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

Using the Extremal Optimization heuristic (EO),\footnote{S. Boettcher \& A.G. Percus, {\it PRL} {\bf 86}, 5211 (2001)} ground states of the SK-spin glass are studied with bonds $J$ distributed according to a Levy distribution $P(J)\propto1/|J|^{1+\alpha}$ with $|J|>1$ and $1<\alpha<4.$ The variation of the energy densities with $\alpha$, their finite-size corrections, their fluctuations, and their local field distribution are analyzed and compared with those for the SK model with Gaussian bonds.\footnote{S. Boettcher, {\it Philosophical Magazine} {\bf 92}, 34 (2012)} We find that the energies attain universally the Parisi-energy of the SK when the second moment of $P\left(J\right)$ exists ($\alpha>2$). They compare favorably with recent one-step replica symmetry breaking predictions well below $\alpha=2$. Near $\alpha=2$, the simulations deviate significantly from theoretical expectations. The finite-size corrections exponent $\omega$ decays from the putative SK value $\omega_{SK}=\frac{2}{3}$ already well above $\alpha=2$. The exponent $\rho$ for the scaling of ground state energy fluctuations with system size decays linearly from its SK value for decreasing $\alpha$ and vanishes at $\alpha=1$.