Hermitian and non-Hermitian topology in active matter

Self-propulsion is a quintessential aspect of biological systems, which can induce nonequilibrium phenomena that have no counterparts in passive systems. Motivated by biophysical interest together with recent advances in experimental techniques, active matter has been a rapidly developing field in physics. Meanwhile, over the past few decades, topology has played a crucial role to understand certain robust properties appearing in condensed matter systems. For instance, the nontrivial topology of band structures leads to the notion of topological insulators, where one can find robust gapless edge modes protected by the bulk band topology. We here review recent progress in an interdisciplinary area of research at the intersection of these two fields. Specifically, we give brief introductions to active matter and band topology in Hermitian systems, and then explain how the notion of band topology can be extended to nonequilibrium (and thus non-Hermitian) systems including active matter. We review recent studies that have demonstrated the intimate connections between active matter and topological materials, where exotic topological phenomena that are unfeasible in passive systems have been found. A possible extension of the band topology to nonlinear systems is also briefly discussed. Active matter can thus provide an ideal playground to explore topological phenomena in qualitatively new realms beyond conservative linear systems.

💡 Research Summary

This review article surveys the rapidly emerging interdisciplinary field that lies at the intersection of active matter physics and band topology, with a particular focus on the distinction between Hermitian and non‑Hermitian (NH) topological phenomena. The authors begin by outlining the defining characteristics of active matter—self‑propelled particles that continuously consume energy and generate non‑equilibrium flows. Classic models such as the Vicsek model, run‑and‑tumble dynamics of bacteria, and the Toner‑Tu continuum theory are presented as the foundational frameworks for describing flocking, bacterial turbulence, and collective cell migration. These models highlight how activity introduces intrinsic dissipation, non‑reciprocity, and nonlinear feedback, which are absent in conventional equilibrium systems.

The manuscript then revisits the essentials of Hermitian band topology, summarizing the quantum Hall effect, Chern numbers, symmetry‑protected topological insulators, and higher‑order phases. It emphasizes that while these concepts were originally developed for electronic systems, they have been successfully transplanted to classical wave platforms such as photonic crystals, acoustic metamaterials, and mechanical lattices, where loss and gain can be engineered in a controlled manner.

A substantial portion of the review is devoted to non‑Hermitian band topology. The authors explain that the complex eigenvalue spectra of NH Hamiltonians require new gap definitions—point gaps and line gaps—leading to a richer classification scheme. They discuss hallmark NH phenomena: the non‑Hermitian skin effect, where an extensive number of bulk states accumulate at system boundaries; exceptional points (EPs), where eigenvectors coalesce and give rise to protected edge modes; and the recently formulated non‑Bloch band theory, which provides a systematic way to compute NH topological invariants (e.g., complex Chern numbers) by redefining Brillouin‑zone concepts in the complex plane.

Connecting these ideas to active matter, the review highlights several concrete examples. Periodic arrays of pillars or passive rods immersed in bacterial suspensions generate effective NH lattices that exhibit skin‑localized modes analogous to the SSH model, thereby realizing an “active topological insulator.” Similarly, the asymmetric stress tensors that arise in Toner‑Tu hydrodynamics produce NH skin effects and EP‑driven edge waves in two‑dimensional active fluids. The authors also discuss how active turbulence can be tamed by structured environments, enabling the observation of topologically protected sound or density waves in otherwise chaotic flows.

In the outlook, the authors identify three promising directions. First, the experimental realization of NH topological phases in three‑dimensional active systems, which would require sophisticated control over activity gradients and boundary conditions. Second, the extension of topological concepts to genuinely nonlinear regimes, where amplitude‑dependent band structures and dynamical bifurcations could host “nonlinear topological solitons.” Third, the exploration of real‑space topological defects (e.g., disclinations in cell monolayers) that may couple to activity‑induced stresses, potentially linking topological protection to biological functions such as tissue morphogenesis or wound healing.

Overall, the review convincingly argues that active matter provides a natural laboratory for non‑Hermitian topological physics, offering experimental accessibility, tunable dissipation, and intrinsic nonlinearity. By bridging these domains, the work opens pathways toward designing metamaterials with unprecedented robustness, understanding biological pattern formation through a topological lens, and discovering new phases of matter that exist only far from equilibrium.