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arXiv:2602.09920

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

Jan Wójcik, Erik Kalz

cond-mat.stat-mechcond-mat.softquant-phSubmitted Feb 10, 2026• Announced Feb 11, 2026
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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.

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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, ppp, 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=1p=1p=1) remarkably persists into the dissipative regime (p<1p<1p<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.
Discrete-time quantum walks (DTQW) have topological phases that are richer than those of time-independent lattice Hamiltonians. Even the basic symmetries, on which the standard classification of topological insulators hinges, have not yet been properly defined for quantum walks. We introduce the key tool of time frames, i.e., we describe a DTQW by the ensemble of time-shifted unitary time-step operators belonging to the walk. This gives us a way to consistently define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using an ``inversion symmetric'' pulse sequence. For one-dimensional DTQW's with CS, we identify the bulk $\mathbb{Z}\ifmmode\times\else\texttimes\fi{}\mathbb{Z}$ topological invariant that controls the number of topologically protected 0 and $\ensuremath{\pi}$ energy edge states at the interfaces between different domains, and give simple formulas for these invariants. We illustrate this bulk-boundary correspondence for DTQW's on the example of the ``4-step quantum walk,'' where tuning CS and particle-hole symmetry realizes edge states in various symmetry classes.
We introduce the discrete angular tetravalent lattice as a minimal classical geometric model. The system consists of a four-component density evolving under strict +90° cyclic permutations and conditional spatial shifts. We construct a minimal non-unitary quantum walk with complex amplitudes and prove that, under a full quantum dephasing channel, its intensity evolution exactly recovers the classical tetravalent process. By analysing the continuum limit of this decohered process we demonstrate a duality: in the passive regime the system yields a dissipative complex wave equation, while in the active regime the continuum dynamics acquire the algebraic skeleton of the two-dimensional Weyl operator ∂x + i∂y. Quantitative scaling tests confirm a wave-speed duality: the absolute information edge propagates at the hyperbolic light-cone speed c = 1, while the macroscopic density ridge propagates at c = 1/2. Furthermore, a spatial domain wall between opposite chiral cycles (A and Aᵀ) traps a robust one-dimensional chiral edge current driven by classical skipping orbits; the current remains stable under strong spatial disorder. We discuss the scope of this construction relative to phenomenological active quantum walks, clarifying the boundary between classical geometric mechanisms and fully coherent multi-sector quantum dynamics.
Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to the discrete nature of time evolution, the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries and identify the anomalous edge states protected by these symmetries.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
Chiral edge states in a quantum Hall effect are the paradigmatic example of a quasiparticle with chirality. In even space-time dimensions, the Nielsen-Ninomiya theorem strictly forbids chiral states in physical isolation. The exceptions to this theorem only occur in the presence of nonlocality, non-Hermiticity, or by embedding the system at the boundary of the higher-dimensional bulk. In this Letter, using the IBM quantum computer platform, we realize a Floquet chiral quantum walk enabled by nonlocality. The unitary time-evolution operator is described by an effective Floquet Hamiltonian with long-ranged coupling. We find that the chiral wave packets lack the common features of conventional wave phenomena such as localization. The absence of localization is witnessed by the robustness against external perturbations. However, the intrinsic quantum errors of the current quantum device give rise to a finite lifetime where the chiral wave packet eventually disperses in the long-time limit. Nevertheless, we observe the stability of the chiral wave by comparing it with a conventional nonchiral model.
Topological insulators are fascinating states of matter exhibiting protected edge states and robust quantized features in their bulk. Here, we propose and validate experimentally a method to detect topological properties in the bulk of one-dimensional chiral systems. We first introduce the mean chiral displacement, and we show that it rapidly approaches a multiple of the Zak phase in the long time limit. Then we measure the Zak phase in a photonic quantum walk, by direct observation of the mean chiral displacement in its bulk. Next, we measure the Zak phase in an alternative, inequivalent timeframe, and combine the two windings to characterize the full phase diagram of this Floquet system. Finally, we prove the robustness of the measure by introducing dynamical disorder in the system. This detection method is extremely general, as it can be applied to all one-dimensional platforms simulating static or Floquet chiral systems.