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Hermitian and non-Hermitian topology in active matter

Kazuki Sone, Kazuki Yokomizo, Kyogo Kawaguchi, Yuto Ashida

Abstract

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.

Hermitian and non-Hermitian topology in active matter

Abstract

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.
Paper Structure (44 sections, 70 equations, 23 figures)

This paper contains 44 sections, 70 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Active matter
  3. Self-propelled particles
  4. Flocking
  5. Cells and bacteria
  6. Molecular motors
  7. Other systems
  8. Biochemical network
  9. Active hydrodynamics
  10. Toner-Tu model
  11. From micro to hydrodynamic models
  12. Hydrodynamics of various self-propelled particles
  13. Hermitian band topology
  14. Band theory and its classical counterpart
  15. Hermitian topology
  16. ...and 29 more sections

Figures (23)

  • Figure 1: Examples of biological active matter. (a) School of fish is a typical example of active matter. Reprinted from Rep. Phys., 517, T. Vicsek and A. Zafeiris, "Collective motion," 71--140, Copyright © (2012) Vicsek2012, with permission from Elsevier. (b) Bacteria exhibit active turbulence even in a low-Reynolds-number regime. This figure is adapted from H. H. Wensink et al., "Meso-scale turbulence in living fluids" Proc. Natl. Acad. Sci. USA 109, 14308--14313 (2012) Wensink2012. (c) The collective dynamics of cells is governed by their topological defects. This figure is adapted from T. B. Saw et al., "Topological defects in epithelia govern cell death and extrusion" Nature 544, 212--216 (2017) Saw2017, Springer Nature. Copyright © 2017. (d) A schematic and a snapshot of actin filaments are shown. This figure is adapted from V. Schaller et al., "Polar patterns of driven filaments" Nature 467, 73--77 (2010) Schaller2010, Springer Nature. Copyright © 2010.
  • Figure 2: (a) Four-state model of a chemical compound. The compound is composed of two types of monomers and takes four internal states labeled A, B, C, and D. If these reactions can be regarded as a single molecule reaction, the rates of reactions are proportional to the concentration of reactants, whose coefficients are $\gamma_{\rm in}$ or $\gamma_{\rm ex}$. (b) Schematic of the reaction network of the four-state model. The blue square shows the unit cell of this model and the numbers show those of monomers attached to the chemical compound at each state. These figures are adapted from E. Tang, J. Agudo-Canalejo, and R. Golestanian, "Topology Protects Chiral Edge Currents in Stochastic Systems" Phys. Rev. X 11, 031015 (2021) Tang2021, licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).
  • Figure 3: (a) Classification of active matter by its symmetry and interaction. Polar active matter shows a directional motion and polar (i.e., ferromagnetic) interaction. Self-propelled rods also exhibit directional motions, while they only have aliging (nematic) interaction. There also exists apolar active matter that can move both forward and backward. (b) Chiral active particles. The chirality of active matter can be induced by self-rotation and/or curved motions.
  • Figure 4: Quantum Hall effect. (a) Schematic of the quantum Hall effect. Cycrotolon motions of electrons are prevented at the edge of the sample, and instead, the skipping orbit can emerge, which induces the chiral edge current. (b) Experimental results of the quantized Hall voltage. The inset shows the top view of the device where the quantized Hall conductance was first observed. Panel (b) is adapted from K. v. Klitzing, G. Dorda, and M. Pepper, "New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance" Phys. Rev. Lett. 45, 494 (1980) Klitzing1980, Copyright © 1980 by the American Physical Society.
  • Figure 5: Haldane's honeycomb model. (a) Schematic of the Haldane model is shown. The arrow represents the direction of the gauge field, i.e., the vector potential. (b) The phase diagram of the Haldane model. $\nu$ corresponds to the Chern number in each phase. The panels (a) and (b) are adapted from F. D. M. Haldane, "Model for a Quantum Hall Effect without Landau Levels; Condensed-Matter Realization of the "Parity Anomaly"' Phys. Rev. Lett. 61, 2015 (1988) Haldane1988, Copyright © 1988 by the American Physical Society. (c) We can obtain gapless edge modes at the zigzag boundary of the Haldane model.
  • ...and 18 more figures