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We study a paradigmatic model of absorbing-phase transition - the Oslo model- on a one-dimensional ring of $L$ sites with a fixed global density$\bar{\rho}$; notably, microscopic dynamics conserve both mass and {\it centerof mass (CoM), but lacks time-reversal symmetry}. Despite having highlyconstrained dynamics due to CoM conservation, the system exhibits diffusiverelaxation away from criticality and superdiffusive relaxation nearcriticality. Furthermore, the CoM conservation severely restricts particlemovement, rendering the mobility to vanish exactly. Indeed the temporal growthof current fluctuation is qualitatively different from that observed indiffusive systems with a single conservation law. Away from criticality,steady-state fluctuation $\langle \mathcal{Q}_i^2(T,\Delta) \rangle$ of current$\mathcal{Q}_i$ across $i$th bond up to time $T$ {\it saturates} as $\langle\mathcal{Q}_i^2 \rangle \simeq \Sigma_Q^2(\Delta) - {\rm const.} T^{-1/2}$;near criticality, it grows subdiffusively as $\langle \mathcal{Q}_i^2 \rangle\sim T^\alpha$, with $0 < \alpha < 1/2$, and eventually {\it saturates} to$\Sigma_Q^2(\Delta)$. The asymptotic current fluctuation $\Sigma_Q^2(\Delta)$is a {\it nonmonotonic} function of $\Delta$: It diverges as$\Sigma_Q^2(\Delta) \sim \Delta^2$ for $\Delta \gg \rho_c$ and$\Sigma_Q^2(\Delta) \sim \Delta^{-\delta}$, with $\delta > 0$, for $\Delta \to0^+$. By using a mass-conservation principle, we exactly determine theexponents $\delta = 2(1-1/\nu_\perp)/\nu_\perp$ and $\alpha = \delta/z\nu_\perp$ via the correlation-length and dynamic exponents, $\nu_\perp$ and$z$, respectively. Finally, we show that, in the steady state, theself-diffusion coefficient $\mathcal{D}_s(\bar{\rho})$ of tagged particles isconnected to activity by $\mathcal{D}_s(\bar{\rho}) = a(\bar{\rho}) /\bar{\rho}$.

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