Abstract Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds J distributed according to a Levy distribution P(J) / 1/|J| 1+α ; |J| > 1; are investigated for a range of powers α. We determine ground state energy density variation with α and their finite-size corrections. We find that the energies attain universally the Parisi-energy of the SK as long as the second moment of P (J) exists (α > 2). They compare favorably with recent one-step replica symmetry breaking predictions well below α = 2. At and just below α = 2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent ω decays from the putative SK value ω SK = = 2/3 already well above α = 2, at which point it reaches a minimum.
Abstract Ground states of Ising spin glasses on fully connected graphs are studied for a broadly distributed bond family. In particular, bonds J distributed according to a Levy distribution P(J) / 1/|J| 1+α ; |J| > 1; are investigated for a range of powers α. We determine ground state energy density variation with α and their finite-size corrections. We find that the energies attain universally the Parisi-energy of the SK as long as the second moment of P (J) exists (α > 2). They compare favorably with recent one-step replica symmetry breaking predictions well below α = 2. At and just below α = 2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent ω decays from the putative SK value ω SK = = 2/3 already well above α = 2, at which point it reaches a minimum.
We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond diluted to an overall density p, such that the average degree diverges as ∼pN with the system size N. Ground state energies are obtained with high accuracy for random instances over a wide range of fixed p. Since this is an NP-hard combinatorial problem, we employ the extremal optimization heuristic to that end. We find that the exponent describing the finite-size corrections ω varies continuously with p, a somewhat surprising result, as one would not expect that gradual bond dilution would change the T=0 universality class of a statistical model. For p→1, the familiar result of ω(p=1)≈2/3 for the SK model is obtained.
Extensive computations of ground-state energies of the Edwards-Anderson spin glass on bond-diluted, hypercubic lattices are conducted in dimensions d =3, ..., 7. Results are presented for bond densities exactly at the percolation threshold, p = p c , and deep within the glassy regime, p > p c , where finding ground states is one of the hardest combinatorial optimization problems. Finite-size corrections of the form 1/ N ω are shown to be consistent throughout with the prediction ω=1− y / d , where y refers to the "stiffness" exponent that controls the formation of domain wall excitations at low temperatures. At p = p c , an extrapolation for d → ∞ appears to match our mean-field results for these corrections. In the glassy phase, however, ω does not approach its anticipated mean-field value of 2/3, obtained from simulations of the Sherrington-Kirkpatrick spin glass on an N -clique graph. Instead, the value of ω reached at the upper critical dimension matches another type of mean-field spin glass models, namely those on sparse random networks of regular degree called Bethe lattices.
We present a large-scale simulation of the ground state interface properties of the diluted Sherrington-Kirkpatrick spin glass of Gaussian disorder for a broad range of the bond occupation probability p using the strong disorder renormalization group and the population annealing Monte Carlo methods. The model is studied in the framework of the diluted one-dimensional Kotliar-Anderson-Stein spin glass with power-law interactions in the mean-field regime. We find that the interface is space-filling independent of p, i.e., the fractal dimension ds=1. The stiffness exponent $\ensuremath{\theta}$ is likely also independent of p, despite that the energy finite-size correction exponent $\ensuremath{\omega}$ varies with p as recently found. The energy finite-size scaling is also analyzed and compared with that of the $\ifmmode\pm\else\textpm\fi{}J$ disorder, finding that the thermodynamic energy is universal in both p and the disorder, and the exponent $\ensuremath{\omega}$ varies with p but is universal in the disorder.
We use heuristic optimization methods in extensive computations to determine with low systematic error ground-state configurations of the mean-field p-spin glass model with p=3. Here, all possible triplets in a system of N Ising spins are connected with a bond. This model has been of recent interest, since it exhibits the "overlap gap condition," which should make it prohibitive to find ground states asymptotically with local search methods when compared, for instance, with the p=2 case better known as the Sherrington-Kirkpatrick model (SKM). Indeed, it proves more costly to find good approximations for p=3 than for the SKM, even for our heuristic. Compared to the SKM, the ground-state behavior for p=3 is quite distinct also in other ways. For the SKM, finite-size corrections for large system sizes N→∞, of both the ensemble average over ground-state energy densities and the width of their distribution, vary anomalously with noninteger exponents. In the p=3 case here, the energy density and its distribution appear to scale with lnN/N and 1/N corrections, respectively. The distribution itself is consistent with a Gumbel form. Even more stark is the contrast for the bond-diluted case, where the SKM has shown previously a notable variation of the anomalous corrections exponent with the bond density, while for p=3 no such variation is found here. Hence, for the 3-spin model, all measured corrections scale the same as for the random energy model (REM), corresponding to p=∞. This would suggest that all p-spin models with p≥3 exhibit the same ground-state corrections as in the REM.
We study the probability distribution function of the ground-state energies of the disordered one-dimensional Ising spin chain with power-law interactions using a combination of parallel tempering Monte Carlo and branch, cut, and price algorithms. By tuning the exponent of the power-law interactions we are able to scan several universality classes. Our results suggest that mean-field models have a non-Gaussian limiting distribution of the ground-state energies, whereas non-mean-field models have a Gaussian limiting distribution. We compare the results of the disordered one-dimensional Ising chain to results for a disordered two-leg ladder, for which large system sizes can be studied, and find a qualitative agreement between the disordered one-dimensional Ising chain in the short-range universality class and the disordered two-leg ladder. We show that the mean and the standard deviation of the ground-state energy distributions scale with a power of the system size. In the mean-field universality class the skewness does not follow a power-law behavior and converges to a nonzero constant value. The data for the Sherrington-Kirkpatrick model seem to be acceptably well fitted by a modified Gumbel distribution. Finally, we discuss the distribution of the internal energy of the Sherrington-Kirkpatrick model at finite temperatures and show that it behaves similar to the ground-state energy of the system if the temperature is smaller than the critical temperature.
The performance of quantum annealing for combinatorial optimization is fundamentally limited by the minimum energy gap Δ encountered at quantum phase transitions. We investigate the scaling of Δ with system size N for two paradigmatic quantum spin-glass models: the two-dimensional Edwards-Anderson (2D-EA) and the all-to-all Sherrington-Kirkpatrick (SK) models. Utilizing a newly proposed unbiased energy-gap estimator for continuous-time projection quantum Monte Carlo simulations, complemented by high-performance sparse eigenvalue solvers, we characterize the gap distributions across disorder realizations. It is found that, in the 2D-EA case, the inverse-gap distribution develops a fat tail with infinite variance as N increases. This indicates that the unfavorable super-algebraic scaling of Δ, recently reported for binary couplings [Nature 631, 749 (2024)], persists for the Gaussian disorder considered here, pointing to a universal feature of 2D spin glasses. Conversely, the SK model retains a finite-variance distribution, with the disorder-averaged gap following a rather slow power law, close to Δ∝N−1/3. This finding provides a promising outlook for the potential efficiency of quantum annealers for optimization problems with dense connectivity.