Apparent bistability from weak long-range interactions

Abstract

Bistability, or the coexistence of two stable phases, can be broken by a bias field $h$ destabilising one of the phases via the nucleation and growth of defects. Strong long-range interactions, $1/r^\alpha$ with $\alpha$ less than the system's dimensionality $d$, can suppress the proliferation of defects and restore bistability. The case of weak long-range interactions $d<\alpha < d+1$ remains instead poorly understood. Here, we show that it supports \emph{apparent} bistability: While the system has in principle a unique stable phase, it appears bistable for all practical purposes for $\alpha < \alpha_c$, with $\alpha_c > d$ behaving like a genuine critical point. At the core of this is an exponential scaling of the critical droplet size $R_c\sim h^{-1/(\alpha - d)}$, which makes nucleating destabilizing droplets extremely unlikely for $\alpha < \alpha_c$, and such that $\alpha_c$ is mostly independent of system size. In support of these conclusions we provide field-theoretical arguments and numerics on a probabilistic cellular automaton. Overall, our results offer a way to rethink phase stability in systems with long-range interactions as well as a new route to achieve practical bistability.