Low temperature dynamics for confined $p=2$ soft spin in the quenched regime

TL;DR

This paper investigates the low-temperature dynamics of confined $p=2$ soft spin systems with quartic and sextic potentials, analyzing how they relax to equilibrium and identifying a critical temperature separating exponential and power-law decay regimes.

Contribution

It provides a detailed analysis of the dynamical behavior of confined $p=2$ soft spins at low temperatures, including the derivation of self-consistent evolution equations and the identification of a critical temperature.

Findings

Above the critical temperature, relaxation is exponential.

Below the critical temperature, relaxation follows a power law.

The study characterizes the transition between different relaxation regimes.

Abstract

This paper aims to address the low-temperature dynamics issue for the $p=2$ spin dynamics with confining potential, focusing especially on quartic and sextic cases. The dynamics are described by a Langevin equation for a real vector $q_i$ of size $N$, where disorder is materialized by a Wigner matrix and we especially investigate the self consistent evolution equation for effective potential arising from self averaging of the square length $a(t)\equiv \sum_i q_i^2(t)/N$ for large $N$. We first focus on the static case, assuming the system reached some equilibrium point, and we then investigate the way the system reach this point dynamically. This allows to identify a critical temperature, above which the relaxation toward equilibrium follows an exponential law but below which it has infinite time life and corresponds to a power law decay.