The chiral random walk: A quantum-inspired framework for odd diffusion

Abstract

Chirality in active and passive fluids gives rise to odd transport properties, most notably the emergence of robust edge currents that defy standard dissipative dynamics. While these phenomena are well-described by continuum hydrodynamics, a microscopic framework connecting them to their topological origins has remained elusive. Here, we present a lattice model for an isotropic chiral random walk that bridges the gap between classical stochastic diffusion and unitary quantum evolution. By equipping the walker with an internal degree of freedom and a tunable chirality parameter, $p$, we interpolate between a standard diffusive random walk and a deterministic, topologically non-trivial quantum walk. We show that the topological protection characteristic of the unitary limit ($p=1$) remarkably persists into the dissipative regime ($p<1$). This correspondence allows us to theoretically ground the robustness of edge flows in classical chiral systems using the bulk-boundary correspondence of Floquet topological insulators. Our results provide a discrete microscopic description for odd diffusion, offering a powerful toolkit to predict transport in confined geometries and disordered chiral media.

Figures (7)

• Figure 1: Conceptual scheme of the chiral random walk model. A charged Brownian particle in a magnetic field experiences Lorentz forces that induce chiral motion. Its diffusive motion, however, cannot be captured by a standard random walk model on a lattice. To model such systems, we introduce an internal degree of freedom (IDF), represented schematically as floors in a building. The time evolution then decomposes into two sequential operations: the coin operation, which acts on the IDF, similar to a lift between the floors, while the step operation shifts the particle's position on the lattice according to which floor, i.e., which IDF state ($\rightarrow, \leftarrow, \uparrow, \downarrow$), it currently has. This structure enables the model to incorporate passive chirality, resulting in a probability distribution that exhibits not only standard gradient flux but also rotational flux, the hallmark of odd diffusion that cannot be reproduced by classical random walks.
• Figure 2: MSD of the chiral random walk. Double-logarithmic plot showing the time dependence of the mean-squared displacement for different values of the chirality parameter $p$. The dashed red lines indicate the analytical prediction of Eq. \ref{['msd']}, demonstrating excellent agreement with numerical simulations. The inset displays typical trajectories for the standard random walk (blue, $p = 0$), the CRW with intermediate chirality (orange, $p = 0.5$), and the deterministic chiral walk (red, $p = 1$). While all cases ($p<1$) exhibit Brownian scaling $\text{MSD} \propto t$, the bare diffusion coefficient decreases with increasing $p$ as given by Eq. \ref{['diff_chiral_random_walk']}. The $p=1$ limit, however, is deterministic and no Brownian motion is observed.
• Figure 3: Emergence of topological edge modes in the chiral random walk. (A, B): Heatmaps of probability distribution after 300 steps on a square lattice with reflective boundaries, starting from a near-edge position. (A) Standard random walk ($p = 0$) shows symmetric diffusive spreading. (B) CRW ($p = 0.7$) exhibits pronounced accumulation along the boundary and a characteristic teardrop shape, with the handedness determined by the sign of $p$. This edge-following behaviour, absent in the bulk, emerges from the antisymmetric odd-diffusion tensor and persists robustly against perturbations. (C) Numerically obtained quasi-energy spectrum for the CRW in the deterministic limit $p = 1$ with reflective boundary conditions. The bulk spectrum (orange) is gapped, while boundary-localized states (blue) close this gap, consistent with the bulk-boundary correspondence for Floquet topological insulators. These edge states are absent under periodic boundary conditions, confirming their topological origin.
• Figure 4: Edge and bulk states fidelity decay. Double-logarithmic plot of the fidelity between the initial probability distribution and the distribution after $t$ steps of CRW evolution with $p = 0.9$ on a two-dimensional grid with reflective boundary conditions (system size $L = 500$). We compare two scenarios: an initial state taken as an edge eigenvector of the $p = 1$ evolution operator (orange) and a bulk eigenvector (blue). Edge states decay as $\simeq t^{-1/2}$, characteristic of effectively one-dimensional diffusion, while bulk states decay faster as $\simeq t^{-1}$, reflecting two-dimensional spreading. Red stars mark the transition from exponential to polynomial decay. Topologically protected edge states remain long-lived even in the dissipative regime $p < 1$. The inset shows the marginal probability distribution summed along one lattice direction (system size $L = 25$ for visibility) after 100 steps for illustration purposes of the initial states.
• Figure 5: Robust topological edge modes in chiral random walk with spatially disordered chirality parameter. Heatmaps of probability distribution after $300$ steps of the CRW on a square lattice, where the chirality parameter $p$ is randomly chosen from the interval $[0.1, 0.9]$ at each lattice site. Initial conditions are (A) near-edge position and (B) centre position. Despite the strong spatial disorder in $p$, the bulk behaviour remains diffusive and qualitatively unchanged, while edge currents show robustness against local variations in chirality strength, consistent with the topological protection hypothesis.