Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter1-6, non-equilibrium systems7-9, networks of neurons10,11, social groups with conformist and contrarian members12, directional interface growth phenomena13-15 and metamaterials16-20. Although wave propagation in non-reciprocal media has recently been closely studied1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points21. We describe the emergence of these phases using insights from bifurcation theory22,23 and non-Hermitian quantum mechanics24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.

Non-reciprocal phase transitions.

Active matter encompasses different nonequilibrium systems in which individual constituents convert energy into non-conservative forces or motion at the microscale. This review provides an elementary introduction to the role of topology in active matter through experimentally relevant examples. Here, the focus lies 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 whose robustness can be traced to a common mathematical origin: the presence of topological invariants. These invariants are typically integer numbers that cannot be changed by continuous deformations of the relevant order parameters or physical parameters of the underlying medium. We first explain the mechanisms whereby topological defects self propel and proliferate in active nematics, leading to collective states which can be manipulated by geometry and patterning. Possible implications for active microfluidics and biological tissues are presented. We then illustrate how the propagation of waves in active fluids and solids is affected by the presence of topological invariants characterizing their dispersion relations. We discuss the relevance of these ideas for the design of robotic metamaterials and the properties of active granular and colloidal systems. Open theoretical and experimental challenges are presented as future research prospects.

pdf/topological-active-matter-3pbg45jcmv.pdf

Topological active matter

Hooke's law states that the forces or stresses experienced by an elastic object are proportional to the applied deformations or strains. The number of coefficients of proportionality between stress and strain, i.e., the elastic moduli, is constrained by energy conservation. In this Letter, we lift this restriction and generalize linear elasticity to active media with non-conservative microscopic interactions that violate mechanical reciprocity. This generalized framework, which we dub odd elasticity, reveals that two additional moduli can exist in a two-dimensional isotropic solid with active bonds. Such an odd-elastic solid can be regarded as a distributed engine: work is locally extracted, or injected, during quasi-static cycles of deformation. Using continuum equations, coarse-grained microscopic models, and numerical simulations, we uncover phenomena ranging from activity-induced auxetic behavior to wave propagation powered by self-sustained active elastic cycles. Besides providing insights beyond existing hydrodynamic theories of active solids, odd elasticity suggests design principles for emergent autonomous materials.

Odd elasticity

Fluids in which both time reversal and parity are broken can display a dissipationless viscosity that is odd under each of these symmetries. Here, we show how this odd viscosity has a dramatic effect on topological sound waves in fluids, including the number and spatial profile of topological edge modes. Odd viscosity provides a short-distance cutoff that allows us to define a bulk topological invariant on a compact momentum space. As the sign of odd viscosity changes, a topological phase transition occurs without closing the bulk gap. Instead, at the transition point, the topological invariant becomes ill defined because momentum space cannot be compactified. This mechanism is unique to continuum models and can describe fluids ranging from electronic to chiral active systems.

https://purehost.bath.ac.uk/ws/files/190307767/PhysRevLett.122.128001_2019.pdf

Topological Waves in Fluids with Odd Viscosity.

Solids built out of active components can exhibit nonreciprocal elastic coefficients that give rise to non-Hermitian wave phenomena. Here, we investigate non-Hermitian effects present at the boundary of two-dimensional active elastic media obeying two general assumptions: their microscopic forces conserve linear momentum and arise only from static deformations. Using continuum equations, we demonstrate the existence of the non-Hermitian skin effect in which the boundary hosts an extensive number of localized modes. Furthermore, lattice models reveal non-Hermitian topological transitions mediated by exceptional rings driven by the activity level of individual bonds.

Non-Hermitian Band Topology and Skin Modes in Active Elastic Media.

In simple fluids, such as water, invariance under parity and time-reversal symmetry imposes that the rotation of constituent ‘atoms’ is determined by the flow and that viscous stresses damp motion. Activation of the rotational degrees of freedom of a fluid by spinning its atomic building blocks breaks these constraints and has thus been the subject of fundamental theoretical interest across classical and quantum fluids. However, the creation of a model liquid that isolates chiral hydrodynamic phenomena has remained experimentally elusive. Here, we report the creation of a cohesive two-dimensional chiral liquid consisting of millions of spinning colloidal magnets and study its flows. We find that dissipative viscous ‘edge-pumping’ is a key and general mechanism of chiral hydrodynamics, driving unidirectional surface waves and instabilities, with no counterpart in conventional fluids. Spectral measurements of the chiral surface dynamics suggest the presence of Hall viscosity, an experimentally elusive property of chiral fluids. Precise measurements and comparison with theory demonstrate excellent agreement with a minimal chiral hydrodynamic model, paving the way for the exploration of chiral hydrodynamics in experiment. A chiral fluid comprising spinning colloidal magnets exhibits macroscopic dynamics reminiscent of the free surface flows of Newtonian fluids, together with unique features suggestive of Hall—or odd—viscosity.

The odd free surface flows of a colloidal chiral fluid

In simple fluids, such as water, invariance under parity and time-reversal symmetry imposes that the rotation of constituent 'atoms' are determined by the flow and that viscous stresses damp motion. Activation of the rotational degrees of freedom of a fluid by spinning its atomic building blocks breaks these constraints and has thus been the subject of fundamental theoretical interest across classical and quantum fluids. However, the creation of a model liquid which isolates chiral hydrodynamic phenomena has remained experimentally elusive. Here we report the creation of a cohesive two-dimensional chiral liquid consisting of millions of spinning colloidal magnets and study its flows. We find that dissipative viscous edge pumping is a key and general mechanism of chiral hydrodynamics, driving uni-directional surface waves and instabilities, with no counterpart in conventional fluids. Spectral measurements of the chiral surface dynamics reveal the presence of Hall viscosity, an experimentally long sought property of chiral fluids. Precise measurements and comparison with theory demonstrate excellent agreement with a minimal but complete chiral hydrodynamic model, paving the way for the exploration of chiral hydrodynamics in experiment.

The free surface of a colloidal chiral fluid: waves and instabilities from odd stress and Hall viscosity

The competition between thermal fluctuations and potential forces is the foundation of our understanding of phase transitions and matter in equilibrium Driving matter out of equilibrium allows for a new class of interactions which are neither attractive nor repulsive but transverse The existence of such transverse forces immediately raises the question of how they interfere with basic principles of material self-organization Despite a recent surge of interest, this question remains open Here, we show that activating transverse forces by homogeneous rotation of colloidal units generically turns otherwise quiescent solids into a crystal whorl state dynamically shaped by self-propelled dislocations Simulations of both a minimal model and a full hydrodynamics model establish the generic nature of the chaotic dynamics of these self-kneading polycrystals Using a continuum theory, we explain how odd and Hall stresses conspire to destabilize chiral crystals from within This chiral instability produces dislocations that are unbound by their self-propulsion Their proliferation eventually leads to a crystalline whorl state out of reach of equilibrium matter

Chiral crystals self-knead into whorls

The competition between thermal fluctuations and potential forces is the foundation of our understanding of phase transitions and matter in equilibrium. Driving matter out of equilibrium allows for a new class of interactions which are neither attractive nor repulsive but transverse. The existence of such transverse forces immediately raises the question of how they interfere with basic principles of material self-organization. Despite a recent surge of interest, this question remains open. Here, we show that activating transverse forces by homogeneous rotation of colloidal units generically turns otherwise quiescent solids into a crystal whorl state dynamically shaped by self-propelled dislocations. Simulations of both a minimal model and a full hydrodynamics model establish the generic nature of the chaotic dynamics of these self-kneading polycrystals. Using a continuum theory, we explain how odd and Hall stresses conspire to destabilize chiral crystals from within. This chiral instability produces dislocations that are unbound by their self-propulsion. Their proliferation eventually leads to a crystalline whorl state out of reach of equilibrium matter.

Motile dislocations knead odd crystals into whorls.

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The odd free surface flows of a colloidal chiral fluid

The free surface of a colloidal chiral fluid: waves and instabilities from odd stress and Hall viscosity

Chiral crystals self-knead into whorls

Motile dislocations knead odd crystals into whorls.