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arXiv:2501.13205 · Physical Review Letters

Ground States of the Mean-Field Spin Glass with 3-Spin Couplings

Stefan Boettcher, Ginger E. Lau

Published Jul 2, 2025• 1 citation

Abstract

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.

DOI: 10.1103/j159-lpfx

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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 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 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 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

{d}_{s}=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 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/r^{2 σ}

, for two values of

σ

,

p

and

0.85

. No de Almeida-Thouless line is expected at these

σ

values. The spin glass correlation length

ξ_{\text{SG}}

varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when

ξ_{\text{SG}}

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 (

\approx 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.

We have studied the diluted Heisenberg spin glass model in a three-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/r^{2σ}, 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 ξ_{SG} varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when ξ_{SG} becomes comparable to the system size L, there are departures which we attribute to the features deriving from the replica symmetry breaking 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. In this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.

We present an Ansatz for the ground states of the quantum Sherrington-Kirkpatrick model, a paradigmatic model for quantum spin glasses. Our Ansatz, based on the concept of generalized coherent states, very well captures the fundamental aspects of the model, including the ground state energy and the position of the spin glass phase transition. It further enables us to study some previously unexplored features, such as the nonvanishing longitudinal field regime and the entanglement structure of the ground states. We find that the ground state entanglement can be captured by a simple ensemble of weighted graph states with normally distributed phase gates, leading to a volume law entanglement, contrasting with predictions based on entanglement monogamy.