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Topological active matter

Abstract

In active matter systems, individual constituents convert energy into non-conservative forces or motion at the microscale, leading to morphological features and transport properties that do not occur in equilibrium and that are robust against certain perturbations. In recent years, a fruitful method for analysing these features has been to use tools from topology. In this Review, we focus on topological defects and topologically protected edge modes, with an emphasis on the distinctive properties they acquire in active media. These paradigmatic examples represent two physically distinct classes of phenomena that are robust thanks to a common mathematical origin: the presence of topological invariants. Beyond active matter, our Review underscores the role of topological excitations in non-equilibrium settings of relevance, from open quantum systems to living matter.

Key points

  • Topology plays a defining role in understanding robust features in active media whose basic constituents convert energy into non-conservative forces and motion.

  • Topological defects in active media can acquire self-propulsion and non-reciprocal interactions.

  • Local stresses and flows generated by active defects can have biological functionality in living systems.

  • When detailed balance is broken, unidirectional density waves emerge that are protected against scattering by the presence of topological invariants in the band structure of the media.

  • Non-Hermitian band theory naturally arises in active materials because energy is both consumed and dissipated, resulting in the presence of skin modes and odd viscoelasticity.

  • The full potential of these ideas extends from the fundamental understanding of topology in non-equilibrium systems to applications including materials design and tissue mechanics.

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Fig. 1: Dynamics of defects under confinement.
Fig. 2: Controlling and patterning defects.
Fig. 3: Topological edge states in fluids far from equilibrium.
Fig. 4: Topology and exceptional points in active and robotic metamaterials.

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Acknowledgements

The work of M.J.B. was supported in part by the National Science Foundation under grant no. NSF PHY-1748958 and the Designing Materials to Revolutionize and Engineer our Future (DMREF) programme, via grant no. DMREF-1435794. M.C.M. was primarily supported by the National Science Foundation under grant no. DMR-2041459, with additional support from DMR-1720256 (iSuperSeed). S.S. is supported by the Harvard Society of Fellows. V.V. was supported by the Complex Dynamics and Systems programme of the Army Research Office under grant W911NF-19-1-0268, by the Simons Foundation and by the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation under award number DMR-2011854. S.S. and A.S. gratefully acknowledge discussions during the 2019 summer workshop on ‘Active and Driven Matter: Connecting Quantum and Classical Systems’ at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. The participation of A.S. at the Aspen Center for Physics was supported by the Simons Foundation. A.S. acknowledges the support of the Engineering and Physical Sciences Research Council (EPSRC) through New Investigator Award no. EP/T000961/1 and of the Royal Society under grant no. RGS/R2/202135. The authors also acknowledge illuminating discussions throughout the virtual 2020 KITP programme on ‘Symmetry, Thermodynamics and Topology in Active Matter’, which was supported in part by the National Science Foundation under grant no. NSF PHY-1748958. The authors thank M. Fruchart, C. Scheibner, G. Baardink and J. Binysh for their inspiring conversations and suggestions.

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Glossary

Reciprocity

The symmetry between perturbation and response.

Detailed balance

The symmetry between the past and the future within the dynamics of microscopic processes.

Circulators

A ring in which air is constantly moved by a fan.

Metamaterials

Materials with properties arising from their macroscopic structure, rather than their chemical constituents. A simple example is a ‘holey sheet’: a slab of rubber with holes that have size and shape tailored to achieve a specific mechanical response.

Colloidal rollers

Micron-sized dielectric spheres suspended in an ionic solvent that can undergo an electrohydrodynamic instability (Quincke instability), causing the spheres to spontaneously roll upon the application of a strong enough DC electric field.

Smectics

Describes a type of liquid crystal in which molecules order in periodic layers; each layer behaves like a fluid in its plane.

Focal conic domains

Characteristic defects seen in smectic liquid crystals. They occur when equidistant layers of the smectic form geometric structures consisting of nested surfaces generated by conic sections, the foci of which lie on a curve given by the conjugate conic section.

Poincaré–Hopf theorem

A theorem in differential geometry and topology (also colloquially called the ‘hairy ball theorem’) that relates the number of zeros of a tangential vector field on a closed surface to the Euler characteristic of the same surface.

Bend and splay elastic constants

Material constants of a liquid crystal that quantify the energy cost of distorting orientational order through bend or splay deformations, respectively.

Morphogenesis

The process by which biological tissues, organs and organisms acquire their distinct shapes over the course of development.

Actomyosin

A complex of biopolymer filaments called actin, molecular motors called myosin and associated proteins. Actomyosin is commonly found in the cytoskeleton and cortex of cells, and is responsible for generating contraction, particularly in muscle.

Myoblasts

A type of embryonic stem cell that gives rise to muscle cells.

Mitotic spindle

A self-assembled cytoskeletal structure, consisting largely of stiff biopolymers called microtubules and a host of molecular motors and proteins, that plays a key role in eukaryotic cell division for segregating chromosomes to the two daughter cells.

Actin treadmilling

A dynamic process relevant to cell motility and crawling, by which cytoskeletal filaments such as actin get continually disassembled at one end, while monomer units are added at the other end.

Oocytes

Immature egg cells or germ cells involved in sexual reproduction.

Bulk gap

The region of frequency space where bulk modes do not exist.

Galilean invariance

The principle that constant boosts in velocity leave the system unchanged.

Dynamical matrix

In the linear approximation, the dynamical matrix Dij defines the potential energy V of a solid as the quadratic form \(V=\frac{1}{2}{\sum }_{ij}{u}_{i}{D}_{ij}{u}_{j}\), where ui are particle displacements and the i and j indices run over all dN degrees of freedom for an N-particle system in d dimensions.

Hermitian

A matrix D is Hermitian if D = D. The † denotes a conjugate transpose, \({({D}^{\dagger })}_{ij}={D}_{ji}^{* }\). A matrix D is anti-Hermitian if D = −D.

Advection

The transport of matter and other quantities, such as momentum, temperature or concentration, by the bulk motion of a fluid.

Gauge fields

Terms that appear in the definition of an objective and covariant derivative (akin to the vector potential in electromagnetism) that capture how specific fields or order parameters transform under the action of local symmetries.

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Shankar, S., Souslov, A., Bowick, M.J. et al. Topological active matter. Nat Rev Phys 4, 380–398 (2022). https://doi.org/10.1038/s42254-022-00445-3

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