Many disordered systems such as spin glasses and supercooled liquids exhibit a glass transition where, below some characteristic temperature, the dynamics slows down drastically, with relaxation time scales either diverging or at least becoming too large to measure. The system then falls out of equilibrium, e.g. because it gets trapped in the local minima of its potential energy landscape. One of the simplest models for the resulting slow relaxation and aging is the trap model by Bouchaud and others: the system is simplified to a point in phase-space hopping between local energy minima. Two main ingredients specify the phase-space evolution: the transition rates between neighbouring minima and the phase-space connectivity, which defines a network of allowed transitions between minima.
Previous studies have explored the analytically tractable mean-field case [JP Bouchaud et al, 1995], where the network is fully connected, and related approximations [P Moretti et al, 2011]. Our work deepens the understanding of the trap model by investigating the effects of more limited phase-space connectivity. We focus on the spectral properties of the master operator, which govern the dynamics of the system, and leave to future investigations the study of e.g. two-time correlation functions.
In our study, transition rates only depend on the departing trap depth as in the original Bouchaud model: every transition effectively involves activation to the top of the energy landscape and then falling into the new state. We consider the paradigmatic case of finite phase-space connectivity given by a random regular graph, where every state has the same number of accessible neighbours.. The main quantities of interest are the average eigenvalue spectrum, or density of states (DOS), and the average degree of localisation of states. We use the inverse participation ratio as a measure for localisation as it allows us to distinguish between two situations: localised eigenstates with a finite number of non zero components, and de-localised eigenstates with an extensive number of non zero components. Similarly one can distinguish between the total DOS, and the extended DOS, which only includes the de-localised eigenstates of the system.
We develop a general approach for the case of sparse phase-space connectivity by means of the cavity method; quantities of interest are evaluated numerically via a population dynamics algorithm. Following the cavity construction we are able to develop a simple analytical “high-temperature” approximation for the eigenvalue spectrum, where effectively one cavity iteration is performed at finite temperature starting from the infinite temperature solution.
Numerical results for the eigenvalue spectrum show good agreement with the high temperature approximation: slow modes exhibit a mean field-like trend, while fast modes appear to be strongly influenced by the network structure. Two localization transitions are found, one each at the left and right edge of the spectrum. The total DOS and the extended DOS indicate the effective coexistence of localised and de-localised states around these localization transitions.
