In mean-field models of the glass transition - mode-coupling theory (MCT) of supercooled liquids and one-step replica-symmetry-breaking (1RSB) spin glasses - all dynamical critical exponents are controlled by a single non-universal exponent parameter λ\lambda. Parisi and Rizzo (Phys. Rev. E 87, 012101 (2013)) proved that, whenever the replicated Gibbs free energy admits a local cubic expansion near the critical point, λ\lambda equals the ratio w2/w1w_2/w_1 of the two cubic vertices of that expansion, and equivalently the ratio ω2/ω1\omega_2/\omega_1 of two static six-point cumulants, making the dynamics a prediction of the statics. The problem is the fate of this statics-dynamics mapping outside its mean-field domain, as posed in that paper: (i) quantitative justification of the cubic expansion and practical computation of w1,w2w_1, w_2 for realistic liquids; (ii) whether, below the upper critical dimension, a renormalized static quantity (or a different static object) still determines the asymptotic dynamics, and how to compute it; (iii) robust protocols to extract ω1,ω2\omega_1, \omega_2 (hence λ\lambda) in the discontinuous (n=1n=1, MCT-like) case, where metastability and enhanced divergences of the intermediate multi-replica cumulants obstruct measurement; (iv) extensions to off-equilibrium aging regimes governed by threshold states (1RSB with breaking parameter x<1x<1), to the marginal λ=1\lambda=1 case where quartic vertices (eight-point functions) become relevant, and to finite-dimensional corrections; (v) the finite-size and estimator methodology needed because ω1\omega_1 and ω2\omega_2 diverge as r−3r^{-3} at criticality. Later literature resolved part of this - off-equilibrium critical aging at continuous transitions, the loop-expansion analysis of the replicated dynamical Landau functional showing the MCT singularity is avoided below an upper critical dimension of 8 and replaced by a stochastic-beta-relaxation crossover depending on the model only through λ\lambda, microscopic computation of those couplings from the static structure factor, and multi-replica estimator protocols for ω1,ω2\omega_1, \omega_2 in spin glasses in a field - but the central core remains open: whether any static quantity determines the asymptotic dynamics below the upper critical dimension, a working protocol for the discontinuous case, the static quartic analysis at λ=1\lambda=1, and a quantitatively validated static prediction of λ\lambda for a realistic three-dimensional liquid.

Background

When a liquid is supercooled, its structural relaxation time grows by many orders of magnitude. Mode-coupling theory (MCT), the standard microscopic mean-field theory of this slowdown, closes the equations for the density-autocorrelation functions on themselves and predicts a dynamical arrest transition at a critical temperature TcT_c: the relaxation time diverges, and the correlator develops a two-step decay in which the approach to and the departure from a plateau are power laws with exponents aa and bb. Götze showed that these power laws are universal: every model-dependent detail enters only through a single non-universal exponent parameter λ\lambda, which fixes both exponents through

λ=Γ2(1−a)Γ(1−2a)=Γ2(1+b)Γ(1+2b),\lambda = \frac{\Gamma^2(1-a)}{\Gamma(1-2a)} = \frac{\Gamma^2(1+b)}{\Gamma(1+2b)},

where Γ\Gamma is the Gamma function. The identical equations describe equilibrium dynamics at the dynamical transition temperature TdT_d of mean-field spin glasses with one step of replica symmetry breaking (1RSB) - the discontinuous case of Parisi's replica scheme, appropriate to systems with an exponential number of metastable states, such as fully connected p-spin models. Before 2013, λ\lambda could be obtained only by solving the dynamics explicitly (possible for a handful of solvable models) or, within MCT, from approximate functionals of the static structure factor S(k)S(k) via uncontrolled approximations. The replica method addresses these transitions purely statically. One expands the replicated Gibbs free energy G[δQ]G[\delta Q] - the Legendre transform of the replicated free energy - near criticality in the fluctuation δQab\delta Q_{ab} of the overlap order parameter (the matrix of overlaps between nn identical copies, or replicas, of the system; QabQ_{ab} generalizes the spin-glass overlap). For the transitions of interest the leading nonlinear terms are cubic: a trace term with coefficient w1w_1 and a diagonal term with coefficient w2w_2. Continuous spin-glass transitions (e.g. along the de Almeida-Thouless line, where a spin glass in a field orders continuously) correspond to the n→0n\to 0 replica limit, while the discontinuous MCT-like dynamical transition corresponds to n=1n=1. Parisi and Rizzo proved, under the sole assumption that this local cubic expansion exists, that in both cases the exponent parameter is λ=w2/w1\lambda = w_2/w_1: a strictly dynamical quantity equals a ratio of static vertices. Because a Gibbs free energy is a Legendre transform, its proper vertices are cumulants of the order parameter, and the same ratio can be written as λ=ω2/ω1\lambda = \omega_2/\omega_1 where ω1\omega_1 and ω2\omega_2 are static six-point cumulants; for spins, ω1=N−1∑ijk⟨sisj⟩c⟨sjsk⟩c⟨sksi⟩c‾\omega_1 = N^{-1}\sum_{ijk}\overline{\langle s_i s_j\rangle_c\langle s_j s_k\rangle_c\langle s_k s_i\rangle_c} and ω2=(2N)−1∑ijk⟨sisjsk⟩c2‾\omega_2 = (2N)^{-1}\sum_{ijk}\overline{\langle s_i s_j s_k\rangle_c^2}, with cc denoting connected thermal averages and the overline an average over disorder or over metastable states. These can be measured quasi-statically by simulating several real replicas (independent copies under the same disorder) and combining eight cubic overlap cumulants W1,…,W8W_1,\dots,W_8. The transition is driven by the vanishing of the replicon mass rr (the inverse spin-glass susceptibility); since w1,w2w_1,w_2 stay finite, ω1\omega_1 and ω2\omega_2 diverge as r−3r^{-3}. In the discontinuous n=1n=1 case the situation is worse: the individual WiW_i diverge as fast as r−5r^{-5} and r−4r^{-4} with exact cancellations required to reach r−3r^{-3}, and the needed averages are only defined inside metastable glassy states. Within mean field (fully connected models, and models above the upper critical dimension), this result makes the dynamics a prediction of the statics. The open questions all concern what survives outside that regime. Below the upper critical dimension - now understood to be du=8d_u = 8 for the MCT/1RSB dynamical transition - the loop expansion of the corresponding field theory diverges at every order and the sharp dynamical transition is expected to be smoothed into a crossover; resumming the leading divergences yields the stochastic beta-relaxation (SBR) equations, an extension of the MCT equation for the critical correlator in which the separation from the transition point becomes a spatially random field, and these describe the crossover. It is therefore unclear whether any renormalized static quantity still determines the asymptotic dynamics. For realistic liquids, the existence of the local cubic expansion itself is an assumption requiring quantitative verification within approximate static frameworks (cloned-liquid theory, which studies nn weakly coupled copies of the liquid using standard liquid-state approximations such as the hypernetted-chain closure; replicated Ornstein-Zernicke equations, the standard integral-equation closure scheme of liquid-state theory applied to the replicated system; cavity message-passing on locally tree-like lattices; and diagrammatic schemes). At the marginal point λ=1\lambda = 1 (the A3 singularity of MCT, where the discontinuous transition degenerates into a continuous one) the cubic analysis breaks down and quartic vertices - eight-point functions - are expected to take over. In the off-equilibrium aging regime below TdT_d, governed by threshold states (the highest-free-energy glassy states, which are marginally stable and dominate the long-time aging dynamics) described by a 1RSB solution with breaking parameter x<1x<1 (the size of the overlap block in the Parisi ansatz), it is unknown whether a static expansion determines the aging exponents. Finally, because the measurable ω1,ω2\omega_1,\omega_2 diverge at criticality, practical estimation of λ\lambda in simulations and experiments requires dedicated finite-size and estimator methodology.

Problem Statement

Overarching objective: establish whether the purely static determination of the dynamical exponent parameter in glassy systems - λ=w2/w1=ω2/ω1\lambda = w_2/w_1 = \omega_2/\omega_1, where w1,w2w_1, w_2 are the cubic vertices of the replicated Gibbs free energy and ω1,ω2\omega_1, \omega_2 are the corresponding static six-point cumulants - survives beyond the mean-field regime in which it was proved, and turn it into a practical, quantitative tool. The problem, as posed by Parisi and Rizzo (Phys. Rev. E 87, 012101 (2013)) and refined by what later literature left unresolved, decomposes into linked targets that jointly serve this objective:

(a) Below the upper critical dimension: determine whether a renormalized static quantity, or a different static object, continues to determine the asymptotic dynamics near the (avoided) dynamical transition, and if so compute it. The currently available statement - that below du=8d_u = 8 the crossover is governed by stochastic beta-relaxation equations whose model dependence reduces to the bare λ\lambda plus a noise amplitude - is derived at the level of leading divergences of the loop expansion; whether it is exact, what happens above du=8d_u = 8, and what the renormalized six-point ratios measured numerically in finite dimensions mean dynamically, are all open.

(b) In the discontinuous (n=1n = 1, MCT-like) case: design and validate robust protocols to extract ω1\omega_1 and ω2\omega_2, and hence λ\lambda, in the presence of metastability (static averages are defined only inside metastable glassy states, requiring quasi-static measurements within a state) and of enhanced divergences of the intermediate multi-replica cumulants (individual Wi∼r−5W_i \sim r^{-5} and r−4r^{-4}, with exact cancellations down to the r−3r^{-3} divergence of ω1\omega_1 and ω2\omega_2), including the finite-size extrapolation methodology this requires. The analysis of the cumulant relations in this case was promised by the source authors but never published, and no measurement in this case is known.

(c) At the marginal λ=1\lambda = 1 point (the A3 singularity where the MCT discontinuous transition becomes continuous): establish the static counterpart of the dynamics, namely whether and how the quartic coefficients of the replicated Gibbs free energy - corresponding to eight-point functions - determine the critical dynamical behavior, completing the program announced as 'work under way' in the source paper.

(d) For realistic structural glasses: justify quantitatively the assumption that the replicated Gibbs free energy of a liquid near its dynamical transition admits the local cubic expansion, and compute w1,w2w_1, w_2 (equivalently ω1,ω2\omega_1, \omega_2, or λ\lambda) in practice from static input within controlled approximations, to the point of predicting the measured dynamical exponents of a three-dimensional liquid without dynamical input.

(e) Off equilibrium: extend the static determination of dynamical exponents to aging in the discontinuous case, i.e. to dynamics below TdT_d governed by threshold states described by a 1RSB solution with breaking parameter x<1x < 1, where mean-field theory predicts two aging exponents linked by λ\lambda (or by the breaking point xx). The continuous-transition case was resolved by the same authors; the discontinuous case remains open.

Each strand is resolved either by the static object that determines the dynamics in that regime together with a way to compute or measure it, or by a controlled argument that no such static determination exists. The strands are independent, and each concerns the regime stated in its own item.

References

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