Exact solution of Dynamical Mean-Field Theory for a linear system with annealed disorder

TL;DR

This paper extends Dynamical Mean-Field Theory to exactly solve a high-dimensional linear system with annealed disorder, revealing complex behaviors like non-monotonic variance and re-entrant phase transitions as interaction correlation time varies.

Contribution

It provides the first exact analytical solution for a linear system with annealed disorder using an extended DMFT framework, uncovering novel dynamical phenomena.

Findings

Stationary variance is non-monotonic with interaction correlation time.

Re-entrant phase transitions occur as correlation time changes.

Analytical expressions for auto-correlation, variance, and spectral density are derived.

Abstract

We investigate a disordered multi-dimensional linear system in which the interaction parameters vary stochastically in time with defined temporal correlations. We refer to this type of disorder as "annealed", in contrast to quenched disorder in which couplings are fixed in time. We extend Dynamical Mean-Field Theory to accommodate annealed disorder and employ it to find the exact solution of the linear model in the limit of a large number of degrees of freedom. Our analysis yields analytical results for the non-stationary auto-correlation, the stationary variance, the power spectral density, and the phase diagram of the model. Interestingly, some unexpected features emerge upon changing the correlation time of the interactions. The stationary variance of the system and the critical variance of the disorder are generally found to be a non-monotonic function of the correlation time of the interactions. We also find that in some cases a re-entrant phase transition takes place when this correlation time is varied.