Zero-temperature dynamics of the spherical model with non-reciprocal interactions

TL;DR

This paper analytically investigates the zero-temperature dynamics of the spherical model with non-reciprocal random interactions, revealing how asymmetry affects relaxation, correlations, and introduces oscillatory regimes.

Contribution

It provides an exact solution for the dynamics of the spherical model with non-reciprocal interactions, highlighting the effects of asymmetry on relaxation and oscillations.

Findings

Long-time correlations depend on both times, indicating broken time-translation invariance.

Relaxation is exponential, contrasting with power-law decay in symmetric models.

For antisymmetric interactions, a transition to oscillatory behavior occurs after a certain time scale.

Abstract

We analytically solve the zero-temperature dynamics of the spherical model with non-reciprocal random interactions drawn from the real elliptic ensemble of random matrices, where a single parameter $\eta$ continuously interpolates between purely symmetric ($\eta=1$) and purely antisymmetric ($\eta=-1$) couplings. We show that the two-time correlation and response functions depend on both times in the presence of non-reciprocal interactions, reflecting the breakdown of time-translation invariance and the absence of equilibrium at long times. Nevertheless, the long-time relaxation of the two-time observables is governed by exponential decays, in contrast to the slow, power-law relaxation characteristic of the model with purely symmetric interactions. We further show that, when the interactions present antisymmetric correlations of strength $\eta <0$, there is a time scale $\tau(\eta)$ above which the dynamics undergoes a transition to an oscillatory regime where the two-time observables display periodic oscillations with an exponentially decaying amplitude. Overall, our results give a detailed account of the dynamics of the spherical model with non-reciprocal interactions at zero temperature, providing a benchmark for the study of complex systems with nonlinear and asymmetric interactions.