Ground-state properties of the two-dimensional SWAP spin-glass ensemble

Abstract

We study the recently introduced SWAP ensemble for Ising spin-glasses, where the spins obtain varying lengths, with length scale $Δ\in [0, 2]$. The lengths can be exchanged in the spirit of the SWAP algorithm for glassy poly-disperse hard-sphere systems. Using an annealing schedule, if the annealing is slow enough, ground states of the corresponding Ising Hamiltonian, where the spin lengths are incorporated into the bonds, can be obtained with high probability. We prove this statement for two-dimensional systems with sizes of up to $L^2=1024^2$ spins by comparison with exact ground states obtained by using graph matching algorithms. In particular, we analyze the nature of the obtained realizations of the disorder by calculating exact zero-temperature domain-wall energies, from exact ground state calculations using periodic and anti-periodic boundary conditions. For medium or large values of $Δ$, e.g. $Δ=1$, the realizations turn ferromagnetic if the annealing is slow enough, i.e., they lose their zero-temperature spin-glass property. If $Δ$ is small, the realizations remain glassy, but the SWAP annealing does not easily find a true ground state.