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Journal of Physics C Solid State Physics

On the ground state of the infinite-range Ising spin glass

Scott Kirkpatrick, A. P. Young

Published Nov 20, 1982• 2 citations
Publisher

Abstract

The authors investigate by direct enumeration of all states for small systems the projection of the ground state of the infinite-range Ising spin glass onto the eigenvectors of the interaction matrix J ij . They find that the largest projection (which is usually for the eigenvector corresponding to the largest eigenvalue) varies as N -0.19 , where N is the number of spins. This is close to Sompolinsky's recent prediction (1981) of N -1 6/.

DOI
10.1088/0022-3719/15/32/009

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The authors investigate by direct enumeration of all states for small systems the projection of the ground state of the infinite-range Ising spin glass onto the eigenvectors of the interaction matrix J ij . They find that the largest projection (which is usually for the eigenvector corresponding to the largest eigenvalue) varies as N -0.19 , where N is the number of spins. This is close to Sompolinsky's recent prediction (1981) of N -1 6/.
The low-temperature behavior of the infinite-range Ising spin-glass model with a Gaussian distribution of interactions is studied by evaluating the statistical mechanics exactly for small systems (up to N=20N=20N=20 spins). Our results can be well understood by a simple picture of elementary cluster excitations from the ground state which overturn, on average, of order N12{N}^{\frac{1}{2}}N21​ spins. The conventional definition of the spin-glass order parameter is shown to be unsatisfactory and a new definition is proposed. It is speculated that ensemble averages may give different results from time averages for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ even in the infinite time limit.
We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field ΓΓΓ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of ΓΓΓ and for system sizes N≤16N \le 16N≤16. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of ΓΓΓ. The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at Γc≃1.5Γ_c \simeq 1.5Γc​≃1.5. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get z=2.1±0.1z=2.1 \pm 0.1z=2.1±0.1 which is in contradiction with a recent estimate (z=4z=4z=4). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for Γ<ΓcΓ< Γ_cΓ<Γc​.
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.
Mean-field models of two-spin Ising spin glasses with interaction matrices taken from ensembles that are invariant under O(N) transformations are studied. A general study shows that the nature of the spin glass transition can be deduced from the eigenvalue spectrum of the interaction matrix. A simple replica approach is derived to carry out the average over the O(N) disorder. The analytic results are confirmed by the extensive Monte Carlo simulations for large system sizes and by the exact enumeration for small system sizes.
The Edwards and Anderson model for spin glasses is investigated by a decimation rescaling transformation applied to a spin- 1 / 2 Ising model. The technique reproduces the exactly known results for a linear chain and, in particular, predicts ordering to a spin-glass type ground state at T=0. For a two-dimensional square lattice it is shown that approximations which give qualitatively correct results in other situations predict no spin-glass transition at any temperature. A comparison is made with results for spin glasses on a Bethe lattice and with an infinite-range spin-glass model, which has been found to exhibit a transition. It is argued that the difference between infinite-range and short-range models is that the former has entirely extended eigenvectors of the random J ij matrix whereas the latter has at least some localized eigenvectors.
In the droplet picture of the scaling theory for the short-range Ising spin glass, the largest eigenvalue of the correlation matrix diverges algebraically with the Edwards-Anderson susceptibility, the exponent being d/(d-\ensuremath{\theta}), where \ensuremath{\theta} is the stiffness exponent. The spectral density for the correlation matrix has a bump near the lower edge of the spectrum. Comparing these predictions with results from numerical studies of the spectrum might provide a unique test of the droplet picture.