Effective critical behavior of the two-dimensional Ising spin glass with bimodal interactions

Katzgraber, Helmut G.; Lee, Lik Wee; Campbell, I. A.

doi:10.1103/physrevb.75.014412

Cited by 32 publications

(72 citation statements)

References 35 publications

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“…The intermediate temperature regime that corresponds to the power-law specific heat only appears for L > 100. In fact, our data for intermediate temperatures, T ≈ 0.35, are entirely consistent with previously published work [10], which saw an effective exponent α ≈ −4.2, with C ∼ T −α , but the effective exponent crosses over to lower values as T decreases. [20].…”

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confidence: 81%

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Zero- and Low-Temperature Behavior of the Two-Dimensional±JIsing Spin Glass

2011

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Recently extended precise numerical methods and droplet scaling arguments allow for a coherent picture of the glassy states of two-dimensional Ising spin glasses to be assembled. The length scale at which entropy becomes important and produces "chaos", the extreme sensitivity of the state to temperature, is found to depend on the type of randomness. For the ±J model this length scale dominates the low-temperature specific heat. Although there is a type of universality, some critical exponents do depend on the distribution of disorder.PACS numbers: 75.10. Nr, Glassy systems, characterized by extremely slow relaxation and resultant complex hysteresis and memory effects, are difficult to study because their dynamics encompass a great range of time scales [1]. Glassy materials include those without intrinsic disorder, such as silica glass, and those where quenched disorder influences the active degrees of freedom. An example of a model of the latter is the Edwards-Anderson spin glass model [2], which includes the disorder and frustration necessary to capture many of the complex behaviors seen in disordered magnetic materials. Though this prototypical model of glassy behavior was originally proposed well over 30 years ago, many aspects of it remain poorly understood. The droplet and replica-symmetry-breaking pictures of spin glasses provide distinct views of spin glass behavior [3]. Analytical results are rare, so numerical approaches are invaluable for both testing and motivating new ideas. But the numerics are also exceedingly difficult: computing spin glass ground states is in general a NP-hard problem [4]. It is believed that these classes of problems require exponential computational time to solve exactly [5].A fortunate special case which is not prone to this computational intractability is the two-dimensional Ising spin glass (2DISG). The Hamiltonian is H = ij J ij s i s j , where the couplings J = {J ij } are independent random variables coupling classical spins s = ±1 at sites i,j on a square toroidal grid with L 2 sites. The randomness of the sign of J ij leads to competing interactions, not all of which can be satisfied. In general, such models have complex (free) energy landscapes and very slow dynamics. The most commonly-used distributions for the J ij are the ±J distribution where each bond value is ±1 with equal probability, or the Gaussian distribution where the J ij are chosen from a univariate Gaussian distribution with zero mean. One apparent difficulty in using the 2DISG as a model system is that truly long-range spinglass order only occurs at zero temperature. However, at low enough temperature such that the correlation length ξ exceeds L one can study a regime of glassy behavior.In this glassy regime, the dominant spin configurations are very sensitive at large length scales to small changes in temperature or other global perturbations: this sensitivity is referred to as "chaos".Highly developed numerical algorithms [4,[6][7][8][9] can efficiently circumvent the complexity due to disorder and fr...

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confidence: 81%

“…Prior to the present work, the combination of scaling ideas and exact and Monte Carlo numerical evidence had been unable to develop a full understanding of the low-temperature glassy regimes [10].…”

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confidence: 99%

Zero- and Low-Temperature Behavior of the Two-Dimensional±JIsing Spin Glass

2011

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“…12 Limitations in the simulation techniques and analysis methods have so far yielded no conclusive results making this proposal controversial. 13,14 In spin glasses it is extremely difficult and numerically very costly to determine critical exponents with high precision. This is mainly due to the following reason: It is difficult to sample the disorder average with good enough statistics, especially for large system sizes, because spin glasses have very long equilibration times in Monte Carlo simulations and as a consequence one has to deal with corrections to scaling due to a very limited range of system sizes at hand.…”

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confidence: 99%

Universality and universal finite-size scaling functions in four-dimensional Ising spin glasses

Jörg

Katzgraber

2008

Phys. Rev. B

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“…There are two limiting regimes, with a size dependent crossover temperature T * (L) [10], a T < T * (L) ground state plus gap dominated regime and an effectively continuous energy level regime T > T * (L). There have been consistent estimates over decades from correlation function measurements [11,12], Monte Carlo renormalization-group measurements [13], transfer matrix calculations [14], numerical simulations [1,[15][16][17], and ground state measurements [18,19] showing that the anomalous dimension critical exponent η ≈ 0.20 in both regimes, indicating that the bimodal model is not in the same universality class as the continuous distribution models. However, it has also been claimed that the bimodal model in the T > T * (L) regime is in the same universality class as the Gaussian model, because for the bimodal model : "fits... lead to values of η that are very small, between 0 and 0.1, strongly suggestive of η = 0" [10], and "the data are not sufficiently precise to provide a precise determination of η, being consistent with a small value η ≤ 0.2, including η = 0" [20,21].…”

Section: Introductionmentioning

confidence: 95%

Ising spin glasses in two dimensions: Universality and nonuniversality

Lundow

Campbell

2017

Phys. Rev. E

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Following numerous earlier studies, extensive simulations and analyses were made on the continuous interaction distribution Gaussian model and the discrete bimodal interaction distribution Ising Spin Glass (ISG) models in dimension two (P.H. Lundow and I.A. Campbell, Phys. Rev. E 93, 022119 (2016)). Here we further analyse the bimodal and Gaussian data together with data on two other continuous interaction distribution 2D ISG models, the uniform and the Laplacian models, and three other discrete interaction distribution models, a diluted bimodal model, an "anti-diluted" model, and a more exotic symmetric Poisson model. Comparisons between the three continuous distribution models show that not only do they share the same exponent η ≡ 0 but that to within the present numerical precision they share the same critical exponent ν also, and so lie in a single universality class. On the other hand the critical exponents of the four discrete distribution models are not the same as those of the continuous distributions, and differ from one discrete distribution model to another. Discrete distribution ISG models in dimension two have non-zero values of the critical exponent η; they do not lie in a single universality class.

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Effective critical behavior of the two-dimensional Ising spin glass with bimodal interactions

Cited by 32 publications

References 35 publications

Zero- and Low-Temperature Behavior of the Two-Dimensional±JIsing Spin Glass

Zero- and Low-Temperature Behavior of the Two-Dimensional±JIsing Spin Glass

Universality and universal finite-size scaling functions in four-dimensional Ising spin glasses

Ising spin glasses in two dimensions: Universality and nonuniversality

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