Controlled index-resolved stationary-point and Hessian-spectrum analysis of the glass transition in structural glasses

The Thouless-Anderson-Palmer (TAP) free energy of a mean-field spin glass is a functional of the local magnetizations whose stationary points, classified by their index (the number of unstable directions of the Hessian), are the candidate metastable states of the system. In 1998 Cavagna, Giardina and Parisi computed, for the p-spin spherical model, the index-resolved complexity, i.e. the exponential rate $\Sigma_k(E)$ of the number of stationary points with $k$ negative Hessian eigenvalues at energy density $E$, showing that below the threshold energy the minima exponentially dominate all higher-index saddles; this retroactively justified the standard practice of dropping the modulus of the Hessian determinant in complexity computations and verified the consistency of the construction with the Morse theorem. The paper acknowledged as unresolved the fully controlled finite-replica-index computation of the determinant-sign prefactors, the extension of the same index-resolved control to other mean-field spin glasses such as the Sherrington-Kirkpatrick (SK) model, and, as its concluding sentence, the wish to describe the glass transition of structural glasses analogously, 'simply in terms of the evolution of the Hessian spectrum'. The two spin-glass directions have since been settled at the level of theoretical physics, through supersymmetric (Becchi-Rouet-Stora-Tyutin, BRST) and cavity computations of the SK TAP complexity, the discovery that SK TAP states are marginal with the leading complexities of minima and index-one saddles coinciding so that the Morse theorem is satisfied, and a 2024 rigorous proof of the Bray-Moore determinant formula together with its outlier-eigenvalue prefactor, while a fully rigorous complexity theory is currently under construction. What remains open is the structural-glass direction: no structural-glass model admits a controlled, index-resolved analysis of the stationary points of an effective free-energy landscape with determinant-sign bookkeeping, and finite-dimensional landscape statistics are known to escape mean-field theory. The open problem is the construction and solution of such an analysis for a structural glass.

Background

In a mean-field spin glass the low-temperature phase contains exponentially many metastable states. The Thouless-Anderson-Palmer (TAP) approach writes mean-field self-consistency equations for the local magnetizations $m_i$; their solutions are the stationary points of a free-energy functional $F_{\mathrm{TAP}}(m)$, so counting metastable states means counting stationary points. The exponential growth rate of that number at energy density $E$, $\mathcal{N}(E)\sim e^{N\Sigma(E)}$ where $N$ is the number of spins, is called the complexity. For the Sherrington-Kirkpatrick (SK) model, i.e. Ising spins with all-to-all random Gaussian couplings, Bray and Moore computed the TAP complexity in 1980 by a replica calculation (a technique averaging over $n$ copies of the system and then taking $n\to 0$); for the p-spin spherical model, i.e. continuous spins on a sphere coupled by random $p$-body terms, Crisanti and Sommers obtained the corresponding result in 1995, and Franz and Parisi rederived it by a constrained-replica method that prepares a system in the same metastable state as a reference copy.

A stationary point is characterized by its index: the number of negative eigenvalues (unstable directions) of the Hessian, the matrix of second derivatives of $F_{\mathrm{TAP}}$. Counting stationary points without the modulus $|\det H|$ in the Jacobian weight assigns each point the sign $(-1)^{\mathrm{index}}$, and the Morse theorem of differential topology forces the alternating sum over all stationary points of a smooth function on a compact manifold to equal a topological constant. Every meaningful complexity computation must therefore control the modulus of the determinant of a random Hessian; the standard trick, introduced by Kurchan in 1991, represents $|\det H|$ by Grassmann (fermionic) integrals, which turn out to possess a Becchi-Rouet-Stora-Tyutin (BRST) supersymmetry, i.e. a symmetry between bosonic and fermionic variables whose Ward identities tightly constrain saddle-point solutions of the resulting theory. The count of stationary points is of Kac-Rice type: the expected number of stationary points of a random function is expressed analytically by the standard integral formula in terms of its gradient and Hessian, and is evaluated at the saddle-point level with controlled approximations.

In 1998 Cavagna, Giardina and Parisi showed for the p-spin spherical model that the replica representation of the TAP complexity contains distinct saddle points whose contributions give an index-resolved complexity $\Sigma_k(E)$, the rate for stationary points with exactly $k$ negative Hessian eigenvalues, ordered as $\Sigma_0(E)\ge\Sigma_1(E)\ge\Sigma_2(E)\ge\ldots$ below the threshold energy $E_{\mathrm{th}}$ at which the spectrum of the typical TAP Hessian touches zero. Hence at fixed energy density below threshold the minima exponentially dominate, and dropping the modulus is harmless in the thermodynamic limit; at threshold all the $\Sigma_k$ coincide and the sign-weighted sum collapses to the Morse topological constant, as it must. The authors acknowledged what they had not achieved: a direct computation keeping the replica index finite and extracting the sign factor $(-1)^{kn}$ from the Gaussian fluctuations around each saddle (they 'did not succeed in performing this quite complex computation'); the extension of such index-resolved control to other mean-field spin glasses, notably the SK model; and, as their concluding sentence, the wish to describe the glassy transition 'also in structural glasses simply in terms of the evolution of the Hessian spectrum, by means of an analysis similar to the one we have performed here'.

Structural glasses, i.e. supercooled liquids (particle systems with no spin variables), present a different landscape problem. There is no exact TAP free energy; the natural objects are the potential-energy surface, its local minima ('inherent structures') and its index-one saddles, and the spectrum of the Hessian of the potential energy sampled by liquid configurations (the 'instantaneous normal modes'). The dynamic glass transition (a mode-coupling-type crossover, where the structural relaxation time grows steeply upon cooling) is empirically correlated with the disappearance of unstable directions, and random-first-order-transition (RFOT) theory describes it with a constrained-replica (Franz-Parisi) free energy whose marginal stability (the Gardner transition) plays the role of the threshold. In the limit of infinite spatial dimension this program is controlled and quantitative, but it counts constrained-replica equilibria rather than stationary points classified by index, and its determinant-sign bookkeeping is not under control; in finite dimensions, even the statistics of the unstable modes of the landscape are known to escape mean-field theory.

Problem Statement

The surviving open part of the acknowledged open directions of Cavagna, Giardina and Parisi (1998) is to implement, for a structural-glass system, the analog of their p-spin stationary-point analysis, at a comparable level of analytic control. Concretely, choose a structural-glass model — a particle system (supercooled liquid) rather than a spin glass such as the p-spin spherical or Sherrington-Kirkpatrick model; the mean-field, infinite-dimensional and finite-dimensional cases are all admissible, and any of them resolves the problem when the analysis is controlled as required below — and: (i) construct the effective landscape functional whose stationary points represent the metastable glass states of that model, with an argument that its stationary points, rather than some other object, are the correct states to count; (ii) compute, with controlled analytic approximations, the averaged spectrum of the Hessian of this functional at its stationary points, including the behavior of its lower edge (threshold, marginality); (iii) obtain the index-resolved complexity of the stationary points, i.e. the exponential rate of the number of stationary points with $k$ unstable directions, together with explicit determinant-sign bookkeeping analogous to the sign-carrying prefactors of the source analysis; where the chosen landscape functional is defined on a compact manifold, or admits a compactification, such that a Morse-type topological identity applies, sign-weighted sums over all indices must be consistent with that identity; and (iv) determine the organizing structure of the landscape, in particular the dominance relations among indices, and show how the evolution of the Hessian spectrum through the glass transition, e.g. the vanishing of the typical index density (the fraction of negative Hessian eigenvalues) below a threshold energy, accounts for the dynamic crossover of the model. A solution must state clearly which structural-glass model it treats and in what sense the analysis is controlled there. The two spin-glass items of the original bundle, the finite-$n$ sign-factor computation for the p-spin spherical model and the index-resolved analysis of the SK TAP free energy, are excluded from this problem because later literature resolved them through supersymmetric, cavity and rigorous computations.

References

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