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.
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.
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 propose and demonstrate numerically a fast classical annealing scheme for the Sherrington-Kirkpatrick (SK) spin glass model, employing the Suzuki-Kubo mean-field Ising dynamics (supplemented by a modified Thouless-Anderson-Palmer reaction field). The resultant dynamics, starting from any arbitrary paramagnetic phase (with local magnetizations m_{i}=±1, for the ith spin, and the global magnetization m=0), takes the system quickly to an appropriate state with small local values of magnetization (m_{i}) commensurate with the (frustrated) interactions. As the temperature decreases with the annealing, the configuration practically remains (in an effective adiabatic way) close to a low-energy configuration as the magnitudes of m_{i}'s and the spin glass order parameter q grow to unity. While the configuration reached by the procedure is not the ground state, for an N-spin SK model (with N up to 10 000), the deviation in the energy per spin E_{N}^{0}-E^{0} found by the annealing procedure scales as N^{-2/3}, with E^{0}=-0.7629±0.0002, suggesting that in the thermodynamic limit the energy per spin of the low-energy configurations converges to the ground state of the SK model (analytical estimate being E^{0}=-0.7631667265⋯), fluctuation σ_{N} in E_{N}^{0} decreases as ∼N^{-3/4}, and the annealing time τ_{N}∼N, making this protocol highly efficient in estimating the ground state energy of the SK model.
We present a collection of simulations of the Edwards–Anderson lattice spin glass at T=0 to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample the ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a variety of accurate exponents for domain wall stiffness and finite-size corrections, which allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattice Ising spin glasses.
We perform large scale quantum annealing of the Sherrington-Kirkpatrick (SK) spin glass up to a system size N=40000 to estimate its ground state energy using the deGennes-Suzuki-Kubo mean-field quantum Ising dynamics, extending the earlier results (reported in Eur. Phys. J. B {\bf 98}, 226 (2025)). Here we numerically solve the deGennes-Suzuki-Kubo annealing dynamics to obtain the spin configurations and subsequently the ground state energy for a given system size at the end of the annealing, starting from a quantum paramagnetic state. The method shows high efficiency, with an overall algorithmic cost of O(N3) in estimating the energy of the ground state. We later extend this quantum annealing study to estimate the ground state energies (starting again from the quantum paramagnetic phase, annealing down to any desired low value of the transverse field) for the Edwards-Anderson (EA) spin glass model on a square lattice.
We have studied the diluted Heisenberg spin glass model in a 3-component random field for the commonly-used one-dimensional long-range model where the probability that two spins separated by a distance r interact with one another falls as 1/r2σ, for two values of σ, 0.75 and 0.85. No de Almeida-Thouless line is expected at these σ values. The spin glass correlation length ξSG varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξSG becomes comparable to the system size L, there are departures which we attribute to the features deriving from the TNT picture of spin glasses. For the case σ=0.85 these features go away for system sizes with L>L∗, where L∗ is large (≈4000−8000 lattice spacings). In the case of σ=0.75 we have been unable to study large enough systems to determine its value of L∗. We sketch a renormalization group scenario to explain how these features could arise. On this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce TNT features and some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.