Replicated liquid theory in $1+\infty$ dimensions

TL;DR

The work develops an inhomogeneous replicated-liquid theory in $1+(d-1)$ dimensions, exact as $d-1\to\infty$, to capture spatial variations of glassy order along a longitudinal axis via a space-dependent order parameter $Δ_{ab}(z)$. It constructs a free-energy functional $F_m[\rho,Δ_{ab}(z)]$ within a density-functional plus replica framework and solves a 1RSB saddle point to study hard-sphere glasses in bulk and in cavities, identifying diverging length scales near dynamical and Kauzmann transitions. The key findings are that three interrelated lengths—the point-to-set dynamical length, the Hessian-based correlation length, and the cavity-profile length—all scale as $δ\hat{\varphi}^{-1/4}$ as the dynamical transition is approached, with the cavity profile length matching the Hessian length and static transition shifts decreasing with cavity size. The results reproduce known mean-field exponents, reveal a nontrivial spatial structure of $Δ(z)$ in confined geometries, and provide a microscopic tool to study surface-like critical phenomena and finite-size effects in glassy systems, with potential extensions to yielding and jamming phenomena.

Abstract

We develop a replicated liquid theory for structural glasses which exhibit spatial variation of physical quantities along one axis, say $z$-axis. The theory becomes exact with infinite transverse dimension $d-1 \to \infty$. It provides an exact free-energy functional with space-dependent glass order parameter $Δ_{ab}(z)$. As a first application of the scheme, we study diverging lengths associated with dynamic/static glass transitions of hardspheres with/without confining cavity. The exponents agree with those obtained in previous studies on related mean-field models. Moreover, it predicts a non-trivial spatial profile of the glass order parameter $Δ_{ab}(z)$ within the cavity which exhibits a scaling feature approaching the dynamical glass transition.

Figures (4)

• Figure 1: Schematic picture of our system in a cylinder
• Figure 2: The spatial profile of the glass order parameter of the hard-spheres in the cavity: $\Delta(\hat{z})$ of $\hat{L}_{\rm cav}=10,40$ at $\hat{\varphi}=4.90$ in (a),(b) and various other volume fractions $\hat{\varphi}$ (c),(d). The dotted line in (a),(b) represents the order parameter $\Delta=\Delta_{\rm bulk}$ in bulk system ($\hat{L}_{\rm cav}=\infty$).
• Figure 3: Scaling properties close to the dynamical transition density of the hard-spheres: Panel (a) shows the spatial profile of $\Delta_{\rm bulk}-\Delta(\hat{z},\delta\varphi))$ for various cavity sizes $\hat{L}_{\rm cav}=10,20,30,40$ at $\hat{\varphi}=4.81$ (or $\delta\varphi=(\hat{\varphi}-\hat{\varphi}_{\rm d})/\hat{\varphi}_{\rm d}=6.70\times 10^{-4}$). The exponential fitting function Eq. (\ref{['eq-fit-exponential-delta-Delta']}) is also shown. Here we find $A=0.448192$ and $\xi_{\rm d}^{\rm profile}=3.44808$. Panel (b) shows a scaling plot of $\delta\Delta(\hat{z},\delta\varphi)$with $\nu=1/4$ using data of $\hat{L}_{\rm cav}=2000$. Here $\delta\varphi\equiv(\varphi-\varphi_{d})/\varphi_{d}$ which represents the distance to the critical point. Panel c) displays various lengths diverging at the dynamical transition point: the PS length $\xi^{\rm PS}_{\rm d}$ (purple circles), the correlation length obtained from the spatial profile of $\delta \Delta(\hat{z},\delta\varphi)$ (see (a)) (red dots) and the correlation length extracted in the analysis of the Hessian (blue line)( see Eq. (\ref{['eq-correlation-length-from-Hessian']})) vs $\delta\varphi$.
• Figure 4: Scaling features of $M_{0}$ and $M_{2}$. Here the dotted lines are linear in $\delta\hat{\varphi}^{1/2}$.