Ground States of the Sherrington-Kirkpatrick Spin Glass with Levy Bonds

TL;DR

This study explores the ground states of the Sherrington-Kirkpatrick spin glass model with Levy-distributed bonds, revealing universal behaviors for certain parameters and deviations near critical points, with implications for understanding disordered systems.

Contribution

It provides a comprehensive analysis of how Levy bond distributions affect the ground state energies and fluctuations in the SK model, extending understanding beyond Gaussian bonds.

Findings

Ground state energies attain SK Parisi energy for >2

Finite-size correction exponent ecays from SK value above =2

Energy fluctuation exponent ecays and vanishes at =1

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)\propto 1/|J|^{1+\alpha}, |J|>1, are investigated for a range of powers \alpha. The results are compared with those for the Sherrington-Kirkpatrick (SK) model, where bonds are Gaussian distributed. In particular, we determine the variation of the ground state energy densities with \alpha, their finite-size corrections, measure their fluctuations, and analyze the local field distribution. We find that the energies themselves at infinite system size attain universally the Parisi-energy of the SK as long as the second moment of P(J) exists (\alpha>2), and compare favorably with recent one-step replica symmetry breaking predictions well below \alpha=2. At and just below \alpha=2, the simulations deviate significantly from theoretical expectations. The finite-size investigation reveals that the corrections exponent \omega decays from the SK value \omega_{SK}=2/3 already well above \alpha=2, at which point it reaches a minimum. This result is justified with a speculative calculation of a random energy model with Levy bonds. The exponent \rho that describes the variations of the ground state energy fluctuations with system size decays monotonically from its SK value over the entire range of \alpha and apparently vanishes at \alpha=1.