Nonreciprocal Spin-Glass Transition and Aging

Abstract

Disordered systems generically exhibit aging and a glass transition. Previous studies have long suggested that non-reciprocity tends to destroy glassiness. Here, we show that this is not always the case using a bipartite spherical Sherrington-Kirpatrick model that describes the antagonistic coupling between two identical complex agents modeled as macroscopic spin glasses. Our dynamical mean field theory calculations reveal an exceptional-point mediated transition from a static disorder phase to an oscillating amorphous phase as well as non-reciprocal aging with slow dynamics and oscillations.

Figures (5)

• Figure 1: Sketch of the non-reciprocal spin-glass model: two ($N$-dimensional, in the plot $N=3$) spherical spin systems, corresponding degrees of freedom are coupled non-reciprocally. The sphere's colors sketch the identical random potential in both systems.
• Figure 2: Real and imaginary part of the diagonal and antisymmetric components of the response (a) and correlation (b) functions at the critical point. In gray we highlight $\omega/\alpha=\pm 1$. $T=T_c=1$. (c) Diagonal component of the correlation function, normalized by the Edward-Anderson order parameter, for different values of the waiting time.
• Figure 3: Projection on the two leading eigenvectors of the interaction matrix $J$ of the trajectories (green and red) of the two systems. Random initial conditions, $T=0$, $\alpha=0.2$, $N=20000$, $t_{max}=2000$.
• Figure 4: Numerical results for the amplitude of the Fourier component at $\omega=\alpha$ of the projection of one of the two clones on the leading eigenvector, $\frac{1}{N}\sum s_i^a(v_{\mu_0})_i$, as a function of temperature. The amplitude goes to zero continuously for $T\to T_c=1$; in grey we show $q_{EA}=1-\frac{T}{T_c}$. Inset: amplitude of the Fourier transform as a function of $\omega$ for $T=0.5$. $\alpha=0.2$
• Figure 5: Correlation functions for different values of the initial time $t'$, from numerical simulations of the equations (\ref{['eq:motion']}), above (top, $T=1.4$) and below (bottom, $T=0.3$) the critical point. $\alpha=1$, $N=20000$, averaged over 5 runs. In black the analytical prediction for the envelope of the oscillations (i.e. $C_0(t, t')$) for $t'=2^4$.