Interaction-free ergodicity-breaking driven by temporally hyperuniform noise

TL;DR

The work demonstrates that a noninteracting spherical spin model under a global norm constraint undergoes a sharp ergodicity-breaking transition when driven by temporally hyperuniform noise of class I, i.e., with a small-frequency spectrum $\tilde{D}(\omega)\propto|\omega|^{\alpha}$ and $\alpha>1$. The transition corresponds to condensation of zero-frequency fluctuations, akin to Bose–Einstein condensation in frequency space, and is captured by dynamical mean-field theory with a self-consistent $\mu$ determined from $\mu=\frac{2T}{\pi}\int_0^{\infty}d\omega\,\frac{\tilde{D}(\omega)}{\mu^2+\omega^2}$. Numerically, the transition is confirmed by measuring the steady-state correlator $C(t)$ and identifying a finite nonergodicity parameter $C_{\infty}$ below $T_c$. The analysis extends to $L_p$ norm constraints and soft global constraints, showing ergodicity breaking as a generic consequence of class-I driving with global constraints. Overall, the paper reveals a novel route to ergodicity breaking without interactions and highlights a frequency-space condensation mechanism with potential connections to constraint-satisfaction and non-equilibrium condensation phenomena.

Abstract

We show that norm-conserving spin models driven by temporally hyperuniform noise exhibit a sharp ergodicity-breaking transition in the absence of interactions. In the nonergodic phase, the dynamics freeze into configurations determined by the initial condition. Our analysis demonstrates that such interaction-free ergodicity breaking arises generically whenever a global constraint is imposed and the driving noise is class-I hyperuniform, the strongest form in Torquato's classification. The transition can also be interpreted as a condensation of fluctuations into the zero-frequency mode, reminiscent of Bose--Einstein condensation in an ideal gas.

Figures (5)

• Figure 1: Correlation function $C(t)$ of the spherical model driven by high-pass filtered noise. Markers denote numerical results, while solid lines show the theoretical predictions. For $T=1.5$ and $1.0$, $C(t)$ decays to zero at long times, indicating ergodicity. For $T=0.5$, $C(t)$ converges to a finite value, indicating nonergodicity.
• Figure 2: $T$ dependence of nonergodicity parameter $C_\infty=\lim_{t\to\infty}C(t)$ of the spherical model driven by high-pass filtered noise for $k=1$. Markers denote numerical results, while solid line shows theoretical prediction.
• Figure 3: $\tilde{D}(\omega)$ for several values of $\alpha$. The case $\alpha=0$ corresponds to thermal white noise. For $\alpha>0$, $\tilde{D}(\omega)$ vanishes in the limit $\omega\to 0$.
• Figure 4: Phase diagram for $\omega_0=1$. Solid line denotes the transition line $T_c$, and shaded region indicates the nonergodic phase.
• Figure 5: External field $h$ dependence of the order parameter $m$ for the spherical model driven by high-pass filtered white noise with $k=1$. Markers denote numerical results, while solid lines denote theoretical predictions. For $T\geq T_c=1$, $m$ increases continuously with $h$, whereas for $T<T_c$, $m$ changes discontinuously at $h=0$.