Asymptotic nature of the glass transition in the three-dimensional short-range p-spin glass: overlap distribution, critical behavior, and the rare-region mechanism

The three-dimensional short-range Ising p-spin glass (the M-p construction with p=4, M=3 spins per site on the cubic lattice) is a canonical lattice realization of the mean-field p-spin model that underlies random-first-order-transition theories of the glass transition. Simulations by Campellone, Coluzzi, and Parisi at linear sizes up to L=6 found a second-order-like transition at $T_c\simeq 2.6$ with a diverging spin-glass susceptibility, an overlap distribution $P(q)$ with continuous weight on $[-q_1,q_1]$ that resembles neither one-step replica symmetry breaking (1RSB) nor standard full replica symmetry breaking (FRSB), no evidence of a dynamical transition above $T_c$, and a conjectured nonperturbative rare-region ('bubble') mechanism. Whether this phenomenology survives the thermodynamic limit, what the critical exponents are, and whether the bubble mechanism quantitatively explains the data remain unresolved: later renormalization-group and fluctuation arguments by Moore, Drossel, and Yeo contend that finite-dimensional fluctuations destabilize 1RSB and may remove the transition entirely below six dimensions, and a 2026 large-scale study of a one-dimensional long-range proxy finds no transition at three-dimensional-like effective dimension, but with explicit finite-size caveats and not on the three-dimensional lattice model itself. The problem is to determine, numerically and/or analytically, the thermodynamic-limit nature of this transition and its mechanism.

Background

Spin glasses are disordered systems whose canonical lattice model places Ising spins on a lattice with random nearest-neighbor couplings; a central open issue is the nature of their glassy low-temperature phase. In the replica formalism the order parameter is the probability distribution $P(q)$ of the overlap $q$ between two independent equilibrium copies of the same disorder realization (for Ising spins, $q$ is the normalized sum of products of the two copies' spins). In 'full' replica symmetry breaking (FRSB) the low-temperature $P(q)$ has continuous weight between symmetric peaks at $\pm q_{EA}$, while in one-step replica symmetry breaking (1RSB) it is a sum of delta peaks (at $0$ and $\pm q_1$ in zero field), corresponding to a discontinuous jump of the order parameter. The mean-field (fully connected) $p$-spin glass, in which each random coupling links $p>2$ spins, is the prototypical 1RSB model: its glass transition at $T_c$ is discontinuous, it is preceded by a purely dynamical ergodicity-breaking temperature $T_d>T_c$ of the mode-coupling type (mode-coupling theory being the standard dynamical theory of the slowdown of supercooled liquids), and this structure underlies the random first-order transition (RFOT) theory of structural glasses. Whether this mean-field picture survives in finite dimensions is a core open question of glass physics. The source paper studies a standard lattice realization: the $M$-$p$ construction puts $M$ Ising spins on each site of a cubic lattice, and every product of $p=4$ spins drawn from the two sites of a nearest-neighbor pair is coupled by a quenched (fixed once per disorder realization) random coupling $J=\pm 1$ of equal probability. The mean-field $p$-spin solution is recovered for $M\to\infty$, and a loop expansion in $1/M$ (Campellone, Parisi, and Ranieri) predicts that in three dimensions the transition remains 1RSB-like, with no divergent static correlation length. Campellone, Coluzzi, and Parisi simulated the $M=3$, $p=4$ case for linear sizes $L\le 6$ (at most 648 spins) using parallel tempering (a Monte Carlo scheme that swaps configurations between temperatures to overcome energy barriers) and found the opposite of the perturbative prediction: an apparently second-order transition at $T_c\simeq 2.6$ with a diverging spin-glass susceptibility $\chi_{SG}=N\langle q^2\rangle$ ($N=3L^3$ the spin number); a $P(q)$ with continuous weight over the whole interval $[-q_1,q_1]$, unlike the two-delta-peak form of 1RSB and also unlike the standard FRSB shape, as shown by the dimensionless cumulant $g(T)=(3-\langle q^4\rangle/\langle q^2\rangle^2)/2$, which is non-monotone and largely negative, in disagreement with both scenarios; and no sign of a dynamical $T_d>T_c$, the energy after a quench relaxing as a single power law $e(t)\propto t^{-\alpha_e}$ with $\alpha_e=d_e/z(T)$, $d_e=d-1/\nu$, where $\nu$ is the correlation-length exponent and $z(T)$ the dynamical critical exponent. To reconcile these observations they conjectured a nonperturbative 'bubble' (rare-region) mechanism: because the disorder is not space-invariant, rare regions that locally freeze above $T_c$ proliferate as $T\to T_c^+$, so that $\chi_{SG}=\chi_B+\chi_R$ with $\chi_B\propto(T-T_c)^{-\gamma}$ and $\chi_R=O(1)$: a transition driven by the growing weight of rare frozen regions rather than by an instability of the mean-field saddle point, whose propagators remain massive. The authors themselves state that more extensive simulations are necessary, that roughly $2^{23}$ parallel-tempering steps would be needed to fully equilibrate $L=6$ down to the lowest temperature studied, that their susceptibility fits give $\nu\simeq 1$ while the companion correlation-length study gives $\nu\simeq 2/3$, and that the bubble interpretation deserves a more careful study. Later work introduced a competing possibility, the 'avoided' transition, in which the correlation length grows to a large but finite value as the temperature is lowered and no true transition exists; in that language, the central unresolved question is whether the three-dimensional short-range $p$-spin glass has a genuine thermodynamic glass transition at all, and if so of which replica-symmetry-breaking type.

Problem Statement

Determine the thermodynamic-limit ($L\to\infty$) nature of the glass transition in the three-dimensional short-range Ising $p$-spin glass — the $M$-$p$ construction with $p=4$, $M=3$ spins per site on the cubic lattice with quenched $J=\pm 1$ couplings — the model simulated by Campellone, Coluzzi, and Parisi at $L\le 6$. All requested results characterize one object, the putative transition at $T_c\simeq 2.6$: (i) Existence and symmetry-breaking structure: does a genuine finite-temperature thermodynamic transition exist, and if so what is the equilibrium overlap distribution $P(q)$ in the thermodynamic limit — does the continuous weight over the interval $[-q_1,q_1]$ between the large-$|q|$ peaks of $P(q)$ survive (an FRSB-like transition), collapse onto delta peaks (a 1RSB transition, as the $1/M$ expansion predicts), or does the apparent transition dissolve into an avoided crossover with no diverging spin-glass susceptibility $\chi_{SG}$? (ii) Critical behavior, if the transition exists: the critical temperature, the correlation-length exponent $\nu$ (resolving the discrepancy between $\nu\simeq 2/3$ and $\nu\simeq 1$), the anomalous-dimension exponent $\eta$, and the temperature-dependent dynamical exponent $z(T)$, with their finite-size corrections controlled. (iii) Mechanism: establish, numerically and/or analytically, whether the proposed nonperturbative rare-region ('bubble') mechanism — the decomposition $\chi_{SG}=\chi_B+\chi_R$ with $\chi_B\propto(T-T_c)^{-\gamma}$, $\chi_R=O(1)$, driven by rare locally frozen regions whose weight grows as $T\to T_c^+$ — quantitatively accounts for the observed $P(q)$ shape, the non-monotone cumulant $g(T)$, and the divergence of $\chi_{SG}$; or refute it in favor of another mechanism (for example a fluctuation-driven instability of the 1RSB state). (iv) Dynamics of the same transition: confirm or refute, over time windows long enough to exclude slow-crossover artifacts, whether the dynamical singularity coincides with the static one ($T_d=T_c$, single-power-law energy relaxation) as the original finite-size data suggested, and characterize the associated aging (the dependence of relaxation on the time since preparation) and generalized fluctuation-dissipation behavior.

References

1. M. Campellone, B. Coluzzi, G. Parisi, 'Numerical study of a short-range p-spin glass model in three dimensions', Physical Review B 58, 12081-12089 (1998). DOI: 10.1103/PhysRevB.58.12081; preprint arXiv:cond-mat/9804291, https://arxiv.org/abs/cond-mat/9804291
2. S. Franz, G. Parisi, 'Critical properties of a three-dimensional p-spin model', The European Physical Journal B 8, 417-422 (1999). DOI: 10.1007/s100510050707; preprint arXiv:cond-mat/9805088, https://arxiv.org/abs/cond-mat/9805088
3. M. Campellone, G. Parisi, P. Ranieri, 'Finite-dimensional corrections to the mean field in a short-range p-spin glassy model', Physical Review B 59, 1036-1045 (1999). DOI: 10.1103/PhysRevB.59.1036; preprint arXiv:cond-mat/9806235, https://arxiv.org/abs/cond-mat/9806235
4. M. Campellone, F. Ritort, 'Complexity and line of critical points in a short-range spin-glass model', Physica A: Statistical Mechanics and its Applications 286, 1-9 (2000). DOI: 10.1016/S0378-4371(00)00060-1; preprint arXiv:cond-mat/9907465, https://arxiv.org/abs/cond-mat/9907465
5. G. Parisi, M. Picco, F. Ritort, 'Continuous phase transition in a spin-glass model without time-reversal symmetry', Physical Review E 60, 58-68 (1999). DOI: 10.1103/PhysRevE.60.58, https://doi.org/10.1103/PhysRevE.60.58
6. M. A. Moore, B. Drossel, 'p-Spin Model in Finite Dimensions and Its Relation to Structural Glasses', Physical Review Letters 89, 217202 (2002). DOI: 10.1103/PhysRevLett.89.217202; preprint arXiv:cond-mat/0201107, https://arxiv.org/abs/cond-mat/0201107
7. M. A. Moore, J. Yeo, 'Thermodynamic Glass Transition in Finite Dimensions', Physical Review Letters 96, 095701 (2006). DOI: 10.1103/PhysRevLett.96.095701; preprint arXiv:cond-mat/0510375, https://arxiv.org/abs/cond-mat/0510375
8. M. Campellone, G. Parisi, M. A. Virasoro, 'Replica Method and Finite Volume Corrections', Journal of Statistical Physics 138, 29-39 (2009). DOI: 10.1007/s10955-009-9891-1, https://doi.org/10.1007/s10955-009-9891-1
9. J. Yeo, M. A. Moore, 'Renormalization group analysis of the M-p-spin glass model with p=3 and M=3', Physical Review B 85, 100405(R) (2012). DOI: 10.1103/PhysRevB.85.100405; preprint arXiv:1111.3105, https://arxiv.org/abs/1111.3105
10. J. Yeo, M. A. Moore, 'Possible instability of one-step replica symmetry breaking in p-spin Ising models outside mean-field theory', Physical Review E 101, 032127 (2020). DOI: 10.1103/PhysRevE.101.032127; preprint arXiv:1911.02719, https://arxiv.org/abs/1911.02719
11. J. Yeo, M. A. Moore, 'Replica symmetry broken states of some glass models', Physical Review E 108, 054134 (2023). DOI: 10.1103/PhysRevE.108.054134; preprint arXiv:2308.14229, https://arxiv.org/abs/2308.14229
12. P. Gupta, A. Sharma, B. Vedula, J. Yeo, M. A. Moore, 'The Phase Transitions in a p spin Glass Model: A Numerical Study', preprint arXiv:2604.15433 (2026), https://arxiv.org/abs/2604.15433