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

TL;DR

The paper tackles the zero-temperature dynamics of the spherical model with non-reciprocal couplings drawn from the real elliptic ensemble, parameterized by $\eta$ to smoothly interpolate between symmetric and antisymmetric interactions. Using biorthogonal eigenvector decomposition and random-matrix theory, the authors derive exact expressions for the two-time autocorrelation $C(t,t')$ and response $G(t,t')$, showing that time-translation invariance is broken for any $\eta<1$ and that the long-time relaxation is exponential with algebraic prefactors, except in the fully antisymmetric limit where oscillations emerge. A detailed analysis across regimes $0 \le \eta \le 1$ and $-1 \le \eta < 0$ reveals plateaued but non-equilibrium dynamics for the former, and a transition to sustained oscillations with a characteristic period $T=\pi/\sqrt{|\,\eta\,|}$ and exponential damping for the latter; the special case $\eta=-1$ yields pure oscillations with a power-law envelope. The results provide a comprehensive benchmark for understanding complex, nonlinear, and asymmetric interactions in driven systems, with implications for neural networks, ecosystems, and other non-reciprocal dynamical networks.

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 $η$ continuously interpolates between purely symmetric ($η=1$) and purely antisymmetric ($η=-1$) 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 $η<0$, there is a time scale $τ(η)$ 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.

Figures (7)

• Figure 1: Autocorrelation function $C(t_w + \tau,t_w)$ of the spherical model with non-reciprocal interactions drawn from the elliptic ensemble of random matrices with symmetry parameter $\eta$ (see Eq. (\ref{['kopo']})). These results are obtained by numerically computing $C(t_w + \tau,t_w)$ from Eqs. (\ref{['jj0']}) and (\ref{['jj2']})-(\ref{['jj111']}), which are valid in the $N \to \infty$ limit. (a) $\eta=0.7$ and different values of the waiting time $t_w$. (b) $t_w=40$ and different $\eta$.
• Figure 2: Response function $G(t_w + \tau,t_w)$ of the spherical model with non-reciprocal interactions drawn from the elliptic ensemble of random matrices with symmetry parameter $\eta$ (see Eq. (\ref{['kopo']})). These results are obtained by numerically computing $G(t_w + \tau,t_w)$ from Eqs. (\ref{['gg0']}) and (\ref{['jj111']}), which are valid in the thermodynamic limit. (a) $\eta=0.7$ and different values of $t_w$. (b) $t_w=40$ and different $\eta$.
• Figure 3: Comparison between numerical results, obtained from Eqs. (\ref{['toq1']}) and (\ref{['toq2']}) for different system sizes $N$, and the analytical expressions for the correlation and response functions in the limit $N \rightarrow \infty$ for $\eta=0.7$ and $t_w = 10$. The error bars in the numerical data represent the standard deviation of the empirical average computed over a large number of independent realizations.
• Figure 4: Absolute value of the autocorrelation function of the spherical model with non-reciprocal interactions drawn from the real elliptic ensemble of random matrices with symmetry parameter $\eta<0$. These results are obtained by numerically computing $|C(t_w + \tau,t_w)|$ from Eqs. (\ref{['jj0']}), (\ref{['jj2232']}) and (\ref{['jj222']}), which are valid in the $N \to \infty$ limit. (a) $\eta=-0.8$ and different values of $t_w$. The vertical dashed line indicates the characteristic time $\tau_{*}$, Eq. (\ref{['jkl']}), which marks the onset of oscillations for $t_w=10$. (b) $t_w=5$ and different values of $\eta < 0$.
• Figure 5: Rescaled correlation function $C_0(t_w + \tau,t_w)$ for negative $\eta$. These results are obtained by numerically computing $C_0(t_w + \tau,t_w)$ from Eqs. (\ref{['jj0']}) and (\ref{['bobo7']}). (a) $C_0(t_w + \tau,t_w)$ for $\eta=-0.7$ and $t_w=40$. The solid red lines show the analytic result for the amplitude extracted from Eqs. (\ref{['uywd']}) and (\ref{['bobo7']}). (b) Long-time periodic oscillations of $C_0(t_w + \tau,t_w)$ for $\eta=-0.7$ and two values of $t_w$. The dashed lines are the analytic prediction of Eq. (\ref{['uywd']}).