Generic power laws in higher-dimensional lattice models with multidirectional hopping

TL;DR

This work shows that conserved-mass lattice models in dimensions $d>1$ with multidirectional hopping and full lattice symmetries exhibit generic scale invariance, producing power-law spatial correlations $C^{mm}({\bf r}) \sim 1/r^{d+2}$ far from any critical point. A fluctuating hydrodynamics framework relates the static structure factor to a density-dependent diffusion coefficient and an Onsager mobility matrix, yielding $S({\bf q}) \simeq S_0(\bar{\rho}) - S_1(\bar{\rho})|{\bf q}|^2 + S_2(\bar{\rho}) \frac{\sum_{\alpha} q_{\alpha}^4}{|{\bf q}|^2}$, with $S_0,S_1,S_2$ expressed in terms of $D$ and the coefficients $\gamma_0,\gamma_1,\gamma_2$. The paper provides exact microscopic calculations of the Onsager matrix for specific models (e.g., MCM I and Oslo model) and shows that $\gamma_1\neq\gamma_2$ drives the PL decay, while $\gamma_1=\gamma_2$ yields SR; in center-of-mass conserving cases, hyperuniformity can coexist with long-range correlations. The results unify a broad class of mass-transport processes and clarify how multidirectional hopping generates scale-invariant structures in equilibrium-like lattice systems.

Abstract

We show that, on a $d-$dimensional hypercubic lattice with $d>1$, conserved-mass transport processes, with {\it multidirectional} hopping that respect all symmetries of the lattice, exhibit power-law correlations for generic parameter values $-$ even {\it far} from phase transition point, if any. The key idea for generating the algebraic decay is the notion of {\it multidirectional} hopping, which means that several chunks of masses, or several particles, can hop out simultaneously from a lattice site in multiple directions, consequently breaking detailed balance. Notably, the systems we consider are described by a continuous-time Markov process, are diffusive, {\it lattice-rotation symmetric}, spatially homogeneous and thus have {\it no} net mass current. Using hydrodynamic and exact microscopic theory, we show that, for spatial dimensions $d > 1$, the steady-state static density-density and ``activity''-density correlation functions in the thermodynamic limit typically decay as $\sim 1/r^{(d+2)}$ at large distance $r=|{\bf r}|$; the strength of the power law is exactly calculated for several models and expressed in terms of the density-dependent bulk-diffusion coefficient and Onsager matrix (or, mobility tensor). In particular, our theory explains why center-of-mass-conserving dynamics, used to model novel disordered {\it hyperuniform} state of matter, result in generic long-ranged correlations. However, in a restricted parameter regime, the correlations can also be short ranged and are characterized through the Onsager matrix.

Figures (7)

• Figure 1: Schematic representation of dynamical update rules in mass transport processes in two-dimensional space for two variants: (a) MCM I and (b) MCM II. The arrows indicate the direction of mass transfer from the central site $(i,j)$ to its neighbors, with color-coded expressions representing the corresponding fractions of masses transferred during the particular update. Panel (a): In MCM I, mass is distributed simultaneously to all four nearest neighbors along both $x-$ and $y-$ axes. Panel (b): On the other hand, in MCM II, mass is transferred to only two neighboring sites along either horizontal (solid line) or vertical (dotted line) axis, with equal probability. For other variants (not shown here), when center of mass (CoM) is conserved, the two chunks of masses transferred from site $(i,j)$, in $+$ve and $-$ve directions along a particular axis, are the same; the respective variants are referred to as MCM-CoMC I and MCM-CoMC II.
• Figure 2: MCM I: The spatial range of fluctuating current strength is illustrated both along the same direction ($\alpha=x$) (panel (a)) and the perpendicular direction ($\alpha =x$ and $\beta =y$) (panels (b) and (c)) for different nearest-neighbor mass transport processes of the model in $d=2$ considered here. In both panels (a) and (b), the same color represents the same magnitude and sign: red for positive and green for negative values.
• Figure 3: The scaled variance of space-time-integrated currents, $\langle [\sum_{\alpha, \textbf{r}}{\cal Q}_\alpha(\textbf{r}, T)]^2 \rangle/(VTd)$ is plotted as a function of global density $\bar{\rho}$ for MCM I (blue cross), MCM II (orange star), and centre-of-mass conserving models - Manna CoMC and Oslo CoMC (green plus). Points are obtained from simulations, and dotted lines represent $\gamma_0(\bar{\rho})$ [as in Eq. \ref{['eq:2chi_gamma_sm']}; the analytic expressions of $\gamma_0(\bar{\rho})$ are given in Table \ref{['tab:Gam_bq_table']} for respective models].
• Figure 4: The range of $B(\textbf{r})$ for various nearest-neighbor mass-transfer rules considered in $d=2$ dimensions in this paper (see Eq. \ref{['eq:Br_rspace']}) is analyzed. The quantity $B(\textbf{r})$ is strictly localized in the sense that it is nonzero on a finite plaque but zero otherwise. Panels (a) (e.g., SSEP, ZRP), (b) (e.g., MCM II, Manna CoMC, Oslo; see Eq. \ref{['eq:BR_oslo']}), and (d) (e.g., MCM I; see Eq. \ref{['eq:BR_MCMI']}) illustrate the range of $B(\textbf{r})$ for different models. In panel (c), mass is transported along two randomly chosen perpendicular directions. The same color in each panel represents the same magnitude and sign, as the lattice reflection symmetry $B(\textbf{r})=B(-\textbf{r})$ holds for the models discussed here.
• Figure 5: In Panels (a)-(c) and (d)-(f), the quantity $\mathcal{B}(\textbf{q})$ and scaled structure factor $2D S(\textbf{q})$, respectively, as defined in Eq. \ref{['eq:sq']}, are plotted for three representative sets of the Onsager transport coefficients (equivalently, the mobility tensor) $\gamma$'s as defined in Eqs. \ref{['eq:Gqaa']} and \ref{['eq:Gqab']}. Panels (a) and (d) are for SSEP- or ZRP-like systems (delta-correlated) where $\gamma_0 \neq 0$ and $\gamma_1 = \gamma_2 =0$. Panels (b) and (e) represent the cases where $\gamma_0 = 0$ and $\gamma_1 = \gamma_2\neq 0$; these systems exhibit short-ranged correlations. Panels (c) and (f) illustrate the cases where $\gamma_0 = 0$ and $\gamma_1 > \gamma_2=0$; these systems exhibit power-law correlations $\sim 1/r^{d+2}$.