Non-self-averaging in Ising spin glasses and hyperuniversality

Abstract

Ising spin glasses with bimodal and Gaussian near-neighbor interaction distributions are studied through numerical simulations. The non-self-averaging (normalized inter-sample variance) parameter U22(T, L) for the spin glass susceptibility (and for higher moments Unn(T, L)) is reported for dimensions 2, 3, 4, 5 and 7. In each dimension d the non-self-averaging parameters in the paramagnetic regime vary with the sample size L and the correlation lengthand so follow a renormalization group law due to Aharony and Harris [1]. Empirically, it is found that the K d values are independent of d to within the statistics. The maximum values [Unn(T, L)]max are almost independent of L in each dimension, and remarkably the estimated thermodynamic limit critical [Unn(T, L)]max peak values are also dimension-independent to within the statistics and so are "hyperuniversal". These results show that the form of the spin-spin correlation function distribution at criticality in the large L limit is independent of dimension within the ISG family. Inspection of published non-self-averaging data for 3D Heisenberg and XY spin glasses the light of the Ising spin glass non-self-averaging results show behavior incompatible with a spin-driven ordering scenario, but compatible with that expected on a chiral-driven ordering interpretation.

Bimodal Ising spin glass in two dimensions: The anomalous dimension η

“…The latter estimate was qualitatively confirmed by a Monte Carlo renormalizationgroup measurement which indicated η ∼ 0.20 [18] again in what can now be recognized as being the T > T * (L) regime [19]. Later numerical simulation estimates were η ∼ 0.20 [20], η ∼ 0.138 [21], η > 0.20 [22], η = 0.20(2) [7].…”

“…For any continuous distribution model including the Gaussian the anomalous dimension critical exponent is analytically known to be η ≡ 0 because the ground state is non-degenerate; accurate and consistent estimates have been made of the correlation length critical exponent ν = 3.52 (2) for the Gaussian model [3][4][5][6][7][8] and for other continuous distribution models [9]. The values of other critical exponents, in particular the magnetization exponent γ = (2 − η)ν, follow.…”

“…Recent estimates for the bimodal correlation function exponent are ν ∼ 5.5 [11] and ν = 4.8(3) [7], both significantly higher than the accepted estimate for the Gaussian model. "Quotient" analyses for the bimodal model [8] are consistent with the higher value [9].…”

“…It has been established that for finite lattice size L there can be considered to be two regimes separated by a crossover temperature T * (L), a "ground state plus gap" regime for T < T * (L), and an "effectively continuous energy level" regime for T > T * (L) [10]. In the infinite-L thermodynamic limit (ThL) T * reaches zero because T * (L) drops with increasing L as T * (L) ∼ 1.1(1)L −1/2 [7,11]. In the strict ground-state limit T ≡ 0 and for finite L, the bimodal 2D ISG exponent η has been estimated to be η = 0.210 (23) [12] from transfer-matrix ground-state measurements, η = 0.14(1) from ground-state spin correlations [13] and η = 0.22(1) from non-zero-energy droplet probabilities [14].…”

Dynamic scaling in the two-dimensional Ising spin glass with normal-distributed couplings

“…However, due to the considerable challenges with MC simulations, especially for large systems at low temperature, there are still significant issues under debate. For example, whether or not the 2DISG with bimodal J = ±1 and Gaussian couplings belong to the same universality class in their equilibrium criticality is still in question [9][10][11][12][13][14]. Undisputed is the fact that the ground-state properties of the two models are different.…”