Extending the full replica symmetry breaking theory of hard-sphere glasses beyond equilibrium infinite dimension: finite-dimensional effects and off-equilibrium dynamics

The exact solution of dense hard spheres in the limit of infinite spatial dimension (Charbonneau, Kurchan, Parisi, Urbani and Zamponi, 2014) shows that glass states at high pressure enter a marginally stable phase described by full replica symmetry breaking (fullRSB), which predicts the isostaticity of jammed packings and the critical exponents of the jamming transition. The paper's conclusions list the follow-up computations needed to turn this static, equilibrium, infinite-dimensional result into a complete theory of the marginal glass. Three of the listed items have since been completed: the state-following (Franz-Parisi) computation of the Gardner transition and fullRSB phase of a prepared glass state (2015-2016), the shear modulus within fullRSB (2014), and the mean-field avalanche-size distribution, whose jamming exponent was derived from the exponents of the exact solution (2017). The open core is two-fold: (i) extend the fullRSB solution to finite-dimensional hard spheres within the effective-potential (replicated liquid theory) scheme, a computation the exact solution explicitly left for future work, and determine how finite-dimensional critical fluctuations, treated by renormalization group below the upper critical dimension of the Gardner transition, modify the mean-field scenario; and (ii) construct the off-equilibrium (aging) dynamics of the fullRSB glass, a task whose standard spin-glass framework has itself recently been shown to be internally inconsistent, so that the correct dynamical mean-field theory must first be identified. Together these two directions decide whether the marginal-stability theory of glasses, exact at infinite dimension, describes real finite-dimensional systems.

Background

Hard spheres are the minimal model of a glass former: above a packing fraction of about 0.58 in three dimensions the liquid falls out of equilibrium and forms an amorphous solid whose relaxation time grows faster than exponentially. The random first-order transition (RFOT) conjecture holds that structural glasses behave, in the mean-field limit, like certain discontinuous spin-glass models, and this makes an exact treatment possible in the limit of spatial dimension $d\to\infty$: there, the virial expansion of the entropy of replicated copies of the system collapses to a single effective interaction between nearest particles within the same cage (the small region in which each particle is trapped by its neighbors), so that all the thermodynamics of glass states becomes exactly computable. The series of papers by Kurchan, Parisi and Zamponi (paper I), Kurchan, Parisi, Urbani and Zamponi (paper II) and Charbonneau, Kurchan, Parisi, Urbani and Zamponi (paper III) carried out this program.

The central technical notion is replica symmetry breaking (RSB). To describe metastable glasses thermodynamically one considers $m$ copies ("replicas") of the system coupled to one another; the glass order parameter is the matrix of mean-square displacements $\Delta_{ab}$ between replicas $a$ and $b$. In the one-step RSB (1RSB) scheme the replicas form a single block: this describes a collection of distinct, isolated amorphous basins and suffices to locate the dynamical (mode-coupling-like) transition, where the relaxation time diverges, and the Kauzmann transition, the entropy crisis where an ideal glass would appear. The full RSB (fullRSB) scheme allows $\Delta_{ab}$ to take a continuum of values organized in an order function $\Delta(x)$, $x\in[0,1]$: a hierarchy of sub-basins inside each basin. This is the structure that solves the Sherrington-Kirkpatrick (SK) model, the fully connected spin glass with Gaussian random couplings and the archetype of a marginally stable disordered system.

Paper II showed that a 1RSB glass of hard spheres in $d\to\infty$ becomes unstable, at high enough pressure, to a Gardner transition: the splitting of a glass basin into a hierarchy of sub-basins, in the same universality class as the spin glass in a magnetic field. Paper III then derived and solved the fullRSB equations. Their marginal solution (the replicon eigenvalue, the instability mode of the RSB ansatz, vanishes identically throughout the phase) predicts that jamming, the point where a hard-sphere glass becomes mechanically rigid at infinite pressure, produces isostatic packings (number of mechanical contacts equal to the number of degrees of freedom) with critical exponents for the distribution of small contact forces, $P(f)\sim f^{\theta}$ with $\theta=0.42311\ldots$, of small gaps, $P(h)\sim h^{-\alpha}$ with $\alpha=0.41269\ldots$ (an exponent often denoted $\gamma$ in the numerical jamming literature), and of the cage size, $\Delta\sim p^{-\kappa}$ with $\kappa=1.41574\ldots$ as a function of pressure $p$, in remarkable agreement with numerical simulations in low dimensions.

The conclusions of paper III list the computations needed to turn this static, equilibrium, $d\to\infty$ result into a complete theory of the marginal glass. Four further notions are needed to state them. First, the Franz-Parisi (state-following) construction: one constrains a copy of the system to remain close to a frozen reference equilibrium configuration and studies the constrained free energy; this allows one to follow a single, protocol-prepared glass state under compression or shear, and is the framework in which the Gardner transition of a given state is computed. Second, static avalanches: a marginally stable glass responds to a small external perturbation through discrete irreversible jumps (in energy, stress, or mean-square displacement) whose sizes are broadly distributed; the reference computation is the avalanche-size distribution of the SK model by Le Doussal, Muller and Wiese. Third, the effective potential scheme: in finite $d$, the dressed interaction between replicated particles can be inserted into standard liquid-theory integral equations to approximate the replicated entropy; this is the finite-dimensional extension developed at 1RSB level in the mean-field theory of hard-sphere glasses and jamming of Parisi and Zamponi, which paper III explicitly leaves open at the fullRSB level. Fourth, weak ergodicity breaking: the hypothesis, central to the standard aging theory of mean-field spin glasses, that an out-of-equilibrium system never reaches equilibrium but eventually loses all memory of its initial condition.

The state-following computation of the Gardner transition and fullRSB phase of a prepared glass state, the shear modulus within fullRSB, and the mean-field avalanche-size distribution (whose jamming exponent is expressed through the exponents $\theta,\alpha,\kappa$ above) have all been computed and validated by later work. What remains open is the passage beyond the setting in which the exact solution was derived. On the equilibrium side, the finite-dimensional extension has two layers: the approximate effective-potential computation in finite $d$, which paper III deferred to future work, and the treatment of truly finite-dimensional critical fluctuations, since the Gardner transition has upper critical dimension $d_u=6$, one-loop renormalization group finds no stable perturbative fixed point below $d_u$, and a two-loop analysis finds a nontrivial fixed point only at strong coupling, so the fate of the transition and of the fullRSB phase in finite dimension is analytically unsettled. On the dynamical side, the off-equilibrium (aging) dynamics of a fullRSB glass is not understood even at mean-field level: the standard fullRSB dynamical solution of spin glasses has recently been shown to violate weak ergodicity breaking, so the structural-glass theory cannot simply import it.

Problem Statement

The exact $d\to\infty$ solution leaves two linked directions open; completing them extends the full replica symmetry breaking (fullRSB) theory of hard-sphere glasses beyond the equilibrium, infinite-dimensional setting in which it was derived. The two directions serve one common objective, establishing whether and how the marginal (fullRSB) phase survives outside its exact solvability limit. The two directions are separate research directions; direction (1) has two layers, (a) and (b).

(1) Finite-dimensional extension. (a) Perform, at the fullRSB level, the effective-potential (replicated liquid theory) computation in finite dimension that paper III explicitly left for future work: insert the replicated effective potential into liquid-theory integral equations with a fullRSB order parameter and solve the resulting equations in finite $d$, delivering quantitative predictions, such as the distribution of mean-square displacements among sub-basins, the equation of state and shear response of followed states, and the Gardner transition line of protocol-prepared glasses, that are directly comparable with simulations and experiments in $d=3$. (b) Characterize how truly finite-dimensional critical fluctuations modify the mean-field scenario: determine the renormalization-group fate of the Gardner transition below its upper critical dimension $d_u=6$, where available perturbative analyses place any nontrivial fixed point at strong coupling, and establish whether the fullRSB phase, or only part of its phenomenology, survives in finite dimension.

(2) Off-equilibrium dynamics. Construct the aging dynamics of the hard-sphere glass inside the fullRSB phase at mean-field level: identify the correct dynamical framework, noting that the standard fullRSB dynamical solution of mean-field spin glasses has been shown to violate weak ergodicity breaking and therefore cannot simply be transplanted; derive and solve the resulting dynamical equations; and extract the dynamical signatures of marginal stability for structural glasses, such as aging laws, effective temperatures, and fluctuation-dissipation relations.

References

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