Existence, nature, and universality class of the random-pinning glass transition in a three-dimensional Lennard-Jones glass former

Karmakar and Parisi (PNAS 2013) introduced a randomly pinned three-dimensional binary Lennard-Jones glass former — a structural glass with built-in quenched disorder in which spin-glass observables become well-defined — and found that the overlap (spin-glass) susceptibility grows dramatically with system size, while a finite-size-scaling collapse works only with a size-dependent apparent critical temperature, leaving the thermodynamic-limit behavior undetermined. They left open four linked questions: whether the susceptibility truly diverges at a finite temperature as $N\to\infty$ (whether the apparent $T_K(L)$ saturates to a non-zero value); whether any transition belongs to the universality class of the three-dimensional short-range $p$-spin glass or is of the random first-order (RFOT) type predicted by mean-field theory; whether a regime exists at higher pinning densities where the two-step relaxation collapses to one step, as mode-coupling theory predicts; and the precise equilibrium phase structure of the model, including the location of any thermodynamic transition line, its endpoint, and its relation to the Kauzmann point. Later equilibrium studies of pinned Lennard-Jones mixtures (Kob and Berthier; Ozawa et al.) found results consistent with the RFOT scenario but only in small systems without systematic finite-size scaling, the dynamical interpretation remains contested, and the predicted random-field-Ising-class critical endpoint has never been systematically studied. The problem — the thermodynamic-limit existence, nature, universality class, and phase structure of the random-pinning glass transition in this canonical model — remains open.

Background

When a liquid is cooled fast enough it forms a glass: a mechanically rigid but amorphous solid whose relaxation time grows by many orders of magnitude with no evident change in structure. In spin glasses — magnetic systems with quenched (frozen) random interactions — the freezing transition is comparatively well understood through the replica method and replica symmetry breaking (RSB). The random first-order transition (RFOT) theory of structural glasses imports the mean-field solution of $p$-spin spin-glass models (spin models with $p$-body interactions) into the liquid problem: it predicts a dynamical crossover temperature $T_d$ at which mode-coupling theory (MCT), an approximate kinetic theory, predicts a power-law divergence of the relaxation time, and a lower Kauzmann temperature $T_K$ at which the configurational entropy — the extensive part of the entropy that counts distinct metastable amorphous packings — vanishes and a 'random first-order' (one-step RSB, 1RSB) thermodynamic transition takes place. Whether this mean-field scenario survives in finite-dimensional liquids, which have no quenched disorder, is a central open question of the field.

A practical obstacle is that spin-glass observables need quenched disorder. The natural order parameter is the overlap $q$ between two independent equilibrium copies (replicas) of the same system, but in a translationally invariant liquid two replicas can be structurally identical while being shifted relative to one another, so $q$ is ill-defined without care. Random pinning removes this obstacle: one equilibrates the liquid, then permanently freezes a fraction $c$ (often denoted $\rho_{\rm imp}$) of the particles at randomly chosen positions of that equilibrium configuration. The frozen particles constitute quenched disorder that breaks translation invariance while leaving the mobile particles in the same equilibrium liquid state, so the replica overlap $q$ (computed with a window function that counts two particles as matched when they lie within a cutoff distance), its distribution $P(q)$, and the spin-glass susceptibility $\chi_{\rm SG}$ — the variance of $q$ scaled by the number of mobile particles, which would diverge at a continuous ordering transition — become well-defined equilibrium observables. Random pinning is also realizable in colloid experiments.

Theory predicts a specific phase diagram for such systems. Mean-field and renormalization-group analyses of pinned systems (Cammarota and Biroli) show a line of Kauzmann (ideal glass) transitions $T_K(c)$ that rises with pinning fraction $c$ and ends at a critical point whose fluctuations are predicted to be in the random-field Ising model (RFIM) universality class; the dynamical MCT line $T_d(c)$ meets it there. On the dynamical side, Krakoviack's MCT for partly pinned fluids predicts a critical pinning concentration (of order 0.15 in the hard-sphere calculation) above which the characteristic two-step relaxation of supercooled liquids (fast rattling followed by slow collective rearrangement) collapses to a single step, while Szamel and Flenner's alternative mode-coupling approach instead reproduces the RFOT scenario.

In this setting Karmakar and Parisi (PNAS 2013) simulated a three-dimensional binary Lennard-Jones glass former (a Kob-Andersen-type mixture, i.e. a binary blend of two particle species interacting through species-dependent Lennard-Jones potentials, at number density 1.2 with a smoothed interaction potential) in which a fraction $\rho_{\rm imp}=0.11$ of particles was pinned from equilibrated configurations, studying $N=100$ to $4000$ particles down to $T\simeq 0.4$; to reach lower temperatures (down to $T\simeq 0.3$) they used parallel tempering (a replica-exchange Monte Carlo scheme that swaps configurations between neighboring temperatures to accelerate equilibration), restricted to systems of at most $N=250$. They measured $P(q)$, $\chi_{\rm SG}$, and the spatial overlap correlation length, compared with a three-dimensional short-range $p$-spin glass, and estimated the Vogel-Fulcher-Tammann (VFT) divergence temperature (an empirical fit $\tau_\alpha\sim\exp[A/(T-T_0)]$ often identified with $T_K$, giving $T_K\simeq 0.17$) and the MCT crossover temperature ($T_d\simeq 0.394$). The overlap susceptibility grows dramatically with system size (by a factor of order 50 for $N=250$), and a finite-size-scaling collapse of $\chi_{\rm SG}=L^{2-\eta}F(L^{1/\nu}|T-T_K|)$ succeeds only with a size-dependent apparent critical temperature $T_K(L)$; the fitted exponents ($\nu\simeq 1.0$, $\eta\simeq -0.1$) are close to those of the 3D $p$-spin glass. But the data cannot establish whether $\chi_{\rm SG}$ diverges in the thermodynamic limit at a finite temperature, whether $T_K(L)$ saturates to a non-zero value, and whether the transition, if any, is spin-glass-like or of the RFOT type; the paper leaves the phase structure of the model undetermined and asks for accurate equilibrium simulations and comparison with theory.

Problem Statement

For the randomly pinned three-dimensional binary Lennard-Jones glass former introduced by Karmakar and Parisi (a Kob-Andersen-type mixture at number density 1.2 with a fraction $\rho_{\rm imp}\simeq 0.11$ of particles frozen at random positions drawn from an equilibrated configuration, extended to other pinning fractions where the question requires), determine the existence, location, and nature of the equilibrium glass transition induced by random pinning, through the following linked facets that jointly constitute this determination:

(1) Thermodynamic-limit behavior of the overlap (spin-glass) susceptibility: does $\chi_{\rm SG}$ diverge at a finite temperature as $N\to\infty$ — in the source's finite-size-scaling analysis, does the size-dependent apparent critical temperature $T_K(L)$ saturate to a non-zero value rather than extrapolating to zero — or is the strong growth of $\chi_{\rm SG}$ a finite-size or crossover effect tied to the mode-coupling crossover near $T\simeq 0.4$? The source's data, limited to $N=4000$ with a size-dependent $T_K(L)$, leave this question unsettled.

(2) Nature and universality class of any such transition: is it continuous and spin-glass-like, belonging (as the fitted exponents $\nu\simeq 1.0$, $\eta\simeq -0.1$ and the moments of $P(q)$ suggest) to the same universality class as the three-dimensional short-range $p$-spin glass, or is it a random first-order (1RSB/Kauzmann) transition of the RFOT type ending in a critical point with random-field-Ising-like fluctuations, as mean-field theory predicts, or is there no true transition in the thermodynamic limit — the growth of $\chi_{\rm SG}$ a finite-size or crossover effect — so that no universality class is defined (the case in which facet (1) resolves negatively)? At present the evidence is a qualitative similarity to the 3D $p$-spin glass (fitted exponents and moments of $P(q)$); a quantitative identification of the universality class — through critical exponents, scaling functions, and order-parameter distributions — has not been achieved.

(3) High-pinning-density regime: is there a critical pinning fraction above which the two-step relaxation of the liquid collapses to one step, as mode-coupling theory predicts ($\rho_{\rm imp}\sim 0.15$ in the hard-sphere calculation cited by the source), and how does the relaxation scenario connect to the endpoint of the transition line?

(4) Phase structure: the equilibrium phase diagram of the model in the (pinning fraction, temperature) plane — the location of any true thermodynamic transition line, its relation to the Kauzmann point where the configurational entropy vanishes, and its endpoint — precisely determined and compared with theoretical predictions.

The common objective uniting these facets is a single, internally consistent determination of whether and where this canonical randomly pinned atomistic glass former exhibits a true equilibrium thermodynamic transition, and of which universality class governs it; each facet is a required component of that determination rather than an independent target. Facets (2) and (3) are conditional on facet (1): a negative finding there (no divergence of $\chi_{\rm SG}$ in the thermodynamic limit) resolves facet (2) as 'no transition, hence no universality class' and removes the transition-line endpoint that facet (3) is asked to connect to, while the mode-coupling collapse question in facet (3) remains well-posed on its own.

References

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