Temporal Griffiths Phases

Disorder is an unavoidable ingredient of real systems. Spatial disorder generates Griffiths phases (GPs) which, in analogy to critical points, are characterized by a slow relaxation of the order parameter and divergences of quantities such as the susceptibility. However, these singularities appear in an extended region of the parameter space and not just at a (critical) point, i.e., there is generic scale invariance. Here, we study the effects of temporal disorder, focusing on systems with absorbing states. We show that for dimensions $d\ge 2$ there are Temporal Griffiths phases (TGPs) characterized by generic power-law scaling of some magnitudes and generic divergences of the susceptibility. TGPs turn out to be a counterpart of GPs, but with space and time playing reversed roles. TGPs constitute a unifying concept, shedding light on the nontrivial effects of temporal disorder.

Figure 1

Schematic phase diagram for the pure contact process (CP) (solid line) the CP with quenched disorder (dashed line), and the CP with temporal disorder (dot-dashed line). For the second, a Griffiths phase appears within the absorbing region, while for the third a temporal Griffiths phase appears (for $d>1$) within the active region. The actual locations of the critical points may depend on noise intensity and dimension.

Figure 2

Main: Log-log plot of the lifetime $\tau$ as a function of system size $N$ for the TDCP in $d=2$ for various $b_0$ and $\sigma =0.4$. There is a finite region, $b_0\in [0.656,0.675]$ with generic algebraic scaling of $\tau (N)$ and continuously varying exponents. Inset: log-log plot of $\tau (N)$ vs $\ln (N)$; from the fit at criticality (dashed line) we estimate $\tau \sim (\ln N{)}^{5.18(5)}$.

Figure 3

Log-log plot of the susceptibility, $\Xi$, as a function of the field $h$, obtained by integrating Eq. (1) (main plot) and Eq. (2) for $d=2$ (inset), for different values of $a$.