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arXiv:2610.05046

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

Alexander K. Hartmann, Leticia F. Cugliandolo, Marco Tarzia

Published Oct 4, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2610.05046

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Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal $(\ifmmode\pm\else\textpm\fi{}J)$ distributions of the disorder are calculated using a ``matching''algorithm, which allows large system sizes of up to N=4802{N=480}^{2}N=4802 spins to be investigated. We study domain walls induced by two rather different types of boundary condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy,'' which we denote by $\ensuremath{\Delta}E.$ For Gaussian disorder, we find a power-law behavior $\ensuremath{\Delta}E\ensuremath{\sim}{L}^{\ensuremath{\theta}},$ with $\ensuremath{\theta}=\ensuremath{-}0.266(2)$ and $\ensuremath{\theta}=\ensuremath{-}0.282(2)$ for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension dc.{d}_{c}.dc​. For the $\ifmmode\pm\else\textpm\fi{}J$ model, we obtain a different result, namely that the domain-wall energy saturates at a nonzero value for $\stackrel{\ensuremath{\rightarrow}}{L}\ensuremath{\infty},$ so $\ensuremath{\theta}=0,$ indicating that the lower critical dimension for the $\ifmmode\pm\else\textpm\fi{}J$ model is exactly dc=2.{d}_{c}=2.dc​=2.
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The statistics of the ground-state and domain-wall energies for the two-dimensional random-bond Ising model on square lattices with independent, identically distributed bonds having probability ppp of ${J}_{ij}=\ensuremath{-}1$ and $(1\ensuremath{-}p)$ of Jij=+1{J}_{ij}=+1Jij​=+1 are studied. We are able to consider large samples of up to 3202{320}^{2}3202 spins by using sophisticated matching algorithms. We study $L\ifmmode\times\else\texttimes\fi{}L$ systems, but we also consider $L\ifmmode\times\else\texttimes\fi{}M$ samples, for different aspect ratios R=L∕MR=L∕MR=L∕M. We find that the scaling behavior of the ground-state energy and its sample-to-sample fluctuations inside the spin-glass region $({p}_{c}\ensuremath{\leqslant}p\ensuremath{\leqslant}1\ensuremath{-}{p}_{c})$ are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusplike singularity at pc{p}_{c}pc​. Inside the spin-glass region, the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding RRR fixed. Here, large finite-size effects are visible, which can be explained for all ppp by a single exponent $\ensuremath{\omega}\ensuremath{\approx}2∕3$, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for $R\ensuremath{\rightarrow}0$: the distribution of the domain-wall energies converges to a Gaussian for $R\ensuremath{\rightarrow}0$, although the domain walls of neighboring subsystems of size $L\ifmmode\times\else\texttimes\fi{}L$ are not independent.
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