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arXiv:2604.15433

The Phase Transitions in a ppp spin Glass Model: A Numerical Study

Prerak Gupta, Auditya Sharma, Bharadwaj Vedula, Joonhyun Yeo, M A Moore

Published Apr 16, 2026• 0 citations
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Abstract

We investigate the balanced $M=4$, $p=4$ spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range $p$-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for $σ= 0, 0.25$, and 0.55. For these values of the long-range exponent $σ$ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the $λ$-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of $λ\equiv ω_2/ω_1 < 1$ suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of $λ> 1$ is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent $σ= 0.85$, which roughly corresponds to a three dimensional system, we find that the renormalized value of $λ$ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature $T_K$ in three dimensions might be zero and the complete absence of phase transitions in structural glasses.

DOI
10.48550/arxiv.2604.15433

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Abstract We present results of numerical simulations on a one-dimensional Ising spin glass with long-range interactions. Parameters of the model are chosen such that it is a proxy for a short-range spin glass above the upper critical dimension (i.e. in the mean-field regime). The system is quenched to a temperature well below the transition temperature and the growth of correlations is observed. The spatial decay of the correlations at distances less than the dynamic correlation length agrees quantitatively with the predictions of a static theory, the ‘metastate’, evaluated according to the replica symmetry breaking (RSB) theory. We also compute the dynamic exponent z ( T ) defined by and find that it is compatible with the mean-field value of the critical dynamical exponent for short range spin glasses.
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