Numerical renormalization of glassy dynamics

TL;DR

This work tackles the challenge of simulating aging glassy dynamics, where no stationary state is reached and memory effects scale with age. It introduces a dynamical renormalization algorithm that adaptively resamples time via an RG-inspired, fixed-cost discretization and sparse two-dimensional Green's function interpolation, enabling sublinear scaling and access to times up to $t\,\sim\,10^6$. Applied to the mixed spherical $p$-spin model under a zero-temperature quench from finite $T$, the method reveals a finite-temperature transition between strong and weak ergodicity breaking, with continuously varying critical exponents and a nonmonotonic effective temperature in the strong glass, indicating nonuniversal criticality. The approach is general and potentially extends to other slow-dissipative systems, including DMFT impurity solvers and neural-network learning dynamics, offering a powerful tool for exploring slow modes in disordered systems.

Abstract

The quench dynamics of glassy systems are challenging. Due to aging, the system never reaches a stationary state but instead evolves on emergent scales that grow with its age. This slow evolution complicates field-theoretic descriptions, as the weak long-term memory and the absence of a stationary state hinder simplifications of the memory, always leading to the worst-case scaling of computational effort with the cubic power of the simulated time. Here, we present an algorithm based on two-dimensional interpolations of Green's functions, which resolves this issue and achieves sublinear scaling of computational cost. We apply it to the quench dynamics of the spherical mixed $p$-spin model to establish the existence of a phase transition between glasses with strong and weak ergodicity breaking at a finite temperature of the initial state. By reaching times three orders of magnitude larger than previously attainable, we determine the critical exponents of this transition. Interestingly, these are continuously varying and, therefore, non-universal. While we introduce and validate the method in the context of a glassy system, it is equally applicable to any model with overdamped excitations.

Figures (4)

• Figure 1: Comparison of the performance of the algorithm presented here for the mixed $p$-spin model against a standard implementation of Eqs. \ref{['eq:EOM_Glass']} (see Ref. Folena2020). Both codes are run on comparable modern desktop computers. For the time simulated in the present work, explicit SSPRK methods with adaptive step sizes can be used to reach a sublinear scaling of the computational cost (both memory and CPU time). At even later times implicit methods will be necessary to maintain sublinear scaling. Inset: Visualization of the time discretization and a typical contour of the memory integration in Eq. \ref{['eq:memory_int']} (blue). The shaded red areas highlight regions that are densely sampled (for further details see supplemental material).
• Figure 2: Asymptotic initial state correlations $q_0$ as a function of the initial temperature $T$ in units of the mode coupling temperature for the mixed $p$-spin model with $p=3$, $s=4$, and $\lambda=1/2$. Below the lower critical temperature $T_{c_l}$, no aging is observed, and the system remains strongly correlated with the initial state. Above the upper critical temperature $T_{c_u}$, one finds a weak long-term memory and weak ergodicity breaking. The strong glass phase between the critical temperatures is characterized by the coexistence of aging and strong ergodicity breaking. The blue line is a numerical fit to the critical behavior near $T_{c_u}$ from which we extract the critical exponent $\eta$. The red line represents the analytic result for the low-temperature phase.
• Figure S1: Contours on which the integrands of $I_1$ and $I_2$ are evaluated. The choice of discretization in terms of $\phi^{(1,2)}$ is indicated. Red and green ellipses denote areas requiring dense sampling. The need for commensurate sampling of $t$ and $t'$ leads to dense sampling of the regions highlighted by dashed ellipses.
• Figure S2: (a) Inverse effective temperature at various times for $p=3$, $s=4$, and $\lambda=1/2$ in the weak glass phase. As the system's age increases, the effective temperature approaches a constant. The dashed line indicates the marginal value given by Eq. \ref{['eq:x']}. (b) In the strong glass phase, the effective temperature remains non-monotonic at all simulated times.