Emergent phenomena in chiral active matter

: In recent years, there has been growing interest in the study of chiral active materials, which consist of building blocks that show active dynamics featuring chiral symmetry breaking, e.g., particles that rotate in a common direction. These materials exhibit fascinating phenomena such as odd viscosity, odd di ﬀ usivity, active turbulence in ﬂuids, vivid dislocation dynamics or odd elasticity in crystals or elastic materials, and hyperuniform states. The systematic study of soft chiral active matter systems is relatively new, starting around 2017, but has already shown promising applications in robust cargo transport, segregation and mixing dynamics, or manipulation of metamaterials. In this review, we summarize recent experimental and theoretical advances in this ﬁeld, highlighting the emergence of anti-symmetric and odd stresses and ensuring e ﬀ ects such as odd viscosity or topologically protected edge modes. We further discuss the underlying mechanisms and provide insights into the potential of chiral active matter for various applications.


INTRODUCTION
Active matter systems are comprised of agents that are either externally actuated or have the ability to convert energy into forces or torques [1], resulting in an intrinsic motion [2,3].Prominent examples are living matter on the mesoscale that are composed of bacteria, sperm, and other organisms.On the other hand, different designs of synthetic active matter with constituent activated particles powered by mechanisms such as chemical reactions [4,5], laser illumination [6], externally applied electric [7] or magnetic fields [8] have become wide spread and a myriad of different approaches have been proposed.Resulting from the interagent interactions, various emergent phenomena such as collective dynamics or structure formation arise [2,9] and novel dynamic materials could be designed for applications ranging from medicines [10,11], display [12], to environment [13].To understand the nonequilibrium physics of mesoscale active matter is of central importance in order to decipher the complex processes of life and conclusively to develop strategies to comprehend and manipulate biological processes, such as cancer invasion [14][15][16][17], in vitro fertilisation [18], the formation of bio-films [19,20], or targeted drug delivery at the microscopic level [21][22][23][24].
In the last years, the focus in active matter studies has been gradually broadened and partially shifted to chiral active matter that are composed of a large number of agents showing active motion with chiral oppositely arranged Hexbug robots mounted on a foam disk (σ ≈ 5 cm) constitute a rotor on the centimeter scale.Reprinted with permission from [41].Copyright c ⃝2020 the American Physical Society.
symmetry breaking as, rotating or particles performing a circular motion [25,26].Rotations are abundant in biological systems crossing multiple length scales, ranging from rotating subunits to collective vortical motion [27].Examples (Figure 1A-E) are rotating motor proteins in membranes [28], such as ATP synthase [29], circular swimming algae [30], co-rotating bacteria [31], and starfish embryos [32], and bound states of Volvox colonies [33].Significant effort has been put in designing synthetic particles with spinning motion (Figure 1F-J), which includes magnetically driven synchronously spinning colloidal [34][35][36] or larger particles [37,38] with a ferromagnetic moment, light-driven asynchronously spinning colloids [39], shaking grains [40] and vibrating robots [41], expanding nearly five decades of length scales.Such rotating particles allow for the systematic involvement of the rotational degrees of freedom as a continuation of active dynamics that exclusively utilise the translational degrees of freedom.
The word chirality is derived from the Greek word χειρ (kheir), for "hand", which is abundant in nature, and is a basic and intrinsic characteristic of many natural and man-made systems [44][45][46][47].A key feature of chirality is that the mirror image of an object can not overlap with itself, with hand the most recognized example [48].For an object rotating in one direction, its mirror image would rotate into the opposite direction, bearing a particular symmetry such that the two are not equivalent.Active matter systems composed of agents that spin or rotate in a common direction are therefore said to be chiral [49].In addition, directed rotation in chiral active matter also breaks the invariance under time-reversal (t → −t) and parity (or coordinate mirror transformation, x → −x) transformations that underlie conventional fluids and solids, brought the system into nonequilibrium steady states with exotic collective phenomena and properties.The interactions between the active agents and the surrounding medium have been identified as the cause of the unique properties of chiral active matter.Disordered hyperuniform states have been observed in circularly swimming algae due to a combination of circular trajectories and repulsive interactions [30] (Figure 2A).Vortex formation is shown in electric-field agitated pear-like Quincke rollers, ascribed to hydrodynamic dissipative coupling/alignment of the particles' inherent rotation [50] (Figure 2B).Active turbulent behaviour emerges in a carpet of standing and rotating magnetic rods at low Reynolds numbers [34] where the rotating particles also exhibit translational motion resembling active self-propelled particles [51,52] due to mutual influence (Figure 2C).On the other hand, unidirectional waves along free surfaces (Figure 2D) [26,32], and melting, "kneading" of crystalline order and vivid dislocation dynamics (Figure 2E) are revealed in dense cohesive chiral active fluids [35] and crystals [36,43], respectively.Interestingly, self-sustaining chiral elastic waves are revealed in overdamped chiral active crystals self-assembled from thousands of swimming starfish embryos [32] ( Figure 2F).
All of these diverse phenomena can be explained with the concept of chiral active matter, where the internal stresses between the rotating units imply the emergence of anti-symmetric transport coefficients that are absent in usual non-chiral matter [53,54].Moreover, these anti-symmetric contributions are even forbidden by energy conservation in equilibrium systems [55].In soft matter physics, the effects of these odd (in the sense of not-even, or anti-symmetric) transport coefficients [56] first received theoretical attention in 2017 [53], which predicted odd viscosity in chiral active fluids and its effect on dissipationless flow and density-vorticity correlation.Experimentally, odd viscosity was first measured in 2019 in a chiral active fluid composed of cohesive spinning colloidal magnets, which displayed free surface flow [35].This has agitated considerable subsequent theoretical [57][58][59][60] and experimental [32,43,61] interests, offering insight into chiral active fluids featuring odd viscosity, but also complementary consequences such as odd diffusivity [62,63] or odd elasticity in chiral active elastic systems [55,64,65].Theoretical studies have also excited first studies on the physics of the swimming behaviour of particles swimming in chiral active baths [66,67], but also topics such as active rheology [68,69], or odd viscoelasticity [70] can be extended to chiral active systems.For the description of these diverse phenomenologies, different approaches have been employed that highlight the aspects under consideration.For a hydrodynamic or continuum description a generalisation of the Navier-Stokes equations, with coarse-grained active stresses that model the interactions between the chiral constituents can be employed [53,71].Such approaches are particularly helpful to describe the dynamics of Nature.(E) Magnetic colloidal spinners with significant magnetic attraction form rotating and "kneading" polycrystalline structures resulting from the combination of magnetic and hydrodynamic interactions.Reprinted with permission from [43].Copyright c ⃝2022 Springer Nature.
the associated vector fields like velocity or density.On the other hand, numerical or simulation studies are capable of focusing on the particular interactions among the chiral particles, which are transmitted when the particles are in touch [40,61,72] such that collective effects at higher densities can be studied where particle collisions are frequent.Alternatively, the interactions can also be incorporated using a hydrodynamic scheme, where an explicit integration of the solvent degrees of freedom allows for hydrodynamic chiral interactions between distant active particles [34] as is customary for colloidal systems, and consequently collective effects can also be studied at intermediate densities of the chiral colloids.Phenomenologically speaking, odd transport coefficients give rise to a response acting in the direction perpendicular to those of the even transport coefficients, e.g., odd shear stresses act perpendicular to the direction of applied shear [34] and odd diffusion spreads perpendicular to the density gradient [63].As a consequence, a system inherent correlation between density and vorticity appears in (weakly) compressible chiral active fluids, which was first predicted in theory [53,71,73], and was directly observed experimentally in our recent report [34], allowing a measurement of odd viscosity from the bulk.However, in complex systems with several even and odd contributions to the dynamics, the system behaviour can be much more complicated [74] and vastly different behaviours can be observed in diverse chiral active systems.

ANTISYMMETRIC AND ODD STRESSES IN CHIRAL ACTIVE MATTER
The hydrodynamic and continuum equations of motion are set up on conservation laws and symmetry arguments.Thus, related systems obey the same set of equations of motion.For example, the dynamics of both liquids and gases can be characterised by the Navier-Stokes equations.This concept can be generalised from usual fluids to any continuous material, or as continuum approximated medium, including different active matter continua [1,[75][76][77][78][79].However, active matter systems differ from usual fluids in that they retain only some symmetries but not entirely, resulting in systematic contributions to the equations of motion, as in the case of the dynamics of active agents, or actively rotating particles in a fluid, which breaks local (angular) momentum conservation.As a consequence, the non-equilibrium breaking of microscopic reversibility leads to a violation of reciprocity, the linear response matrix between stresses and applied strain is no longer symmetric. 1)In three dimensions, the definition of a common axis of rotation breaks isotropy and no odd transport coefficients are possible in an isotropic three-dimensional (3D) fluid [56].The framework of chiral active systems can thus only be extended to three dimensions in anisotropic situations and transport coefficients may depend strongly on the system setup [57].Here, we restrict ourselves to the study of two-dimensional (2D) dynamics, in order to generalise the hydrodynamic approach to a chiral active system.We focus on a (quasi-) 2D layer of a chiral active fluid, an ensemble of magnetically actuated rotating colloids trapped at an interface between two phases, as sketched in Figure 3A.However, the concept can also be generalised to chiral active elastic solids, where the elastic stresses takes a similar role as the viscous stresses in fluids and deformation gradients in the elastic medium play the role of shear rates in the fluid system.A combination of both is also possible, leading to odd viscoelasticity [70].The viscosity tensor 2) η αβγδ in two dimensions, as any other tensor, can be written as the sum of its symmetric (η S ) and anti-symmetric (η A ) part with respect to the index exchange (αβγδ) ↔ (γδαβ).
where η S αβγδ = (η αβγδ + η γδαβ )/2 and η A αβγδ = (η αβγδ − η γδαβ )/2.The viscous energy dissipation in the fluid per unit time and unit volume can be calculated as [80] Ėkin = −σ vis αβ ∂ β v α , where σ vis αβ is the viscous stress tensor and ∂ β v α is the shear rate.Accordingly, only the symmetric contributions of the viscosity are associated with dissipation because the anti-symmetric parts cancel in the summation.
1) In a liquid system, the response matrix of applied shear is the viscosity tensor η αβγδ that couples stresses σ αβ to shear rates ∂ α v β , via σ αβ = η αβγδ ∂ δ v γ , while in an elastic system, the response matrix of applied deformation is the elastic modulus C αβγδ that couples stresses to deformation gradients ∂ α U β , via 2) Greek indices are used for the spacial dimensions and summation over repeated indices is implied, in other words, Note, that the non-dissipative nature of the odd, or anti-symmetric viscous stress contributions can also be shown by deriving these contributions with a microscopic Hamiltonian [81], an energy conserving approach.The 16-element rank-four tensor η αβγδ can be represented in another basis as a 4 × 4-matrix η i j , where i, j = 0, 1, 2, 3.In this basis, the shear rate ∂ α v β and stress tensors can be expressed as the vectors ėi and σ i , respectively.Then, the linear relationship between viscous stress and shear rates in an isotropic fluid can be expressed as [54,56] In this representation, σ vis 0 and ė0 can be interpreted with dilation or compression, σ vis 1 and ė1 with rotational stresses, σ vis 2 and ė2 with shear according to horizontal elongation and vertical compression, and σ vis 3 and ė3 shear along an axis rotated by 45 • in contrast to σ vis 2 and ė2 [54].Note that the symmetry η αβγδ = η γδαβ is equivalent to η i j = η ji .On the one hand, the symmetric and dissipative shear η, rotational η R , and bulk ζ viscosities appear as in any compressible viscous fluid, leading to normal viscous dissipation acting upon (rotational) shear disturbances and compression.The possible anti-symmetric contributions comprise an odd shear viscosity η odd that couples independent shear modes, and viscosities η A and η B that couple rotations to compressions and vice versa.Note that η A and η B can have dissipative and non-dissipative, or symmetric and anti-symmetric contributions according to Equation (1).For the sake of simplicity, we assume η A = η B = 0 in the following.
The equation of motion for a chiral active fluid is then obtained by taking the divergence of the total stress tensor and balancing it with the fluid inertia ρ . The stress tensor is composed of the contributions stemming from the viscosity tensor but also accounts for stresses in the fluid in the absence of shearing as the pressure σ p αβ = −pδ αβ .In a chiral active fluid, the intrinsic rotation of the constituents gives rise to the angular velocity density Ω and thus another shear-independent contribution to the stress tensor σ Ω αβ = 2η R Ωε αβ .Since ε αβ = −ε βα , this term is also anti-symmetric [82], which however is not directly associated with the odd viscosity term.
The generalised Navier-Stokes equation is then obtained as The left-hand side describes inertial contributions, equivalently to the ordinary Navier-Stokes equation.The first three terms on the right-hand side denote force densities due to pressure gradients, shearing, and compression, respectively.The 2D fluid vorticity ω = ε αβ ∂ α v β measures twice the local circulation of the fluid.
Accordingly, the third term on the right-hand side of Equation ( 4) represents force densities in the fluid that try to synchronise the intrinsic angular velocity density Ω with the fluid vorticity, and vanishes if the local angular velocity density Ω equals the local circulation of the fluid particles ω/2.This term thus couples the intrinsic rotation of the constituent particles Ω to the fluid vorticity and thus to the fluid velocity.The last term on the right-hand side of Equation ( 4) describes force densities proportional to odd viscosity, acting perpendicular to the direction of local shear flows, since To exemplify this, consider the following simplified shear experiment sketched in Figure 3A.Two infinitely extended parallel no-slip boundaries confining a 2D chiral active fluid start to translate into the x and −x directions.The fluid is coupled to a substrate, such that the steady-state velocity profile decays exponentially from the boundaries, as shown in Figure 3B.Dissipative stresses, as a result of ordinary viscosity η, act (anti-) parallel to the direction of shear (red arrows in Figure 3A) with force density η∂ y ∂ y v x , while stresses resulting from odd viscosity act perpendicular to the direction of shear (blue arrows in Figure 3A) with corresponding force density −η odd ∂ y ∂ y v x .Accordingly, unless the flow has reached a profile of vanishing curvature, force densities resulting from odd stresses point into the directions of higher shear rates, as shown in Figure 3B.So far, we tacitly assumed a constant parameter η odd which is not necessarily the case.Instead, the odd viscosity transport coefficient is proportional to the local intrinsic angular momentum density [53,81] and thus η odd ∝ Ω.The angular momentum density field follows an evolution equation balancing input torque, frictional dissipation, and advection and diffusion of angular momentum [35].This equation is then coupled to the dynamics of the flow v via the rotational stress, that is the term proportional to η R in Equation ( 4).However, for a homogeneous system and a constant energy input, Ω ≡ const.and thus η odd ≡ const can be assumed.
Equation ( 4) can be closed by supplying a relation between the density and the pressure.A common approach is to assume incompressiblity [80] such that Equation (4) together with ∂ α v α = 0 fully determines the dynamics.Note that the compression term in Equation ( 4) to ζ then also vanishes.For an incompressible chiral active fluid (∂ α v α = 0), the odd viscosity term in Equation ( 4) can be rewritten as [53] We can thus interpret the effect of odd viscosity in incompressible fluids as an additional pressure resulting in forces pointing into the direction of the gradient of vorticity.In systems in which density inhomogeneities play a crucial role, such as in systems featuring shock waves, an alternative route could be to explicitly allow for weak density inhomogeneities and close Equation ( 4) with the continuity equation [71].
If the Reynolds number (the ratio of inertial to viscous forces in the fluid) is sufficiently small as is typical for soft matter systems on the micrometer scale, the left-hand side of Equation ( 4) can be typically neglected [80].In an incompressible chiral active fluid with sufficiently high rotational and odd viscosities, we thus arrive at the closed Stokes equation for chiral active fluids [34]: If additionally, the 2D fluid layer dissipates momentum into a frictious substrate with linear friction coefficient Γ, the right-hand side of Equation (7a) then has to be balanced by the friction term −Γv α .This is especially of interest in numerical or analytical studies of true 2D systems with vanishing Reynolds number to prevent the occurrence of unphysical behaviour resulting from the negligence of small but finite inertia terms, similar to the Stokes' paradox [83,84].

Odd viscosity
The implications of chiral activity on solvent dynamics can be very different depending on the setup and which terms dominate in the equations of motion.For incompressible systems in which the odd viscosity dominates over the rotational stresses, the fourth and fifth terms on the right-hand side of Equation ( 4) can be neglected.In such a fluid, if the boundary conditions on the flow only depend on velocity field constraints (e.g., no-slip boundary conditions where v = 0 on the surface), then the flow is unaffected by odd viscosity and the force acted on a closed contour is independent of η odd , even in the presence of forces applied to the contour [85].However, the torque exerted on a closed contour resulting from odd viscosity is non-zero and is proportional to the rate of change of the area of the contour, where the odd viscosity is the proportionality constant.This relation may be of potential interest for the reorientation of an active swimmer in a fluid with odd viscosity, where the scallop theorem [86] for the swimming mechanism remains unaffected by the presence of odd viscosity [66].Conversely, the relation between the rate of change of the contour area and odd viscosity might constitute a setup for a measurement of η odd .On the other hand, for no-stress boundary conditions or stress continuity across the boundary (such as a slip boundary), the flow will in general depend on the value of η odd [85].This situation is of interest, e.g., for fluid membranes hosting rotor proteins such as ATP synthase where the rotors may accumulate in a particular domain or droplet leading to differences of η odd in the droplet and in the hosting membrane [84].Another example is the unidirectional flows and edgepumping waves along a free surface of a cohesive chiral active droplet.The spectral decomposition of the shape fluctuations bears a signature of odd viscosity and in 2019 allowed for the first explicit measurement of η odd in a soft matter system [35].
While incompressibility is a good approximation for usual fluids, assuming incompressibility for chiral active fluids is not always appropriate.On one hand, a semi-dilute ensemble of rotors suspended to an incompressible fluid where odd and shear stresses between the rotors are transmitted via hydrodynamic interactions can be regarded as a chiral active fluid.However, since typically only the colloidal degrees of freedom are tracked in experiments, the coarse-grained fluid consisting out of rotating colloids can exhibit density inhomogeneities and should be regarded as compressible, and the osmotic pressure tries to attain a homogeneous rotor distribution.On the other hand, dense rotor suspensions at a fluid solid interface can also exhibit finite compressibility as the result of mass exchange with fluid layers further away from the interface.If only weak compressibility is assumed, we might still conjecture Equation (7a) to be valid and allow for weak density inhomogeneities only for the final results [53].Then, the effective pressure imposed by vorticity leads to an inherent correlation between density and vorticity in chiral active fluids due to odd viscosity, which can be employed to measure η odd in chiral active fluids with sustained vortex flow [34,71,73], as shown in Figure 4.
More generally, to understand the physics of forces acting in compressible fluids with odd viscosity, an Oseen-type mobility tensor for a point force in a 2D fluid with odd viscosity is derived [58], revealing the occurrence of transverse flows with respect to the direction of the applied force F, as shown in Figure 5A and  B. Further studies have generalised low Reynolds number Stokesian [57,87] and time-dependent linear [88] dynamics and microswimmer propulsion mechanisms, pusher-and puller-like force dipoles, to fluids with odd viscosity [89,90], which shows that a single pusher type force dipole will perform a circle swimmer trajectory, possibly allowing for a measurement of η odd by means of the persistence length or the rotation frequency [91].
In contrast to incompressible fluids, a tracer with no-slip boundary conditions can experience lift forces when being dragged through a compressible fluid with odd viscosity resulting from density relaxation and a coupling of the chiral active fluid layer to the third dimension [92].A finite size circular disk moving with velocity V through the fluid experiences drag and lift forces, leading to anti-symmetric contributions to the friction tensor and accordingly to transverse forces F α = −Γ αβ V β and a Hall angle between V and F up to a value of 45 • , depending on the magnitude of the applied force and the value of odd viscosity [58,72].The energy dissipated during the dragging of the disk of mass M is Ė = M∂ t V 2 = MV • V, and the applied force is M V = −Γ • V such that the dissipated energy can be written as Ė = −V α Γ αβ V β , where the anti-symmetric parts cancel in the summation and, accordingly, do not contribute to dissipation.However, the chiral activity increases the overall damping of the disk, such that the effective mobility of the disk decreases with activity [72].

Hydrodynamic interactions
In dry granular systems the transverse interactions between rotors only take place when the particles are physically in touch [40,41,61,93] (Figure 6A).Accordingly, granular systems are qualitatively different from wet hydrodynamic systems, in the way that the transverse interactions can only be observed at sufficient high densities, where interparticle collisions and an almost negligible compressibility impedes the emergence of several characteristics of chiral active systems as density vorticity correlations [34].For bulk The density and vorticity plots show that areas of positive vorticity tend to be higher populated than the average density in the system, while the density in areas of negative vorticity tends to be lower.Averaging for each value of the density inhomogeneity over all given values of the corresponding vorticity reveals the linear relationship above, such that ν odd can be extracted from the measurement.In the presented system the odd viscosity at an area fraction of ϕ = 0.075 is estimated as ν odd ≈ 1.5 × 10 −2 µm 2 /s.Accordingly, density inhomogeneities resulting from odd shear stresses can only be perceived in long-lived vortical flows, since viscous stresses are transported much faster then odd shear stresses.Reprinted with permission from [34].Copyright c ⃝2023 The Author(s).
effects, this circumstance partially extends to cohesive chiral active fluids [32,35], where the particles attract each other, by virtue of electromagnetic interactions [35,43,94].When chiral active agents do not bear attractive interactions that lead to crystalline [32] or cohesive [35] states of matter, long-ranged hydrodynamic interactions typically become a dominating effect.However, it should be noted that externally actuated rotation and active self-rotation are different.Actively self-rotating swimmers like algae [30,33], bacteria [31], or starfish embryos [32] exert a torque on the surrounding fluid that is balanced by the torque exerted on the swimmer by the fluid.In order to rotate, the thrust centre of the torque exerted on the swimmer has to be located outside the swimmer's drag centre.The resulting cycle-averaged azimuthal flow remains finite only to octupolar order and decays as r −4 [95] for increasing distance from the swimmer r.The transverse forces thus decay very fast with increasing distance [33] (Figure 6B) and are only relevant when the self-rotating

A B C
Figure 6 Transverse interactions between rotors.(A) In granular chiral active systems, the rotors solely interact when directly in contact.
(B) Actively swimming rotors use cilia [33] or flagella [30] in order to exert a force in the thrust centre leading to an active torque M = r × F. particles are very close to each other.
On the other hand, externally actuated rotation of colloidal particles, by virtue of external rotating electromagnetic fields that induce a rotation of the colloids carrying an electromagnetic dipole, inject angular momentum from an external source into the fluid.As a consequence, the excited co-rotating azimuthal fluid flow decays like r −1 for disks or rods and r −2 for spheres.In comparison to self-rotating swimmers, long-ranged hydrodynamic interactions among the rotors are then possible ( Figure 6C).
According to Faxén's law, particles in the vicinity of the rotors will be advected with the flow leading to a mutual orbital translation (Figure 1C-H) that decays with increasing interparticle distance and can be explicitly calculated and measured [34,96].In a rotor ensemble suspended to a solvent the average interparticle distance decreases with increasing particle density and thus the velocity of the pairwise mutual orbital translation also increases with increasing density.However, eventually, the point is reached that the increase of the effective solvent viscosity resulting from interparticle collisions experienced by the individual rotors dominates over the transverse forces, such that a further increase of density leads to a slow down of the translational dynamics [34].
At intermediate densities, the mutual translational actuation of the rotors leads to a cascade of orbital rotation and multi-scale vortices emerge.The energy injected on the particle level is then transported to larger scales until it is taken out of the system at the dissipation scales, due to friction, and the dynamics is reminiscent of 2D high Reynolds number turbulence.However, in chiral active systems, the Reynolds number can be exceedingly small such that inertial contributions may be neglected and the phenomenon is called active turbulence [34,97,98].

Odd diffusivity
Similar to stresses acting perpendicular to applied shear rates, fluxes perpendicular to concentration gradients ∇C appear in chiral active fluids as a result of time reversal and parity symmetry breaking [62].Then, the diffusivity tensor in the flux-concentration equation j α = −D αβ ∂ β C is no longer diagonal, but shows the emergence of off-diagonal terms D αβ = D ∥ δ αβ − D ⊥ ε αβ , analogous to the friction tensor in Equation ( 8) (Figure 7A).The fluxes proportional to D ⊥ are divergence free, such that the continuity equation yields the unaltered diffusion equation ∂ t C = D ∥ ∇ 2 C and the concentration is not altered by D ⊥ when the boundary conditions only involve concentrations [62].Figure 7B and C depict simulation results for D ⊥ and D ∥ , where D ⊥ flips sign upon changing the direction of the chiral activity.
In the presence of impenetrable boundaries as obstacles, odd diffusivity leads to curved fluxes along the boundary, where the direction of the fluxes can be flipped by the sign of D ⊥ [63].Moreover, while two normal diffusive particles mutually hinder the exploration of space because the particles will separate after collision, two odd diffusive particles will move around each other due to the probability fluxes along the particles surfaces, leading to a "mutual rolling effect" [63] (Figure 7D).It is thus possible to enhance selfdiffusion in a chiral active system by collisions, such that diffusion can increase with increasing density in contrast to normal diffusive systems in which the self-diffusion coefficient D s decreases in the low density limit with increasing density ϕ as D s = D 0 (1 − 2ϕ) [99], where D 0 is the individual particles diffusivity.The authors of reference [63] established a chiral active system with κ = D ⊥ /D ∥ .Accordingly, self-diffusion in a chiral active system depends on the ratio of transverse to longitudinal diffusive transport coefficients.When the transverse contributions to the dynamics cannot compensate the mutual steric diffusion obstruction, self-diffusion decreases with increasing ϕ for κ < κ c = 1/ √ 3, similar to a normal diffusive system.On the other hand, for κ = κ c , the decreasing mobility of the particles with increasing density is exactly balanced by the transverse transport contributions, such that D s is density independent.When the transverse diffusion D ⊥ dominates over D ∥ , i.e., κ > κ c , self-diffusion can be enhanced by increasing the density ϕ.
It should be noted, however, that even though the self-diffusion coefficient can be controlled with κ, it is only a measure for how fast a particle escapes the cage set by surrounding particles and explores space within the fluid.From a collective perspective, the indistinguishable particles just changed their positions.This could be of special interest for density or concentration relaxation processes in chiral active fluids, e.g., for mixing.On the other hand, the collective diffusion coefficient is unaffected from odd diffusion [63].However, introducing a periodic array of boundaries, such as obstacles, can still have an impact on the collective diffusion coefficient [100].Experimental evidence of odd diffusion has been found for a granular chiral active system [101] but a full characterisation of the phenomenon on the micro scale is still lacking.

CHIRAL ACTIVE CRYSTALS AND ODD ELASTICITY
In a direct analogy to the fluid continuum systems with odd viscosity there are also the elastic continuum systems with odd elasticity [55].In theory, they may consist out of interconnected beads with springs, where the forces between the beads are not just longitudinal, but also include transverse contributions (Figure 8A-C).In reality, chiral active elastic systems are typically composed of spinning objects with significant attractive interactions [31,32,36,43,94,102].Then, depending on the strength of the cohesive forces and the strength of activity, the material might either form a crystalline structure [32,36], or will maintain some active fluidity and the formation of smaller subunits can be observed [43,94].In principle, odd viscosity and odd elasticity could also appear together in the framework of odd viscoelasticity [70].
From a microscopic point of view, the forces in the interactions between the masses that compose an odd elastic solid can be expressed as a Hookean spring and a chiral transverse force, F = (−kr + k a φ)(r − r 0 ), where r and φ are the unit vectors in radial and azimuthal direction, respectively [55].The radial contribution is the normal term for a harmonic solid, while the transverse or azimuthal part can be linked directly to antisymmetric contributions in the elasticity tensor by a coarse-graining procedure.The elasticity tensor takes a form analogous to the viscosity tensor in Equation (2), where the odd elastic modulus ±K odd takes the role of ±η odd and couples different shear deformations.Due to a non-potential nature of the microscopic transverse forces, odd elastic solids may show a non-zero work balance for deformations over a closed circle.When integrating the previously mentioned force over a closed cycle then the radial part vanishes and the work is W = ∂A dr • F = k a A, where A and ∂A are the area and contour of the cycle [55].
A phenomenological consequence of odd elasticity is the emergence of self-sustaining vibrational dynamics in overdamped solids [55].A 2D odd elastic solid grown on a substrate follows the overdamped equation of motion: where U and σ αβ entail the material's deformation and stresses, respectively, and C is the elastic modulus tensor.A 90 • phase shift between stress and strain arises resulting from the anti-symmetric shear coupling K odd , similar to the phase delay between stress and velocity in underdamped solids [55].However, in the odd elastic material closed circles in deformation space convert internal energy into mechanical work.Then, depending on the ratio between odd and even stresses, either no waves, or exponentially attenuated waves propagate leading to periodically repeating deformations as shown in Figure 8D.The dynamics bears a signature of the underlying non-Hermitian dynamical matrix 3) .While for k a = 0, the system is passive and Hermitian, meaning that the eigenvectors are perpendicular, with increasing k a this is no longer the case.In fact, when |k a |/k = 1/ √ 3, the eigenvectors become co-linear, and the system reaches an exceptional point, a telltale sign of non-Hermitian dynamics [55,103].In the limit of dominating odd contributions, |k a |/k ≫ 1, the waves become self-sustaining.
Experimentally, self-sustaining chiral displacement waves have been found in a living chiral crystal that consists of rotationally swimming starfish embryos [32].The autonomously developing multicellular organisms hydrodynamically attract and form a sheet of a chiral crystal at the fluid surface that spans thousands 3) The dynamical matrix is the q-space representation of the right hand side of Equation (10b) which carries anti-symmetric contributions that break Hermiticity. of spinning organisms and persists for tens of hours and mutually exert hydrodynamic transverse forces on each other.The behaviour of the crystal is shown in Figure 2F.Clearly, the displacement versus time plot in the inset shows a periodic and phase shifted wave behaviour between x-and y-directions.Moreover, the authors conclude that the crystal effectively does work on the surrounding fluid.
In chiral active systems, the non-reciprocal transverse forces lead to destabilisation of active crystals or the propagation of free phonons [104].Grain boundaries mutually glide over one another and individually rotating grain boundary domains emerge, that "knead" the odd crystal [43], as is shown in Figure 8E.Moreover, the competition between non-reciprocal and elastic forces leads to self-propelled dislocations and defects gliding through a chiral active crystal.The ambient torque density stemming from the rotation of the active units exerts forces on the dislocations [64].The direction of propulsion of the dislocation is determined from the Burgers vector b α = C dr β ∂ β u α , where C is a counter-clockwise closed contour around the dislocation, and thus depends on the displacement field.It is thus possible, that the self-propelled dislocations can either attract, or repel, depending on the initial conditions, whereas in the absence of chiral activity, defects normally attract and annihilate [43].

CHIRAL ACTIVE MATTER IN COMPLEX GEOMETRIES Topologically protected edge modes
The mutual rotational stresses among the rotating units in homogeneous chiral active fluids cancel on average such that no net flow is generated.However, the situation is remarkably different at the boundaries of the system.At a bounding wall, the rotors directly at the boundary will experience asymmetric rotational stresses leading to the formation of a flow along the edge.Note, that the exact form of the edge flow depends on the boundary conditions between the chiral active fluid and the wall.Given the frictional damping with a substrate, the resulting very robust and unidirectional flow decays exponentially into the bulk on a scale determined by the strength of the substrate friction and is thus localised at the boundary [35,40,61].This behaviour has been connected to the concept of topological insulators with conducting surface and insulating bulk states, emerging due to a twisted band gap in the electronic dispersion relation [105].The starting point for the analysis is the vorticity evolution equation for a slowly varying chiral active fluid at low Reynolds number with constant Ω.
where λ = √ (η + η R )/Γ.Performing Fourier transformation in space and time 4) reveals for the dispersion relation between inverse dissipative timescale and wavenumber ω( q) = −(ν + ν R ) . The friction scale λ guarantees ω( q) 0 (confer Figure 9A), in other words, there is a finite timescale even for the longest wavelength modes.This situation is different for the modes directly at and parallel to confining walls wall, which fulfil ω( q = 0) = 0 [105].Accordingly, a steady-state flow in the chiral active fluid can exist only at the boundary (Figure 9B and C).On the other hand, if the friction scale λ diverges, then ω( q = 0) → 0, that is the "band gap" closes, and a faster delocalised current forms [106].
The connection between topological insulators and edge flows in chiral active fluids can be drawn more rigorously [71,107].In a weakly compressible chiral active fluid in a circular container with odd viscosity and without substrate friction, a stationary linear solid body-like velocity profile establishes.The solid body rotation gives rise to a Coriolis term to the equation of motion, which leads to a band gap at q = 0 in the dispersion relation of the sound modes.In close analogy to topological insulators, one can then calculate a topological invariant, the Chern number which characterises the geometric and topological properties of the band structure ω( q) [26].It is calculated by an integration over q-space, and a non-zero odd viscosity is necessary for a regularisation such that the Chern numbers are well defined5) [71].Going from the bulk of the fluid to the edge, the Chern number undergoes a transition from its value in the bulk to the zero value outside the material.This change cannot occur smoothly, due to the integer nature of the Chern number.Alternatively, the requirement for a non-zero Chern number, namely, a gapped band structure, ceases validity at the edge.Accordingly, modes with frequencies in the gap can only be excited at the edge [26].The edge states resulting from this mechanism show topological protection, such that the modes are unaffected from material changes and impurities as defects or obstacles, as long as the gap is not closed.As a consequence, the edge modes propagate unimpeded along the boundary through and around obstructions without backscattering, since the edge modes are unidirectional and cannot penetrate into the bulk.
Figure 9D shows the propagation of a topologically protected sound wave travelling along the boundary of a circular container in finite-element simulations of the underlying hydrodynamic equations of motion [71].The density waves are excited at the boundary with frequencies from within the band gap and the travelling shock wave decays exponentially into the bulk.Irrespective of container deformations, the wave travels unidirectionally and no backscattering occurs.However, the authors of reference [71] neglected the usual dissipative viscosity.Taking dissipative viscosity into account renders the dynamical matrix of the problem non-Hermitian, and the shock waves' frequencies become complex valued, where the real parts still account for the travelling wave, while the imaginary parts lead to attenuation and associated decay rates.For small ratios of ordinary to odd viscosity η/η odd , attenuated shock waves occur [71].Dissipative, active, or non-reciprocal systems are in general not Hermitian and the corresponding systems may not only exhibit topologically protected boundary modes, but the dynamics may additionally delicately depend on the boundary conditions [108].Then, a large number of skin modes localised at the boundary may be introduced which are characterised by a topological invariant different from the Chern number, the winding number [26].While topologically protected boundary modes do not effect the bulk dynamics, the full mode spectrum in a non-Hermitian system can be modified by the boundary conditions [109].Skin modes thus might serve as an alternative design for scattering-free edge flows and energy localisation at the boundaries [110].
The topological protection of edge modes makes them immune against disorder and such modes might thus provide a possibility to robustly transport material or information on the microscale.For example, a passive inert particle which itself does not reinforce the boundary mode can be transported along a boundary [41,61].Such a cargo particle in a chiral active fluid consisting out of rotating units which are slightly smaller than the cargo itself experiences depletion interactions at the boundaries, leading to an effective attraction of the cargo and the boundary.Additionally, the effective attraction is aided by odd viscosity and the flows created at the surface of the cargo, such that the cargo additionally experiences odd stresses leading to an significantly increased dwelling probability of the cargo at the boundary for the active system with odd viscosity in comparison to a passive system without odd viscosity [61].As a result, the cargo stays at the boundary and is transported robustly in the emerging edge current, as shown in Figure 9E.

Complex geometries and material design
The singular flow behaviours exhibited by chiral active fluids are closely related not only to transport phenomena in condensed matter physics, such as quantum Hall fluids and topological insulators, but also contribute to understanding collective motion and self-organisation in biological systems.This understanding holds significance in the context of constructing new materials and microfluidic devices with distinctive transport properties.The substantial challenge in designing chiral active functional materials and devices lies in the controlled manipulation of the flow behaviour of chiral active fluids by external factors.Physical boundaries or spatial confinement, together with the robustly emerging edge flow evidently provide powerful means for achieving such control.The key scientific inquiry in this context revolves around understanding how emerging flows and stresses, odd shear coefficients, and spatial confinement conditions synergistically influence the stability and transport behaviour of chiral active matter.
The robust edge currents that emerge in chiral active fluids can be controlled by the particle density and the direction of the local net flow is set by the chirality of the system [111].In a linear and symmetric channel no net flux is created, because the flows on both sides of the corridor are of equal strength and opposite direction [112].However, in a curved channel, such as an annular ring as shown in Figure 10A, the different curvatures of the inner and outer walls lead to an asymmetric flow profile and net flow along the channel can be obtained.Moreover, in the limit of a narrow channel where the rotors cannot overtake one another, unidirectional transport is consequently obtained [113].In systems involving rotors, which can rotate either clockwise or counter-clockwise, the binary mixture tends to separate into domains with opposite chirality.This phenomenon has been observed in granular binary rotor systems, where rotating gears with different rotational directions segregate into distinct domains [114][115][116][117]. Interestingly, the behaviour of these rotors can be controlled by introducing active soft boundaries, which consist of interconnected particles with both clockwise and counter-clockwise rotation [118] (Figure 10B).Additionally, the interconnection of these particles in different geometries leads to fascinating self-organising behaviours, reminiscent of amphiphilic behaviour seen in surfactants, such as double-stranded soft asymmetric boundaries (confer Figure 10C) show affinity to clockwise/counter-clockwise interfaces [114] which could be employed for segregation or ordering in rotor systems.
By combining computer simulations and theoretical calculations, driven granular gears have been shown to exhibit transverse transport when flowing through a square matrix of frictionless obstacles [69] (confer Figure 10D).The transverse transport is similar to the Hall effect and is controlled by the driving force, the driving torque, and the gear density.Moreover, when gears of opposite chirality are employed, this mechanism can be used to separate the particles by chirality, as the transverse transport changes direction with the gears chirality [122].On the other hand, when a particle translates in chiral trajectories through an obstacle lattice without external forcing, the chirality of the particle motion can lead to an enhanced effective diffusive behaviour (Figure 10E).While the influence of the obstacles acts on the one hand constraining, on the other hand, it provides an energy injection into the system resulting from the flow that emerges along its boundaries.As a result, there is an optimal tradeoff between transport facilitation and restriction by the obstacles at intermediate obstacle density, leading to a significant increase of effective diffusive transport which is controlled by obstacle density or spacing, the swimmers trajectory persistence, and disorder such as noise, polydispersity, or irregularity in the obstacle array [100].However, with increasing obstacle density, the restricting influence of the boundaries eventually dominates and leads to diminished effective diffusive transport.If the obstacles additionally bear a chiral structure themselves, the interactions between geometry and particle may depend on the chirality of the particle such that only particles of certain chirality are trapped in the geometry [119].
Active collective dynamics in complex geometries allow for the study of effects of chirality, even in the absence of inherently rotating or circularly moving particles.The creation of vortex lattices by the introduction of pillars or boundaries in bacterial flows can lead to the emergence of locally or globally chiral flows, as has been shown in the experiments depicted in Figure 10G [120].A polar active flow of the collective aligning overdamped dynamics [1,123] of bacteria in an annulus geometry leads to a chiral flow in the confinement [121].Moreover, in the case of interconnected annuli, the fluid in neighbouring annuli circulates in opposite directions.If the annuli are arranged in a Lieb lattice (confer Figure 10H), then the unit cell has a net circulation of steady-state flow and thus is chiral.Density waves on top of the chiral flow then show the emergence of topologically protected sound modes (confer Section "Topologically protected edge modes").The chiral net flow in such systems leads to Coriolis forces [71] and is an analogy to static magnetic fields in the Hall effect leading to Lorentz forces.However, easier to realise geometries typically do not give rise to a net vorticity or chirality in the unit cell (confer Figure 10I), but can still give rise to topologically protected edge modes [107].Then, the locally chiral steady-state flow can still serve as an analogue to the anomalous Hall effect, where spin-orbit coupling replaces the requirement of the external magnetic field, and topologically protected edge modes may emerge even in the absence of net vorticity in the system.
In granular chiral active matter, interactions between a confining geometry and the chiral active system can lead to a chirality transition resulting from the friction between the rotors and the boundary [124][125][126].For few interactions between the granular fluid and the boundaries, the vorticity of the fluid is of the same sign as the constituents inherent rotation.Edge currents then emerge resulting from occasional particle collisions and particle shielding at the boundary [127].In this state, particle collisions and the associated mutual orbital translation dominates the dynamics.However, at large heat dissipation at the boundaries, such as at a highly frictious container wall, the overall vorticity chirality changes to a phase of opposite chirality compared to the internal rotation.In this state, the particles roll along the boundary of the container, and the continuity of the flow then dictates a chirality transition in the interior.

CIRCLE SWIMMERS AND HYPERUNIFORMITY
When an individual active translating particle [4,5,128] is further subject to torque, the linear self-driven motion is coupled with rotation, causing the individual to perform a continuous circular motion with specific chirality.Such active particles are termed circle swimmers and their dynamics can be regarded as a superposition of Brownian motion and an active circular motion [129].The torque acting on the body can be a consequence of particle asymmetry which is relatively common in biological systems, such as E. coli [130] (Figure 11A), sperm cells [131], V. cholerae [132] (Figure 11B), or algae [30] swim in circular chiral trajectories at planar surfaces or fluid interfaces, but also synthetic asymmetric self-phoretic particles [129,133] can show a similar behaviour ( Figure 11C-E).Exemplarily, the circle swimming mechanism of E. coli at planar surfaces relies on hydrodynamic interactions between the flow field initiated by the bacterium and a no-slip boundary.The bacterium swims without the aid of an external force or torque application, thus it swims by applying a force and torque on the fluid which results in a counteracting force and torque on the cell body [2,134].This is achieved by a rotating helicoidal bundle of flagella, which are anchored to the cell body.While in the unconfined fluid, the cell would be propelled straight, the bacterium experiences hydrodynamic forces on the rotating cell body and the counter-rotating flagella bundle acting into opposite directions resulting in a torque from the hydrodynamic interactions between the rotating bacterium and a no-slip wall [3,130].On the other hand, E. coli at planar surfaces with slip boundary conditions give rise to a torque into the opposite direction and thus a circular swimming path of opposite chirality [2].A circular particle trajectory can also be imposed by the application of electromagnetic fields to artificial self-propelling particles carrying a electric or magnetic moment [135,136], but also magnetotactic bacteria move in circles in a rotating magnetic field [137].Chiral microswimmers can be classified according to their swimming characteristics by using some simple static patterns in their environment, or a patterned microchannel acting as a sieve to capture microswimmers [119].When a circle swimmers is confined by an external potential, the interplay of the potential landscape and the persistence of the circular motion can lead to an effective extra confinement mechanism and the particle distribution thus bears a signature of the chirality of the swimmer [138].
Interactions among circle swimmers at higher concentrations lead to the emergence of collective phenomena such as pattern formation and enhanced flocking [140].An dense ensemble of circle swimmers can be regarded as a chiral active fluid exhibiting odd viscosity [72] and may thus be employed as an active chiral bath.Furthermore, such an active bath can be used in order to power a gear submerged in a chiral active bath [141].When the particle density is moderate, stiff self-propelled polymers with intrinsic curvature and chiral circular dynamics can self-assemble into vortex structures such as closed rings, arising from only steric interactions [142,143].Moreover, when the chiral active swimmers are L-shaped, the steric interactions lead to dissimilar collisions and aggregation mechanisms provoke the emergence of an oscillatory dynamic clustering of repeating merging, splitting, and reformation of dynamic clusters [144].Furthermore, circle swimmer systems show the emergence of disorder or flocking states and also motility induced phase separation, governed by the interplay of non-reciprocal interactions among the swimmers, finite size, and chirality [145].

Hyperuniformity
A further collective phenomenon observable in circle swimmer systems is the emergence of disordered hyperuniform states that display vanishing long-wavelength density fluctuations akin to crystalline structures [30,146].Crystals exhibit long-range order and the structure factor S q and density fluctuations ⟨ ∆ρ 2 ⟩ behave like S q→0 = 0 and ⟨ ∆ρ 2 ⟩ ∝ L −λ , respectively, where q is the wavenumber, L the size of the domain under consideration, and λ = d + 1 with d the dimensionality.On the other hand, conventional liquids and gases exhibit S q→0 = const.and λ = d [146,147].When a system shows density fluctuations with λ > d and a structure factor S q→0 = 0, then the system is said to be hyperuniform and the particles are distributed more uniformly in comparison to disordered systems [148].Typically, active matter systems show vivid collective dynamics accompanied by large density fluctuations [83,[149][150][151].However, recently, chiral active fluids have been shown to exhibit hyperuniformity [30,146,148,152], leading to the suppression of large-scale density fluctuations similar to crystals, while a liquid like local isotropic behaviour is retained [30].Such systems could find different practical applications as a crossover material consisting out of a disordered fluid without long-range density fluctuations [147].
In chiral active circle swimmer systems, hyperuniformity can be obtained at intermediate densities when the radius R of the circular trajectory is sufficiently large such that the particle can be well distinguished from merely non-interacting spinning particles [148].On the other hand, for too large R, the system will rather resemble an active Brownian particle system [146] featuring an active gas phase at low and intermediate densities.Accordingly, hyperuniform states can be obtained when R is approximately a few particle diameters The system exhibits a R-dependent length scale that controls the hyperuniform behaviour.On length scales r ≪ R, the particles move effectively straight and behave like an active gas, while for r ≫ R, the particles exhibit a chiral trajectory and the hyperhuniform behaviour can be observed.Reprinted with permission from [146].Copyright c ⃝2019 The Author(s).
but still significantly smaller than the system size ( Figure 12).Then, large density fluctuations as typical for dynamic cluster forming active systems are obtained on length scales comparable and smaller than 2R, and density fluctuations approaching those in crystals are obtained at larger length scales [146,148], where the individual circular trajectories effectively repel each other.A similar behaviour is also obtained for spinning dumbbell particles [153,154].For a circle swimming algae with long-ranged repulsive hydrodynamic interactions, large density fluctuations are even suppressed on short length scales and the hyperuniform behaviour can also be observed at low swimmer densities resulting from the long-ranged interactions [30], as has also been observed for an ensemble of point vortices [152].Without long-ranged hydrodynamic interactions at low densities, in a system where hydrodynamic interactions are not dominant, the particles will barely interact and will not show any cooperative behaviour [146] and the system shows the fluid-like behaviour of a non-interacting spinner fluid [148].

SUMMARY AND PERSPECTIVE
This article has discussed recent developments in chiral active matter, highlighting the collective behaviour arising from chiral activity and emergent phenomena like anomalous density fluctuations, anti-symmetric stresses, odd diffusion, topologically protected edge modes, and non-dissipative transport.These systems proffers a promising foundation for the creation of tunable active materials with explicit control over the rotational degrees of freedom where energy and angular momentum is introduced at the microscopic level.Despite the progress made in the description of chiral active systems, a detailed understanding of the subject is still in its early stages.There is still lack in understanding how odd viscosity can be employed in order to navigate particles [155] through chiral active baths, or whether phenomenologies such as hyperuniformity [30,144,152], active turbulence [34], or odd viscosity [53] can appear together, show interdependencies, or whether these seemingly disparate behaviours in chiral active systems can be captured in a unifying theory.Undeterred by the several different chiral active systems that have been designed successfully in experiments there are still myriads of theoretical predictions that have not been proven in experiments, particularly in colloidal systems, such as the emergence of transverse forces experienced by actively translating colloids suspended to a chiral active bath [58,72,[89][90][91].Another question is, whether phenomena observed in granular materials on the macroscale can be straightforwardly generalised to the colloidal or micro level, such as chiral separation [122,156].This behaviour could be of interest for the separation of two species of the same chemical composition, but with opposite chirality, for instance, in order to promote the chiral separation and analysis of racemic drugs in pharmaceutical industry as well as in clinic such that the unwanted isomer can be eliminated from the preparation to find an optimal treatment and a right therapeutic control for the patient.Chiral activity studies involving complex environment have realised trapping [119,138], separation [114,118], or unidirectional transport [26,105].However, little is known so far about the mutual influences of emergent flows and odd viscosity, and what is the consequent impact on objects suspended to chiral active fluids.
Possible applications for chiral active matter are as diverse as the phenomenology.The emergence of topologically protected edge modes is a very clear candidate for robust cargo transport processes [61] on the microscale.However, it is not yet clear, whether the robust cargo transport in a granular system consisting out of rotating gears reported in reference [61] can be directly extended to colloidal systems [34,35], and depends on the experimentally realisable values of η odd , the relative importance of thermal fluctuations, or the rotor density.The intrinsic correlations between density and vorticity in weakly compressible chiral active fluids could be used for segregation or purification processes [37], in which the introduction of rotating particles into a contaminated fluid leads to the aggregation of rotors and exclusion of impurities in the regions of high vorticity.

Figure 1
Figure 1 Biological (A-E) and synthetic (F-J) chiral active matter over several length scales.(A) The chemical potential difference for protons across the membrane in the biological rotary machine ATP synthase (diameter σ ≈ 10 nm) is converted into chemical energy of ATP synthesis causing a rotation.Reprinted with permission from [29].Copyright c ⃝2001 The Author(s).(B) Marine algae Effrenium voratum (σ ≈ 10 µm) with superimposed trajectory showing chiral circular swimming behaviour at the air-liquid interface.Reprinted with permission from [30].Copyright c ⃝2021 National Academy of Sciences.(C) Bacteria Thiovulum majus (σ ≈ 10 µm) on a surface induce

Figure 2
Figure 2 Collective behaviours in chiral active systems.(A) The circle swimming algae E. voratum generates a period-averaged outgoing radial flow leading to a dispersion of the cells in a disordered hyperuniform state.Streak image averaged over 10 s.Reprinted with permission from [30].Copyright c ⃝2021 National Academy of Sciences.(B) Anisotropic pear-shaped Quincke rollers powered by a static electric field favour rotations around the symmetry axis due to viscous drag leading to curved trajectories.Hydrodynamic alignment interactions then induce emergent patterns like vortices (image) or rotating flocks.Reprinted with permission from [50].Copyright c ⃝2020 The Author(s).(C) Hydrodynamic interactions in an ensemble of isotropic rotors leads to a cascade of transverse dynamics and the formation of multi-scale clock-wise and counter-clock-wise vortices.Reprinted with permission from [34].Copyright c ⃝2023 The Author(s).(D) Viscous edge pumping effect in a cohesive magnetic spinner fluid gives rise to unidirectional surface waves.Spectral decomposition of the surface fluctuations allowed the first experimental measurement of odd viscosity in a soft matter system.Reprinted with permission from [35].Copyright c ⃝2019 Springer

Figure 3
Figure 3 Stresses in chiral active fluids.(A) Sketch of the direction of the stress forces resulting from odd viscosity in shear flow.(B) The corresponding fluid velocity profile (red) and its Laplacian (blue) (proportional to the force densities due to odd viscosity) assuming substrate friction, such that the steady-state velocity profile decays exponentially from the boundaries.

Figure 4
Figure 4 Vorticity (top) and density (bottom) correlations resulting from odd viscosity in particle based hydrodynamic simulations (left) andexperiments (right).The weak compressibility and the presence of a radial effective pressure resulting from odd viscosity amounts to density inhomogeneities (ϕ(r) − ⟨ϕ⟩)/ ⟨ϕ⟩ = ν odd ω/c 2 , where ν odd is the kinematic odd viscosity, and c is the propagation velocity of a colloidal density inhomogeneity.The density and vorticity plots show that areas of positive vorticity tend to be higher populated than the average density in the system, while the density in areas of negative vorticity tends to be lower.Averaging for each value of the density inhomogeneity over all given values of the corresponding vorticity reveals the linear relationship above, such that ν odd can be extracted from the measurement.In the presented system the odd viscosity at an area fraction of ϕ = 0.075 is estimated as ν odd ≈ 1.5 × 10 −2 µm 2 /s.Accordingly, density inhomogeneities resulting from odd shear stresses can only be perceived in long-lived vortical flows, since viscous stresses are transported much faster then odd shear stresses.Reprinted with permission from[34].Copyright c ⃝2023 The Author(s).

Figure 5
Figure 5 Streamlines of the flow created by a point force into the x-direction in a quasi-two-dimensional compressible fluid layer coupled to frictious substrate without (A) and with (B) odd viscosity.The friction between the fluid and the substrate introduces the hydrodynamic cutoff length κ −1 .Without odd viscosity, this amounts to a screened version of the stokeslet.The presence of odd viscosity adds a transverse component to the created flows.Reprinted with permission from [58].Copyright c ⃝2021 the American Physical Society.(C) Trajectory (red) of a particle (red) subject to a constant force into the x-direction in a chiral active bath consisting out of circle swimmers (instantaneous positions depicted as blue circles).The trajectory shows the emergence of a Hall angle of 20 • between the direction the force is applied into and the direction of the particle motion.Reprinted with permission from [72].Copyright c ⃝2019 the American Physical Society.
This torque is balanced by the torque exerted onto the fluid.The corresponding cycle averaged flow field can have azimuthal and radial components, where the radial component decays like r −4 .(C) When the rotation of the particles is excited from an external infinite angular momentum reservoir, then the surrounding fluid co-rotates with the rotor and the azimuthal flow profile decays like r −1 for disks or r −2 for spheres.

Figure 7
Figure 7 (A) Linear density gradient (colour) leading to flux (arrows) with transverse component arising from D ⊥ .(B, C) Diffusion coefficients D ⊥ and D ∥ as obtained from molecular dynamics simulation of a passive tracer particle in a chiral active bath of rotating dumbbells with density ρ bath .The Péclet number Pe is proportional to a force applied to the dumbbell particles causing their rotation, such that a different sign of Pe amounts to an opposite rotation.The coefficients are numerically calculated in simulations with generalised Green-Kubo relations and the relation between flux and concentration by maintaining a constant density gradient (boundary flux).Reprinted with permission from [62].Copyright c ⃝2021 the American Physical Society.(D) Comparison of two nearby particles subject to normal (top) and odd (bottom) diffusion.Odd diffusivity circumvents the mutual steric hindrance of configuration space exploration and the particles mutually roll around each other.Reprinted with permission from[63].Copyright c ⃝2022 the American Physical Society.

FFigure 8
Figure 8 (A) Non-potential force between two masses acting transverse ( φ) and radial (r) to the conncting spring.(B) Compressing the spring results in a radial force, while extension results into an opposite radial force.The closed cycle (∂A) of deformation gives rise to the extracted work W = k a A. (C) A continuum of springs with transverse and radial contributions can be regarded as a material with odd elasticity.(D) Deforming such a material can result in unusual behaviour, such as self-sustaining deformation waves in overdamped media.A 90 • phase shift between stress and strain facilitates wave propagation.The colour gradient indicates time.The work done by a full cycle in deformation space offsets dissipation.Reprinted with permission from [55].Copyright c ⃝2020 Springer Nature.(E) Map of the bond orientational parameter in an active crystal consisting out of cohesive spinning colloidal magnets with magnetic attraction.The crystal is knead or broken up in smaller pieces with high local hexagonal order and like orientational order.(F) The dislocations move through the crystal in a ballistic manner.Colour map same as in (E).(E, F) Reprinted with permission from [43].Copyright c ⃝2022 Springer Nature.

Figure 9
Figure 9 (A) Band structure ω± ( q) for for the bulk with a bandgap (grey) connected to edge states (green) with frequencies laying in the bandgap.(B) Localised edge modes of a chiral active fluid in a linear channel.The rotors move with the emerging edge flow.(C) The topologically protected edge modes are unidirectional and robust such that they do not scatter off sharp edges at the boundary, but navigate along the edge.(D) Topologically protected sound modes in a compressible chiral active fluid.The wave is excited at frequencies laying in the band gab at the star symbol and subsequently move unidirectionally along the boundary without scattering off edges.(A, D) Reprinted with permission from [71].Copyright c ⃝2019 the American Physical Society.(B, C) Reprinted with permission from [105].Copyright c ⃝2018 National Academy of Sciences.(E) Cargo (larger particle) transport in a granular chiral active fluid consisting out of vibrational gears aided by odd viscosity and topologically protected edge modes.Reprinted with permission from [61].Copyright c ⃝2021 the American Physical Society.

Figure 10
Figure 10 Chiral active matter in complex geometries.(A) Rotating particles in an annular channel of width D < 2σ.Reprinted with permission from [113].Copyright c ⃝2010 IOP Publishing Ltd.(B) Rotor binary mixture with soft active boundaries for varying boundary composition.Reprinted with permission from [118].Copyright c ⃝2015 National Academy of Sciences.(C) Binary mixture of vibrational granular rotors with a double-stranded rotor chain in the evolution of time.Reprinted with permission from [114].Copyright c ⃝2021 The Author(s).(D) A chiral active fluid forced through a grid of fixed obstacles.Reprinted with permission from [69].Copyright c ⃝2021 The Author(s).(E) Circle swimmer transport can be facilitated by the introduction of a periodic lattice of obstacles.Reprinted with permission from [100].Copyright c ⃝2022 The Author(s).(F) Circle swimmers can be sorted and caged by chiral surroundings.The active particle changes chirality midway and only shows trapping for one sign of chirality.Reprinted with permission form [119].Copyright c ⃝2013 The Royal Society of Chemistry.(G) Complex or chiral geometries can be employed in order to create chiral flows from polar active fluids.Reprinted with permission from [120].Copyright c ⃝2018 The Author(s).(H) Polar active fluid without inherent internal chirality in a Lieb lattice shows the emergence of chiral flows and a net chiral flow in the unit cell.The material exhibits topologically protected edges modes resulting from a non-zero chirality.Reprinted with permission from [121].Copyright c ⃝2017 Springer Nature.(I) Emergent local chiral flows can lead to the emergence of topologically protected edge modes even in the absence of net vorticity in the unit cell, as shown here.Reprinted with permission from [107].Copyright c ⃝2019 the American Physical Society.

Figure 11
Figure 11 Circle swimming in biological and synthetic active matter.Near surface dynamics show swimming in circular trajectories for E. coli cells (A) and V. cholerae cells (B).(A) Superimposed microscopy images and (B) tracked trajectories.Inset in (B) shows bright-field image of V. cholerae.(A) Reprinted with permission from [130].Copyright c ⃝2006 Elsevier.(B) Reprinted with permission from [132].Copyright c ⃝2014 Springer Nature.(C) Janus colloids coated with a Ni/Ti cap and a protective SiO 2 layer and sandwiched between two coverslips are energised with an AC vertical electric field and perform circular trajectories with tunable radius R resulting from an externally applied in plane rotating magnetic field.The image shows reconstructed trajectories.The inset shows an experimental image revealing that the particles spontaneously orient in opposite directions along to the magnetic field.Reprinted with permission from [135].Copyright c ⃝2017 National Academy of Sciences.(D) Asymmetric Zn/Au rods (inset) show self-electrophoresis exhibiting four different modes (ballistic, linear, circular, helical).By controlling UV light intensity and fuel concentration, the rods can be transformed from ballistic motion to continuous rotating motion, and by adjusting the angle of incident light, these rods can be switched from circular motion to spiral, and eventually to linear motion.The image shows circular motion mode.Reprinted with permission from [139].Copyright c ⃝2020 American Chemical Society.(E)Asymmetric L-shaped colloids exhibit self-phoretic circular motion where the radius of the trajectory depends only on the shape of the object, but is unaffected by the propulsion strength.Reprinted figure with permission from[129].Copyright c ⃝2013 the American Physical Society.

Figure 12
Figure 12 Density fluctuations (A) and structure factor (B) for an ensemble of simulated active Langevin circle swimmers.The particles are of diameter σ and interact via excluded volume interactions.The radius of the circular trajectory is R.The system exhibits a R-dependent length scale that controls the hyperuniform behaviour.On length scales r ≪ R, the particles move effectively straight and behave like an active gas, while for r ≫ R, the particles exhibit a chiral trajectory and the hyperhuniform behaviour can be observed.Reprinted with permission from[146].Copyright c ⃝2019 The Author(s).