Anomalous relaxation and hyperuniform fluctuations in center-of-mass conserving systems with broken time-reversal symmetry

Abstract

We study the Oslo model, a paradigm for absorbing-phase transition, on a one-dimensional ring of $L$ sites with a fixed global density $\overline{\ensuremath{\rho}}$; we consider the system strictly above critical density ${\ensuremath{\rho}}_{c}$. Notably, microscopic dynamics conserve both mass and center of mass (CoM), but lack time-reversal symmetry. We show that, despite having highly constrained dynamics due to CoM conservation, the system exhibits diffusive relaxation away from criticality and superdiffusive relaxation near criticality. Furthermore, the CoM conservation severely restricts particle movement, causing the mobility---a transport coefficient analogous to the conductivity for charged particles---to vanish exactly. Indeed, the steady-state temporal growth of current fluctuation is qualitatively different from that observed in diffusive systems with a single conservation law. Remarkably, far from criticality where the relative density $\mathrm{\ensuremath{\Delta}}=\overline{\ensuremath{\rho}}\ensuremath{-}{\ensuremath{\rho}}_{c}\ensuremath{\gg}{\ensuremath{\rho}}_{c}$, the second cumulant, or the variance, ${\ensuremath{\langle}{\mathcal{Q}}_{i}^{2}(T,\mathrm{\ensuremath{\Delta}})\ensuremath{\rangle}}_{c}$, of current ${\mathcal{Q}}_{i}$ across the $i\mathrm{th}$ bond up to time $T$ in the steady-state saturates as ${\ensuremath{\langle}{\mathcal{Q}}_{i}^{2}\ensuremath{\rangle}}_{c}\ensuremath{\simeq}{\mathrm{\ensuremath{\Sigma}}}_{Q}^{2}(\mathrm{\ensuremath{\Delta}})\ensuremath{-}\mathrm{const}\phantom{\rule{0.16em}{0ex}}{T}^{\ensuremath{-}1/2}$; near criticality, it grows subdiffusively as ${\ensuremath{\langle}{\mathcal{Q}}_{i}^{2}\ensuremath{\rangle}}_{c}\ensuremath{\sim}{T}^{\ensuremath{\alpha}}$, with $0<\ensuremath{\alpha}<1/2$, and eventually saturates to ${\mathrm{\ensuremath{\Sigma}}}_{Q}^{2}(\mathrm{\ensuremath{\Delta}})$. Interestingly, the asymptotic current fluctuation ${\mathrm{\ensuremath{\Sigma}}}_{Q}^{2}(\mathrm{\ensuremath{\Delta}})$ is a nonmonotonic function of $\mathrm{\ensuremath{\Delta}}$: It diverges as ${\mathrm{\ensuremath{\Sigma}}}_{Q}^{2}(\mathrm{\ensuremath{\Delta}})\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}^{2}$ for $\mathrm{\ensuremath{\Delta}}\ensuremath{\gg}{\ensuremath{\rho}}_{c}$ and ${\mathrm{\ensuremath{\Sigma}}}_{Q}^{2}(\mathrm{\ensuremath{\Delta}})\ensuremath{\sim}{\mathrm{\ensuremath{\Delta}}}^{\ensuremath{-}\ensuremath{\delta}}$, with $\ensuremath{\delta}>0$, for $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}{0}^{+}$. Using a mass-conservation principle, we exactly determine the exponents $\ensuremath{\delta}=2(1\ensuremath{-}1/{\ensuremath{\nu}}_{\ensuremath{\perp}})/{\ensuremath{\nu}}_{\ensuremath{\perp}}$ and $\ensuremath{\alpha}=\ensuremath{\delta}/z{\ensuremath{\nu}}_{\ensuremath{\perp}}$ via the correlation-length and dynamic exponents, ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ and $z$, respectively. Finally, we show that in the steady state the self-diffusion coefficient ${\mathcal{D}}_{s}(\overline{\ensuremath{\rho}})$ of tagged particles is connected to activity through the relation ${\mathcal{D}}_{s}(\overline{\ensuremath{\rho}})=a(\overline{\ensuremath{\rho}})/\overline{\ensuremath{\rho}}$.

Topics & Concepts