Analytic theory of the quenched overlap distribution and inherent-structure energy of the Sherrington-Kirkpatrick spin glass (statistics of the minima reached by zero-temperature quenching of equilibrium configurations)

Coluzzi, Marinari, Parisi and Rieger (J. Phys. A 33, 3851, 2000) measured, for the Sherrington-Kirkpatrick (SK) mean-field spin glass, the statistics of the energy minima (inherent structures, IS) reached by quenching equilibrium configurations at temperature $T$ to zero temperature with single-spin-flip dynamics, weighting each minimum by the Boltzmann factor of its basin. They found a trivially shaped but anomalously slowly converging quenched overlap distribution $P_{\mathrm{quen}}(q,T)$ in the paramagnetic phase ('weak' replica symmetry breaking, subexponential decay with exponent $\nu = 0.68 \pm 0.05$), the structure of full replica symmetry breaking below $T_C$, a nearly temperature-independent IS energy $e_{\mathrm{quen}} \approx -0.7145$ for $T > T_C$ with strong finite-size corrections, and a continuous descent toward the ground-state value $-0.7633$ as $T \to 0$. Their study is purely numerical: the published conclusions summarize the measured results, while caveats and suggestions in the body flag the 'nearest inherent structure' question, the possible dependence of the threshold energy on the chosen zero-temperature dynamics, and 'very important' finite-size corrections; no analytic theory of $P_{\mathrm{quen}}(q,T)$ or of the IS energy is given. Later work has refined the numerics, modeled the zero-temperature dynamics by Fokker-Planck and dynamical mean-field methods, and connected quench energies to marginal stability and a critical energy of the metastable-state ensemble, but no analytic account of $P_{\mathrm{quen}}(q,T)$, of $e_{\mathrm{quen}}(T)$ (especially below $T_C$), of the origin and universality of $\nu$, or of the protocol dependence of the reached minima exists. The problem is to construct such a theory.

Background

The Sherrington-Kirkpatrick (SK) model is the mean-field spin glass: $N$ Ising spins $\sigma_i = \pm 1$ coupled all-to-all through independent Gaussian couplings $J_{ij}$ of variance $1/N$, with Hamiltonian $H = -\sum_{i<j} J_{ij} \sigma_i \sigma_j$ at zero field. Its equilibrium is exactly described by Parisi's full replica symmetry breaking (FRSB) solution: above the critical temperature $T_C = 1$ the system is paramagnetic, while below $T_C$ spin-glass order is characterized not by a single number but by the probability distribution $P(q)$ of the overlap $q = N^{-1} \sum_i \sigma_i^{(1)} \sigma_i^{(2)}$ between two independent equilibrium replicas with the same couplings, which equals $\delta(q)$ above $T_C$ and develops the characteristic FRSB shape below $T_C$.

In the physics of supercooled liquids, inherent structures (IS) are the local minima of the potential energy landscape, each surrounded by its attraction basin; quenching a configuration to zero temperature maps it onto one such minimum, and basin-weighted statistics of these minima underlie the configurational-entropy description of the glass transition (an entropy that counts metastable basins rather than microscopic configurations). For glass formers the overlap between minima is in fact a more sharply defined observable than the overlap between configurations, and it can be measured without full equilibration at low temperature, which motivated transplanting the construction to a spin glass. Coluzzi, Marinari, Parisi and Rieger (2000) equilibrated SK samples of $N = 64$ to $1024$ spins by parallel tempering (a Monte Carlo technique in which replicas of the system at different temperatures are simulated simultaneously and neighboring replicas are swapped, which greatly accelerates equilibration in the cold phase), then quenched each stored equilibrium configuration to $T = 0$ by single-spin-flip zero-temperature dynamics (an 'intermediate' rule: a randomly chosen spin is proposed and flipped whenever the flip lowers the energy, and the quench stops only after at least $20N$ consecutive rejected proposals, so the final configurations are stable against any single spin flip). Other standard zero-temperature protocols are the greedy rule, which flips at each step the spin whose flip decreases the energy the most, the reluctant rule, which flips the spin whose flip decreases it the least, and the sequential rule, which flips a randomly chosen spin among those whose flip would lower the energy. Weighting each reached minimum by the Boltzmann factor of its basin at the starting temperature $T$, they measured the quenched overlap distribution $P_{\mathrm{quen}}(q,T)$ (overlaps between minima reached from two independent equilibrium replicas) and the inherent-structure energy $e_{\mathrm{quen}}(T)$.

Their numerical findings define the problem. (i) For $T > T_C$, $P_{\mathrm{quen}}(q,T)$ is trivially shaped but approaches its $\delta(q)$ limit anomalously slowly: introducing $f(q) = -\lim_{N \to \infty} N^{-1} \ln P_{N,\mathrm{quen}}(q)$, replica symmetry is broken only in a 'weak' sense, $f(q) = 0$ on an interval (possibly all of $[0,1]$), with $P_{N,\mathrm{quen}}(q)$ decaying slower than exponentially, governed by an exponent $\nu = 0.68 \pm 0.05$ extracted from the weak $N$-dependence of $r_N(q,T) = N^{-\nu} \ln \int_q^1 P_{N,\mathrm{quen}}(q',T)\,dq'$ (the source's Eq. (6): $\nu$ is fixed by requiring $r_N$ to be nearly independent of $N$ across system sizes); this refines the $\nu \simeq 2/3$ found by Parisi for quenches from $T = \infty$ (random initial conditions). (ii) For $T < T_C$, $P_{\mathrm{quen}}(q,T)$ shows FRSB-like structure, qualitatively mirroring the equilibrium $P(q)$. (iii) The IS energy is nearly temperature-independent above $T_C$, $e_{\mathrm{quen}} \to e_{\mathrm{th}} = -0.7145 \pm 0.004$ (compatible with the 'threshold' energy of the metastable-state ensemble, i.e., the energy level near which the marginally stable states separating basins concentrate in mean-field landscapes), with anomalously slow finite-size corrections $e_N = e_{\mathrm{th}} + C/N^{\alpha}$, $\alpha \approx 0.34$--$0.43$; below $T_C$ it decreases continuously toward the ground-state energy $e_0 = -0.7633$ (known to high precision from the exact Parisi solution) as $T \to 0$. (iv) Quenched minima remain correlated with their starting configurations (mean configuration-minimum overlap $q_t(T) \to 1$ as $T \to 0$, $q_t(T_C) \sim 0.4$), and the zero-temperature dynamics used may not reach the 'nearest' inherent structure that a steepest-descent quench would select, so the result depends on the quench protocol.

The problem sits between out-of-equilibrium dynamics and energy-landscape statistics. For one-step-RSB (1RSB) $p$-spin mean-field models (mean-field glass models whose metastable states are organized in a single step of replica symmetry breaking, so that their stationary points can be enumerated analytically) the analogous IS picture is analytically controlled (threshold energy, complexity, basin statistics; reviewed by Crisanti and Ritort), and the ensemble of all one-spin-flip-stable SK states has long had classical calculations of its complexity (the entropy-like density counting metastable states by energy), exhibiting a critical energy separating trivial-overlap from RSB-overlap states. What has no theory is the FRSB SK case with dynamics: which minima a given local-descent protocol actually selects from thermal states, and with what basin weights. The existing quantitative characterization of quench-reached minima in the SK model is purely numerical, from random and from thermal initial conditions.

Problem Statement

Construct a complete theoretical description - analytic, or a controlled asymptotic theory - of the statistics of the inherent structures reached by quenching equilibrium configurations of the zero-field Sherrington-Kirkpatrick model at temperature $T$ to zero temperature. The theory must predict, in the thermodynamic limit including leading finite-size corrections, for a precisely specified zero-temperature single-spin-flip dynamics and basin weighting:

1. The inherent-structure energy $e_{\mathrm{quen}}(T)$ for all $T > 0$: the plateau value above $T_C$ (numerically $e_{\mathrm{th}} = -0.7145 \pm 0.004$ for the intermediate single-spin-flip protocol, claimed to coincide with the threshold energy of the metastable-state ensemble), the crossover at $T_C = 1$, the interpolation toward the ground-state energy $e_0 = -0.7633$ (known from the exact Parisi solution) as $T \to 0$, and the anomalously slow finite-size corrections ($N^{-\alpha}$ with $\alpha \approx 0.34$--$0.43$, or $1/\ln N$).

2. The quenched overlap distribution $P_{\mathrm{quen}}(q,T)$: its weak-replica-symmetry-broken structure in the paramagnetic phase - establish or refute that $f(q) = -\lim_{N \to \infty} N^{-1} \ln P_{N,\mathrm{quen}}(q)$ vanishes on an interval, and derive (or refute the universality of) the exponent $\nu$ governing the subexponential approach to $\delta(q)$ ($\nu = 0.68 \pm 0.05$ for thermal starts, $\nu \simeq 2/3$ for $T = \infty$ starts) - together with its FRSB-like form for $T < T_C$ and its relation to the equilibrium Parisi overlap distribution.

3. The dependence of these statistics on the quench protocol (greedy, reluctant, sequential, or steepest descent) and on the starting temperature, including the 'nearest inherent structure' question (whether the quench reaches the minimum that steepest descent from the same configuration would select), and the status of the suggestion of Erba et al. (2024) that the reluctant single-spin-flip dynamics converges to the ground-state energy density in the thermodynamic limit (their numerics are, in their words, 'compatible' with this possibility and they leave its theoretical status explicitly open).

These three components serve one overarching objective: to replace the existing purely numerical characterization with a predictive theory of which minima a local descent selects from thermal states of the SK landscape. The theory is to be given for a definite single-spin-flip protocol and basin weighting, and must be consistent with the known anchors: the ground-state energy, $T_C = 1$, the trivial limit $P_{\mathrm{quen}} \to \delta(q)$ above $T_C$, and the $T = \infty$ results. A theory covering items 1-3 jointly resolves the overarching objective; independently, refuting one of the following specific load-bearing empirical claims (with (a)-(c) from the source and (d) from Erba et al. (2024)) settles that claim alone and leaves the others standing: (a) that $f(q) = -\lim_{N \to \infty} N^{-1} \ln P_{N,\mathrm{quen}}(q)$ vanishes on an interval above $T_C$, i.e. that $P_{\mathrm{quen}}(q)$ does not decay exponentially in $N$ (component 2); (b) that the approach to $\delta(q)$ above $T_C$ is governed by a single universal exponent $\nu = 0.68 \pm 0.05$ (component 2); (c) that $e_{\mathrm{quen}}(T)$ is temperature-independent above $T_C$ and plateaus at the metastable-state threshold energy $e_{\mathrm{th}} = -0.7145 \pm 0.004$ (component 1); (d) that the reluctant single-spin-flip dynamics converges to the ground-state energy density in the thermodynamic limit (component 3). The other enumerated findings of the source (for example the correlation remark $q_t(T_C) \sim 0.4$ of finding (iv)) are context and are not by themselves resolution-bearing sub-claims.

References

1. Barbara Coluzzi, Enzo Marinari, Giorgio Parisi, Heiko Rieger, 'On the energy minima of the Sherrington-Kirkpatrick model', J. Phys. A: Math. Gen. 33 (2000) 3851-3862, DOI 10.1088/0305-4470/33/21/301, https://doi.org/10.1088/0305-4470/33/21/301; author preprint arXiv:cond-mat/0003287 (preprint titled 'On the Energy Minima of the SK Model'), https://arxiv.org/abs/cond-mat/0003287
2. Giorgio Parisi, 'On the statistical properties of the large time zero temperature dynamics of the SK model', Fractals 11 (suppl. 1) (2003) 161-171, DOI 10.1142/S0218348X03001823, https://doi.org/10.1142/S0218348X03001823; author preprint arXiv:cond-mat/9501045, https://arxiv.org/abs/cond-mat/9501045
3. Vittorio Erba, Freya Behrens, Florent Krzakala, Lenka Zdeborova, 'Quenches in the Sherrington-Kirkpatrick model', J. Stat. Mech. (2024) 083302, DOI 10.1088/1742-5468/ad685a, https://doi.org/10.1088/1742-5468/ad685a; arXiv:2405.04267, https://arxiv.org/abs/2405.04267
4. Auditya Sharma, Joonhyun Yeo, M. A. Moore, 'Self-organized critical behavior and marginality in Ising spin glasses', J. Stat. Mech. (2018) 053302, DOI 10.1088/1742-5468/aabc85, https://doi.org/10.1088/1742-5468/aabc85; arXiv:1704.08700, https://arxiv.org/abs/1704.08700
5. Heinz Horner, 'Time dependent local field distribution and metastable states in the SK-spin-glass', Eur. Phys. J. B 60 (2007) 413-422, DOI 10.1140/epjb/e2008-00017-1, https://doi.org/10.1140/epjb/e2008-00017-1; arXiv:0707.2714, https://arxiv.org/abs/0707.2714
6. P. R. Eastham, R. A. Blythe, A. J. Bray, M. A. Moore, 'Mechanism for the failure of the Edwards hypothesis in the Sherrington-Kirkpatrick spin glass', Phys. Rev. B 74 (2006) 020406(R), DOI 10.1103/PhysRevB.74.020406, https://doi.org/10.1103/PhysRevB.74.020406; author preprint arXiv:cond-mat/0601402 (preprint titled '... in the SK spin glass'), https://arxiv.org/abs/cond-mat/0601402
7. L. Bussolari, P. Contucci, M. Degli Esposti, C. Giardina, 'Energy-decreasing dynamics in mean-field spin models', J. Phys. A: Math. Gen. 36 (2003) 2413-2421, DOI 10.1088/0305-4470/36/10/303, https://doi.org/10.1088/0305-4470/36/10/303; arXiv:cond-mat/0210545, https://arxiv.org/abs/cond-mat/0210545
8. Seung-Yeon Kim, S. J. Lee, Jooyoung Lee, 'Ground-state energy and energy landscape of the Sherrington-Kirkpatrick spin glass', Phys. Rev. B 76 (2007) 184412, DOI 10.1103/PhysRevB.76.184412, https://doi.org/10.1103/PhysRevB.76.184412
9. Stefan Schnabel, Wolfhard Janke, 'Distribution of metastable states of Ising spin glasses', Phys. Rev. B 97 (2018) 174204, DOI 10.1103/PhysRevB.97.174204, https://doi.org/10.1103/PhysRevB.97.174204; arXiv:1802.04514, https://arxiv.org/abs/1802.04514
10. M. Baity-Jesi, G. Parisi, 'Inherent structures in m-component spin glasses', Phys. Rev. B 91 (2015) 134203, DOI 10.1103/PhysRevB.91.134203, https://doi.org/10.1103/PhysRevB.91.134203; arXiv:1410.2163, https://arxiv.org/abs/1410.2163
11. Marco Baity-Jesi, Victor Martin-Mayor, 'Precursors of the spin glass transition in three dimensions', J. Stat. Mech. (2019) 084016, DOI 10.1088/1742-5468/ab333c, https://doi.org/10.1088/1742-5468/ab333c; arXiv:1901.05581, https://arxiv.org/abs/1901.05581
12. C. M. Newman, D. L. Stein, 'Metastable states in spin glasses and disordered ferromagnets', Phys. Rev. E 60 (1999) 5244-5260, DOI 10.1103/PhysRevE.60.5244, https://doi.org/10.1103/PhysRevE.60.5244; arXiv:cond-mat/9908455, https://arxiv.org/abs/cond-mat/9908455
13. A. Crisanti, F. Ritort, 'Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence', J. Phys. A: Math. Gen. 36 (2003) R181-R290, DOI 10.1088/0305-4470/36/21/201, https://doi.org/10.1088/0305-4470/36/21/201; arXiv:cond-mat/0212490, https://arxiv.org/abs/cond-mat/0212490
14. T. Aspelmeier, M. A. Moore, 'Free-energy barriers in the Sherrington-Kirkpatrick model', Phys. Rev. E 105 (2022) 034138, DOI 10.1103/PhysRevE.105.034138, https://doi.org/10.1103/PhysRevE.105.034138; arXiv:2111.06753, https://arxiv.org/abs/2111.06753
15. Z. Burda, A. Krzywicki, O. C. Martin, 'Network of inherent structures in spin glasses: scaling and scale-free distributions', Phys. Rev. E 76 (2007) 051107, DOI 10.1103/PhysRevE.76.051107, https://doi.org/10.1103/PhysRevE.76.051107; arXiv:0707.1965, https://arxiv.org/abs/0707.1965
16. Michele Castellana, Andrea Decelle, Elia Zarinelli, 'Extreme value statistics distributions in spin glasses', Phys. Rev. Lett. 107 (2011) 275701, DOI 10.1103/PhysRevLett.107.275701, https://doi.org/10.1103/PhysRevLett.107.275701; arXiv:1107.1795, https://arxiv.org/abs/1107.1795
17. Elia Zarinelli, 'Spin-glass models and interdisciplinary applications', doctoral dissertation, Universite Paris-Sud 11 (2012), DOI 10.70675/6419cc05zcfd4z4410z9c69z007f5b64217d, https://theses.fr/2012PA112003
18. Auditya Sharma, Joonhyun Yeo, M. A. Moore, 'The metastable minima of the Heisenberg spin glass in a random magnetic field', Phys. Rev. E 94 (2016) 052143, DOI 10.1103/PhysRevE.94.052143, https://doi.org/10.1103/PhysRevE.94.052143; arXiv:1607.04724, https://arxiv.org/abs/1607.04724