Hyperuniformity in mass transport processes with center-of-mass conservation: Some exact results

We characterize steady-state static and dynamic properties in a broad class of mass transport processes on a periodic hypercubic lattice of volume $L^d$, where both mass and {\it center-of-mass} (CoM) remain conserved and detailed balance is violated in the bulk; we specifically consider these models in $d=1$ and $2$ dimensions. Using a microscopic approach, we exactly determine the decay (or, growth) exponents for various dynamic and static correlation functions. We show that, despite constrained dynamics due to the CoM conservation (CoMC), the density relaxation is indeed diffusive. However, fluctuation properties are strikingly different from that in the diffusive systems with a single (mass) conservation law. In the thermodynamic limit, the steady-state variance $\langle {\cal Q}^2(T) \rangle_c$ of time-integrated bond current ${\cal Q}(T)$ across a bond in time interval $T$ exhibits the following long-time behavior: $\langle {\cal Q}^2(T) \rangle_c \simeq A_1 T + A_2 + A_3 T^{-d/2}$. Remarkably, depending on dimensions and microscopic details, the prefactor $A_1$ can vanish (e.g., for $d=1$), causing the variance to eventually {\it saturate}. The exponents governing the small-frequency behavior of the power spectrum $S_J(f) \sim f^{\psi_J}$ for bond current are exactly determined as $\psi_J=3/2$ and $2$ in $d=1$ and $2$ dimensions, respectively, implying a ``dynamic hyperuniformity''. We also compute the static structure factor $S(q)$, which, in the small-$q$ limit, varies as the square of wave number $q$, i.e., $S(q) \sim q^2$. Indeed, both dynamic and static fluctuations are anomalously suppressed, resulting in an extreme form of (``class I'') hyperuniformity in the systems.