Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0311657

Ground-state energy and frustration of the Sherrington-Kirkpatrick model and related models

S. Kobe

Published Nov 28, 2003• 6 citations
arXivHTMLPDF

Abstract

Exact ground states are calculated for the Sherrington-Kirkpatrick (SK) spin-glass containing up to N=90 spins. A ground-state energy per spin $e^{\infty}_0 = - 0.7637 \pm 0.0004$ is found from the $N$ dependence of the misfit parameter, which is a measure of frustration of the system. The results are compared with those of two related models, which can be introduced by replacing all interactions of the SK model by ferromagnetic or antiferromagnetic ones of the same strengths. A parameter $x$ is introduced, which describes the fraction of antiferromagnetic interactions in these types of models. From the $x$ dependence of finite models it is concluded, that the the SK model $(x=0.5)$ assigns a transition between a ferromagnetic state $(x<0.5)$ to a spin-glass state $(x>0.5)$.

DOI
10.48550/arxiv.cond-mat/0311657

Related papers

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

Stefan Boettcher

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.

arXiv:1912.04974•Physical Review Letters•Apr 28, 2020•11 citations

Classical annealing of the Sherrington-Kirkpatrick spin glass using Suzuki-Kubo mean-field Ising dynamics

Soumyaditya Das, Soumyajyoti Biswas, Bikas K. Chakrabarti

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.

arXiv:2412.18358•Physical review. E•Jun 16, 2025•9 citations

Numerical simulations on spin glasses (invited)

A. P. Young

We discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.

Journal of Applied Physics•Apr 15, 1985•14 citations

Is the Sherrington–Kirkpatrick model relevant for real spin glasses?

A. P. Young

I describe the results of numerical simulations which test whether the Sherrington–Kirkpatrick model, which was solved by Parisi, applies to short-range spin glasses. I conclude that it probably does for dimension d greater than 6, but there appear to be some differences in lower dimensions. In particular, there does not appear to be the line of transitions in a magnetic field (AT line) which is seen in the SK model.

Journal of Physics A Mathematical and Theoretical•Jul 30, 2008•9 citations

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

Stefan Boettcher, Ginger E. Lau

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.

arXiv:2501.13205•Physical Review Letters•Jul 2, 2025•1 citation

Ground State of Quantum Spin Glass with Infinite Range Interactions

Hidetoshi Nishimori, Y‐h. Taguchi, Takehiko Oguchi

We have investigated the spin-1/2 random Heisenberg model with infinite range interactions by a numerical method. Extrapolation of finite size ( N ≦16) properties to the infinite system has yielded the following results. The average ground state energy is lower than that of the SK model (Ising spin) by about 25%. In the ground state, a critical point between the spin glass and ferromagnetic (mixed) phases exists just around the critical point (\(\tilde{J}_{0}/\tilde{J}{=}1\)) of the corresponding classical random Heisenberg model. Quantum fluctuations are concluded not to change the ground state properties qualitatively.

Journal of the Physical Society of Japan•Feb 15, 1986•5 citations
Exact ground states are calculated for the Sherrington-Kirkpatrick (SK) spin-glass containing up to N=90 spins. A ground-state energy per spin e0∞=−0.7637±0.0004e^{\infty}_0 = - 0.7637 \pm 0.0004e0∞​=−0.7637±0.0004 is found from the NNN dependence of the misfit parameter, which is a measure of frustration of the system. The results are compared with those of two related models, which can be introduced by replacing all interactions of the SK model by ferromagnetic or antiferromagnetic ones of the same strengths. A parameter xxx is introduced, which describes the fraction of antiferromagnetic interactions in these types of models. From the xxx dependence of finite models it is concluded, that the the SK model (x=0.5)(x=0.5)(x=0.5) assigns a transition between a ferromagnetic state (x<0.5)(x<0.5)(x<0.5) to a spin-glass state (x>0.5)(x>0.5)(x>0.5).
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 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 discuss the results of numerical simulations of two spin-glass models, the infinite range Sherrington–Kirkpatrick (SK) model, and a nearest–neighbor model in three dimensions. The probability distribution PN(q) for the spin glass order parameter q has been evaluated for the SK model below Tc with N, the number of spins, up to 192. It shows the same basic features as the Parisi function dx/dq as expected, but with sizeable finite size corrections. Results for the short-range model show evidence for a finite temperature transition but further work is needed to confirm this.
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 have investigated the spin-1/2 random Heisenberg model with infinite range interactions by a numerical method. Extrapolation of finite size ( N ≦16) properties to the infinite system has yielded the following results. The average ground state energy is lower than that of the SK model (Ising spin) by about 25%. In the ground state, a critical point between the spin glass and ferromagnetic (mixed) phases exists just around the critical point (J~0/J~=1\tilde{J}_{0}/\tilde{J}{=}1J~0​/J~=1) of the corresponding classical random Heisenberg model. Quantum fluctuations are concluded not to change the ground state properties qualitatively.