Freezing (Kauzmann) transition and low-temperature phase of the hierarchical random energy model and its p-spin generalization

The hierarchical random energy model (HREM) of Castellana, Decelle, Franz, Mézard and Parisi (2010) places Derrida's random energy model on a hierarchical lattice, so that the interaction energy between subsystems scales subextensively with volume, making it a non-mean-field spin glass. Perturbative (orders up to $m=10$) and numerical (recursion up to $k=12$) analyses gave evidence for a REM-like freezing transition at the temperature where the configurational entropy vanishes, supported by a one-step replica-symmetry-breaking ansatz and predicting a non-analytic high-temperature free-energy branch. Rigorous work in 2014 proved the thermodynamic limit, self-averaging, and upper bounds on the Kauzmann temperature, and introduced the hierarchical p-spin model, but left the central questions open. The problem is to settle them: prove or refute a finite-temperature entropy-crisis transition for $0<\sigma<1$, determine the low-temperature free energy and its replica structure, and establish the transition character of the hierarchical p-spin model.

Background

A spin glass is a disordered magnetic system in which randomly signed interactions prevent any spin configuration from satisfying all of them at once; its mean-field thermodynamics is computed with the replica method, in which the free energy averaged over the frozen disorder is obtained by introducing $n$ identical copies ('replicas') of the system and analytically continuing $n \to 0$, with the pattern of replica symmetry breaking (RSB) encoding the glassy phase. The random energy model (REM) of Derrida is the simplest spin glass: the $2^N$ spin configurations are assigned independent, identically distributed random energies. It undergoes a 'freezing' transition at a critical temperature $T_c$ at which the configurational entropy (the logarithm of the number of thermodynamically accessible states) vanishes — an entropy crisis of the kind Kauzmann hypothesized for supercooled liquids — below which the Gibbs probability condenses onto $O(1)$ configurations. In replica language this condensation is a one-step RSB (1RSB) solution: replicas cluster, with cluster-size fraction $x = T/T_c$ below $T_c$.

Whether this entropy-crisis scenario survives outside mean field is a decades-old question of glass theory. Castellana, Decelle, Franz, Mézard and Parisi introduced the simplest non-mean-field random-energy model, a hierarchical REM (HREM): a system of $N = 2^k$ Ising spins built recursively, starting from single spins with independent random energies, by merging two independent subsystems of $2^{k-1}$ spins and adding an independent random interaction between them whose variance is proportional to $2^{(k+1)(1-\sigma)}$ — a Dyson-type hierarchical lattice, in which the interaction strength decays with the hierarchical distance between spins. For $0 < \sigma < 1$ the variance of the interaction energy between the two halves of the system scales as $N^{1-\sigma}$, subextensively in the volume, so the model is non-mean-field ($\sigma \le 0$ recovers mean-field behavior; $\sigma > 1$ an asymptotically free system). A small-coupling replica expansion of the free energy in $g = 2^{1-\sigma}$ carried to order $m = 10$, together with an exact numerical recursion for the microcanonical entropy (entropy at fixed energy) up to hierarchy depth $k = 12$ — with coarse-grained approximations in the bulk of the spectrum and an assumed self-averaging entropy density — gave evidence that the configurational entropy vanishes at a finite temperature $T_c(\sigma)$ for every $0 < \sigma < 1$, that a 1RSB ansatz reproduces the REM-like transition at $T_c$, and that, at variance with mean field, the high-temperature branch of the free energy is non-analytic at the transition: $F(T) - F(T_c) \propto (T - T_c)^{(2-\sigma)/(1-\sigma)}$, corresponding to a specific-heat exponent $\alpha = -\sigma/(1-\sigma)$. The authors explicitly identified the replica structure of the low-temperature phase and the extension to $p$-body interactions ($p \ge 3$) as themes for future research.

Subsequent rigorous work (Castellana, J. Stat. Phys. 157, 219 (2014)) proved for the HREM the existence of the thermodynamic limit and self-averaging of the quenched free energy (its independence from the particular realization of the disorder as $N \to \infty$), and proved that the infinite-volume entropy is strictly positive above a threshold temperature proportional to the mean-field critical temperature; hence, if a Kauzmann transition exists, its temperature is bounded above by an explicit bound, later improved by exploiting the hierarchy. The same paper introduced the hierarchical p-spin model (HPS) — the hierarchical analogue of the mean-field $p$-spin glass, which for $p \ge 3$ also has an entropy-crisis transition — and proved its thermodynamic limit and self-averaging. What is still missing is a lower bound on the transition temperature, i.e. proof that the entropy actually vanishes at some finite temperature; the determination of the low-temperature phase and its replica structure; and the transition character of the HPS. Related activity on hierarchical spin glasses — renormalization-group computations (coarse-graining transformations relating the model at successive length scales), free-energy bounds, and Monte Carlo studies — has concerned the hierarchical Edwards-Anderson model (random hierarchical pair couplings rather than random energies) and has left the HREM questions untouched.

Problem Statement

Give a controlled (rigorous or exact-analytic) determination of the low-temperature thermodynamic behavior of the hierarchical random energy model of Castellana, Decelle, Franz, Mézard and Parisi (2010) — the HREM with $N = 2^k$ Ising spins, built by merging two half-systems with an independent random interaction of variance proportional to $2^{(k+1)(1-\sigma)}$, in the non-mean-field regime $0 < \sigma < 1$ — replacing its perturbative and numerical evidence by established results. Concretely:

1. Transition existence and character. Prove that the infinite-volume configurational entropy — the canonical entropy $s(\beta) = \lim_{k\to\infty} s_{k+1}(\beta)$, with $s_{k+1}(\beta) = -\beta\, df_{k+1}/d\beta + f_{k+1}$ and $f_k$ the finite-volume quenched free energy, as defined in Castellana, J. Stat. Phys. 157, 219 (2014), whose positivity above a threshold temperature yields the known upper bound on the Kauzmann temperature — vanishes at some finite inverse temperature, equivalently establish a lower bound on the transition temperature complementing the known upper bound, thereby proving a finite-temperature freezing (Kauzmann) transition; or prove that no such transition occurs for some or all of $0 < \sigma < 1$. Determine the character of the transition, in particular whether the high-temperature branch of the free energy is non-analytic at $T_c$ with $F(T) - F(T_c) \propto (T - T_c)^{(2-\sigma)/(1-\sigma)}$ as predicted.

2. Low-temperature phase. Determine the thermodynamic solution below the transition: the free energy for $T < T_c$ and its replica structure — whether the one-step replica-symmetry-breaking REM-like condensation with $x = T/T_c$ is the correct description of the glassy phase, or whether other replica-symmetry-breaking patterns or more complex low-temperature solutions arise in this non-mean-field regime.

3. p-body generalization. For the hierarchical p-spin model with $p \ge 3$ introduced in Castellana, J. Stat. Phys. 157, 219 (2014), in its non-mean-field regime $\sigma > 1/2$, determine whether it undergoes an entropy-crisis transition analogous to its mean-field counterpart and, if so, establish its transition character and critical behavior.

These components serve one objective: establishing whether, and how, the mean-field entropy-crisis (ideal-glass) scenario survives in non-mean-field random-energy models on hierarchical lattices — component 1 is the rigorous core of component 2, and component 3 tests the generality of the mechanism. The three components are posed jointly.

References

1. Michele Castellana, Aurélien Decelle, Silvio Franz, Marc Mézard, Giorgio Parisi, 'Hierarchical Random Energy Model of a Spin Glass', Physical Review Letters 104, 127206 (2010). DOI: 10.1103/PhysRevLett.104.127206; arXiv:0912.3634. https://doi.org/10.1103/PhysRevLett.104.127206
2. Michele Castellana, 'Rigorous Results for Hierarchical Models of Structural Glasses', Journal of Statistical Physics 157, 219-233 (2014). DOI: 10.1007/s10955-014-1085-9; arXiv:1403.5667. https://doi.org/10.1007/s10955-014-1085-9
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