are studied for L × L square lattices with L ≤ 48, and p = 0.5, where p is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. The behavior of the density of states near the ground state energy is analyzed as a function of L, in order to obtain the low temperature behavior of the model. For large finite L there is a range of T in which the heat capacity is proportional to T 5.33±0.12 . The range of T in which this behavior occurs scales slowly to T = 0 as L increases.
Physical Review B, 2007
The statistics of the ground-state and domain-wall energies for the two-dimensional random-bond Ising model on square lattices with independent, identically distributed bonds of probability p of Jij = −1 and (1 − p) of Jij = +1 are studied. We are able to consider large samples of up to 320 2 spins by using sophisticated matching algorithms. We study L × L systems, but we also consider L×M samples, for different aspect ratios R = L/M . We find that the scaling behavior of the groundstate energy and its sample-to-sample fluctuations inside the spin-glass region (pc ≤ p ≤ 1 − pc) are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusp-like singularity at pc. Inside the spin-glass region the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding R fixed. Here, large finite-size effects are visible, which can be explained for all p by a single exponent ω ≈ 2/3, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for R → 0: the distribution of the domain-wall energies converges to a Gaussian for R → 0, although the domain walls of neighboring subsystems of size L × L are not independent.
Frustration and ground-state degeneracy in spin glasses
Physical Review B, 1977
The problem of an Ising model with random nearest-neighbor interactions is reformulated to make manifest Toulouse's recent suggestion that a broken "lattice gauge symmetry" is responsible for the unusual properties of spin glasses. Exact upper and lower bounds on the ground-state energy for models in which the interactions are of constant magnitude but fluctuating sign are obtained, and used to place restrictions on possible geometries of the unsatisfied interactions which must be present in the ground state. Proposed analogies between the ferromagnetspin-glass phase boundary at zero temperature and a percolation threshold for the "strings" of unsatisfied bonds are reviewed in the light of this analysis. Monte Carlo simulations show that the upper bound resulting from a "one-dimensional approximation" to the spin-glass ground-state energy is reasonably close to the actual result. The transition between spin glass and ferromagnet at 0 K appears to be weakly first order in these models. The entropy of the ground state is obtained from the temperature dependence of the internal energy, and compared with the density of free spins at very low temperatures. For a two-dimensional spin glass in which half the bonds are antiferromagnetic, S(0)-0.099 k~; for the analogous three-dimensional spin glass the result is S(0)-0.062 k~. Monte Carlo kinetic simulations are reported which demonstrate the existence and stability of a fieldcooled moment in the spin-glass ground state.
Four-dimensional Ising spin glass: scaling within the spin-glass phase
1999
Abstract. We investigate the nature of the spin-glass phase in the four-dimensional king spin glass. We study he probability dishibutions of overlaps, of energy overlaps and ullrmetricity for several sizes. We discover the existence of finite-size scaling in the fails of the first and second of these. This allows us to exmct mame exponents within the spin-glass phase. We also perform studies on quenched thermalization of large samples. Our results together with previous work favour the mean-field picfure of the spin-glass phase.
Ground-state properties of the three-dimensional Ising spin glass
Physical Review B, 1994
We study zero-temperature properties of the 3d Edwards-Anderson Ising spin glass on finite lattices up to size 12 3. Using multicanonical sampling we generate large numbers of groundstate configurations in thermal equilibrium. Finite size scaling with a zerotemperature scaling exponent y = 0.74 ± 0.12 describes the data well. Alternatively, a descriptions in terms of Parisi mean field behaviour is still possible. The two scenarios give significantly different predictions on lattices of size ≥ 12 3 .
Critical scaling of the mutual information in two-dimensional disordered Ising models
Journal of Statistical Mechanics: Theory and Experiment, 2018
Rényi Mutual information (RMI), computed from second Rényi entropies, can identify classical phase transitions from their finite-size scaling at the critical points. We apply this technique to examine the presence or absence of finite temperature phase transitions in various two-dimensional models on a square lattice, which are extensions of the conventional Ising model by adding a quenched disorder. When the quenched disorder causes the nearest neighbor bonds to be both ferromagnetic and antiferromagnetic, (a) a spin glass phase exists only at zero temperature, and (b) a ferromagnetic phase exists at a finite temperature when the antiferromagnetic bond distributions are sufficiently dilute. Furthermore, finite temperature paramagnetic-ferromagnetic transitions can also occur when the disordered bonds involve only ferromagnetic couplings of random strengths. In our numerical simulations, the "zero temperature only" phase transitions are identified when there is no consistent finite-size scaling of the RMI curves, while for finite temperature critical points, the curves can identify the critical temperature Tc by their crossings at Tc and 2 Tc.
Physica A: Statistical Mechanics and its Applications, 2012
We study the two-dimensional Edwards-Anderson spin-glass model using a parallel tempering Monte Carlo algorithm. The ground-state energy and entropy are calculated for different bond distributions. In particular, the entropy is obtained by using a thermodynamic integration technique and an appropriate reference state, which is determined with the method of high-temperature expansion. This strategy provide accurate values of this quantity for finite-size lattices. By extrapolating to the thermodynamic limit, the ground-state energy and entropy of the different versions of the spin-glass model are determined.