Ground-state energy and energy landscape of the Sherrington-Kirkpatrick spin glass

Abstract

We apply the conformational space annealing to the study of the energy landscape of the $\ifmmode\pm\else\textpm\fi{}J$ Sherrington-Kirkpatrick (SK) model. We have investigated the model up to the system size of $N=2047$ which is the largest size ever investigated. We find that the average ground-state energy and the standard deviations depend on $N$ as ${⟨{e}_{0}⟩}_{N}={⟨{e}_{0}⟩}_{\ensuremath{\infty}}+a{N}^{\ensuremath{-}\ensuremath{\omega}}$ with $\ensuremath{\omega}=0.672(4)$ $[{⟨{e}_{0}⟩}_{\ensuremath{\infty}}=\ensuremath{-}0.76321(3)]$ and $\ensuremath{\sigma}({e}_{0})=b{N}^{\ensuremath{-}\ensuremath{\rho}}$ with $\ensuremath{\rho}=0.746(14)$. We have also investigated the characteristics of the energy landscape for a given SK realization by plotting low-energy local minimum states in terms of their energy values versus the overlap of them to the ground state. We observe that the flat-bottom (plateau) region in the floor of the energy landscape for the lowest global minimum energy realization grows as the system size increases, indicating that, in the thermodynamic limit, the energy landscape defined here becomes flat for the SK model, as expected in the presence of continuous replica symmetry breaking. We also obtain the histogram of spin overlaps between local minimum configurations. We observe that there exists a peak at nonzero overlap value for realizations with low ground-state energy while the peak is located at zero overlap for realizations with high ground-state energy.