Issue 
Natl Sci Open
Volume 3, Number 4, 2024
Special Topic: Active Matter



Article Number  20230086  
Number of page(s)  29  
Section  Physics  
DOI  https://doi.org/10.1360/nso/20230086  
Published online  09 April 2024 
REVIEW
Emergent phenomena in chiral active matter
^{1
}
Institute for Advanced Study, Shenzhen University, Shenzhen 518060, China
^{2
}
College of Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China
^{*} Corresponding authors (email: yongxiang.gao@szu.edu.cn)
Received:
21
December
2023
Revised:
1
February
2024
Accepted:
26
February
2024
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 diffusivity, 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 antisymmetric and odd stresses and ensuring effects 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.
Key words: active matter / odd viscosity / odd elasticity / odd diffusivity / hyperuniformity / topologically protected edge modes
© The Author(s) 2024. Published by Science Press and EDP Sciences.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 [1417], in vitro fertilisation [18], the formation of biofilms [19, 20], or targeted drug delivery at the microscopic level [2124].
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 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], corotating 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 [3436] or larger particles [37, 38] with a ferromagnetic moment, lightdriven 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.
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©2001 The Author(s). (B) Marine algae Effrenium voratum (σ≈10 m) with superimposed trajectory showing chiral circular swimming behaviour at the airliquid interface. Reprinted with permission from [30]. Copyright©2021 National Academy of Sciences. (C) Bacteria Thiovulum majus (σ≈10 m) on a surface induce a chiral tornadolike flow that leads to an attraction and mutual orbital rotation of neighbouring cells. Reprinted with permission from [31]. Copyright©2015 American Physical Society. (D) Actively spinning starfish embryos (σ≈200 m) form a corotating pair by flow generated by each other. Reprinted with permission from [32]. Copyright©2022 Springer Nature. (E) Volvox colonies (σ≈500 m) have a ciliated surface of beating flagella pairs on each somatic cell (small dots), leading to directed motion and chiral rotation. Reprinted with permission from [33]. Copyright©2009 the American Physical Society. (F) Silica rodlike colloids (σ≈1 m) with an adhered magnetic tip perpendicular to the symmetry axis [42] orient perpendicular to the substrate and rotate in sync with an externally applied rotating magnetic field. The colloids excite a rotational flow field advecting nearby colloids. Shown streamlines are obtained from simulations. Reprinted with permission from [34]. Copyright©2023 The Author(s). (G) Chiral magnetic colloidal spinners (σ≈2 m) drag the surrounding fluid and exert hydrodynamic transverse and magnetic attractive forces. Upper image and lower image are reprinted with permission from [43, 35], respectively. Copyright©2019 and 2022 Springer Nature. (H) Vaterite colloidal particles (σ≈212 m) asynchronously rotate in circularly polarised light resulting from birefringence, leading to hydrodynamic spinorbit coupling. Reprinted with permission from [39]. Copyright©2023 The Author(s). (I) 3Dprinted granular gearlike rotors (D1=16 mm, D2=21 mm) with tilted bristles at the bottom can be brought into a state of active rotation powered by vertical vibration. Reprinted with permission from [40]. Copyright©2020 National Academy of Sciences. (J) Two 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©2020 the American Physical Society. 
The word chirality is derived from the Greek word (kheir), for “hand”, which is abundant in nature, and is a basic and intrinsic characteristic of many natural and manmade systems [4447]. 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 timereversal () and parity (or coordinate mirror transformation, ) 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 electricfield agitated pearlike 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 selfpropelled particles [51, 52] due to mutual influence (Figure 2C). On the other hand, unidirectional waves along free surfaces (Figure 2D) [32, 26], 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, selfsustaining chiral elastic waves are revealed in overdamped chiral active crystals selfassembled from thousands of swimming starfish embryos [32] (Figure 2F).
Figure 2 Collective behaviours in chiral active systems. (A) The circle swimming algae E. voratum generates a periodaveraged 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©2021 National Academy of Sciences. (B) Anisotropic pearshaped 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©2020 The Author(s). (C) Hydrodynamic interactions in an ensemble of isotropic rotors leads to a cascade of transverse dynamics and the formation of multiscale clockwise and counterclockwise vortices. Reprinted with permission from [34]. Copyright©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©2019 Springer 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©2022 Springer Nature. (F) Spontaneous assembly of swimming starfish embryos (σ≈200 m) into a chiral active crystal featuring sustained overdamped odd elastic waves. Reprinted with permission from [32]. Copyright©2022 Springer Nature. 
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 antisymmetric transport coefficients that are absent in usual nonchiral matter [53, 54]. Moreover, these antisymmetric contributions are even forbidden by energy conservation in equilibrium systems [55]. In soft matter physics, the effects of these odd (in the sense of noteven, or antisymmetric) 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 densityvorticity 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 [5760] and experimental [61, 32, 43] 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 NavierStokes equations, with coarsegrained 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 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 NavierStokes equations. This concept can be generalised from usual fluids to any continuous material, or as continuum approximated medium, including different active matter continua [1, 7579]. 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 nonequilibrium 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 threedimensional (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 twodimensional (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].
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 steadystate velocity profile decays exponentially from the boundaries. 
The viscosity tensor^{2)} in two dimensions, as any other tensor, can be written as the sum of its symmetric () and antisymmetric () part with respect to the index exchange .(1)where and . The viscous energy dissipation in the fluid per unit time and unit volume can be calculated as [80] , where is the viscous stress tensor and is the shear rate. Accordingly, only the symmetric contributions of the viscosity are associated with dissipation because the antisymmetric parts cancel in the summation.
Note, that the nondissipative nature of the odd, or antisymmetric viscous stress contributions can also be shown by deriving these contributions with a microscopic Hamiltonian [81], an energy conserving approach. The 16element rankfour tensor can be represented in another basis as a 4×4matrix η_{ij}, where i, j=0, 1, 2, 3. In this basis, the shear rate and stress tensors can be expressed as the vectors and σ_{i}, respectively. Then, the linear relationship between viscous stress and shear rates in an isotropic fluid can be expressed as [56, 54](2)In this representation, and can be interpreted with dilation or compression, and with rotational stresses, and with shear according to horizontal elongation and vertical compression, and and shear along an axis rotated by 45° in contrast to and [54]. Note that the symmetry is equivalent to η_{ij}=η_{ji}. On the one hand, the symmetric and dissipative shear η, rotational η_{R}, and bulk viscosities appear as in any compressible viscous fluid, leading to normal viscous dissipation acting upon (rotational) shear disturbances and compression. The possible antisymmetric 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 nondissipative, or symmetric and antisymmetric 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 . The stress tensor is composed of the contributions stemming from the viscosity tensor(3)but also accounts for stresses in the fluid in the absence of shearing as the pressure . In a chiral active fluid, the intrinsic rotation of the constituents gives rise to the angular velocity density and thus another shearindependent contribution to the stress tensor . Since , this term is also antisymmetric [82], which however is not directly associated with the odd viscosity term.
The generalised NavierStokes equation is then obtained as(4)The lefthand side describes inertial contributions, equivalently to the ordinary NavierStokes equation. The first three terms on the righthand side denote force densities due to pressure gradients, shearing, and compression, respectively. The 2D fluid vorticity measures twice the local circulation of the fluid. Accordingly, the third term on the righthand side of Equation (4) represents force densities in the fluid that try to synchronise the intrinsic angular velocity density with the fluid vorticity, and vanishes if the local angular velocity density equals the local circulation of the fluid particles ω/2. This term thus couples the intrinsic rotation of the constituent particles to the fluid vorticity and thus to the fluid velocity. The last term on the righthand side of Equation (4) describes force densities proportional to odd viscosity, acting perpendicular to the direction of local shear flows, since(5)To exemplify this, consider the following simplified shear experiment sketched in Figure 3A. Two infinitely extended parallel noslip 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 steadystate 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 , while stresses resulting from odd viscosity act perpendicular to the direction of shear (blue arrows in Figure 3A) with corresponding force density . 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 [81, 53] and thus . 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, and thus 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 fully determines the dynamics. Note that the compression term in Equation (4) to ζ then also vanishes. For an incompressible chiral active fluid (), the odd viscosity term in Equation (4) can be rewritten as [53](6)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 lefthand 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]:(7a)(7b)If additionally, the 2D fluid layer dissipates momentum into a frictious substrate with linear friction coefficient , the righthand side of Equation (7a) then has to be balanced by the friction term . 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].
CHIRAL ACTIVE FLUIDS
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 righthand 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., noslip 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 nonzero 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 nostress 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 semidilute 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 coarsegrained 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 (7) 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.
Figure 4 Vorticity (top) and density (bottom) correlations resulting from odd viscosity in particle based hydrodynamic simulations (left) and experiments (right). The weak compressibility and the presence of a radial effective pressure resulting from odd viscosity amounts to density inhomogeneities , where v^{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 v^{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 . Accordingly, density inhomogeneities resulting from odd shear stresses can only be perceived in longlived vortical flows, since viscous stresses are transported much faster then odd shear stresses. Reprinted with permission from [34]. Copyright©2023 The Author(s). 
More generally, to understand the physics of forces acting in compressible fluids with odd viscosity, an Oseentype 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 [87, 57] and timedependent linear [88] dynamics and microswimmer propulsion mechanisms, pusher and pullerlike 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].
Figure 5 Streamlines of the flow created by a point force into the xdirection in a quasitwodimensional 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©2021 the American Physical Society. (C) Trajectory (red) of a particle (red) subject to a constant force into the xdirection 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©2019 the American Physical Society. 
In contrast to incompressible fluids, a tracer with noslip 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 antisymmetric contributions to the friction tensor(8)and accordingly to transverse forces 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 [72, 58]. The energy dissipated during the dragging of the disk of mass M is , and the applied force is such that the dissipated energy can be written as , where the antisymmetric 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 effects, this circumstance partially extends to cohesive chiral active fluids [35, 32], where the particles attract each other, by virtue of electromagnetic interactions [94, 35, 43]. When chiral active agents do not bear attractive interactions that lead to crystalline [32] or cohesive [35] states of matter, longranged hydrodynamic interactions typically become a dominating effect. However, it should be noted that externally actuated rotation and active selfrotation are different. Actively selfrotating 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 cycleaveraged 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 selfrotating particles are very close to each other.
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. 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 corotates with the rotor and the azimuthal flow profile decays like r^{1} for disks or r^{2} for spheres. 
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 corotating azimuthal fluid flow decays like r^{1} for disks or rods and r^{2} for spheres. In comparison to selfrotating swimmers, longranged 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 multiscale 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 [97, 34, 98].
Odd diffusivity
Similar to stresses acting perpendicular to applied shear rates, fluxes perpendicular to concentration gradients appear in chiral active fluids as a result of time reversal and parity symmetry breaking [62]. Then, the diffusivity tensor in the fluxconcentration equation is no longer diagonal, but shows the emergence of offdiagonal terms , analogous to the friction tensor in Equation (9) (Figure 7A). The fluxes proportional to are divergence free, such that the continuity equation yields the unaltered diffusion equation and the concentration is not altered by when the boundary conditions only involve concentrations [62]. Figure 7B and C depict simulation results for and , where flips sign upon changing the direction of the chiral activity.
Figure 7 (A) Linear density gradient (colour) leading to flux (arrows) with transverse component arising from . (B, C) Diffusion coefficients and 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 GreenKubo relations and the relation between flux and concentration by maintaining a constant density gradient (boundary flux). Reprinted with permission from [62]. Copyright©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©2022 the American Physical Society. 
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 [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 selfdiffusion coefficient D_{s} decreases in the low density limit with increasing density φ as D_{s}=D_{0}(12φ) [99], where D_{0} is the individual particles diffusivity. The authors of reference [63] established a chiral active system(9)with . Accordingly, selfdiffusion 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, selfdiffusion decreases with increasing φ for , similar to a normal diffusive system. On the other hand, for , 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 dominates over , i.e., , selfdiffusion can be enhanced by increasing the density φ.
It should be noted, however, that even though the selfdiffusion 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 [94, 31, 102, 36, 32, 43]. Then, depending on the strength of the cohesive forces and the strength of activity, the material might either form a crystalline structure [36, 32], or will maintain some active fluidity and the formation of smaller subunits can be observed [94, 43]. In principle, odd viscosity and odd elasticity could also appear together in the framework of odd viscoelasticity [70].
Figure 8 (A) Nonpotential force between two masses acting transverse () and radial () to the conncting spring. (B) Compressing the spring results in a radial force, while extension results into an opposite radial force. The closed cycle () 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 selfsustaining 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©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©2022 Springer Nature. 
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, , where and 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 coarsegraining 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 nonpotential nature of the microscopic transverse forces, odd elastic solids may show a nonzero 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 , where A and are the area and contour of the cycle [55].
A phenomenological consequence of odd elasticity is the emergence of selfsustaining vibrational dynamics in overdamped solids [55]. A 2D odd elastic solid grown on a substrate follows the overdamped equation of motion:(10a)(10b)where U and 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 antisymmetric 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 nonHermitian 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 , the eigenvectors become colinear, and the system reaches an exceptional point, a telltale sign of nonHermitian dynamics [55, 103]. In the limit of dominating odd contributions, , the waves become selfsustaining.
Experimentally, selfsustaining 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 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 ydirections. Moreover, the authors conclude that the crystal effectively does work on the surrounding fluid.
In chiral active systems, the nonreciprocal 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 nonreciprocal and elastic forces leads to selfpropelled 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 , where is a counterclockwise closed contour around the dislocation, and thus depends on the displacement field. It is thus possible, that the selfpropelled 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 [40, 35, 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 .(11)where . Performing Fourier transformation in space and time^{4)} reveals for the dispersion relation between inverse dissipative timescale and wavenumber . The friction scale λ guarantees (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 [105]. Accordingly, a steadystate 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 , that is the “band gap” closes, and a faster delocalised current forms [106].
Figure 9 (A) Band structure 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©2019 the American Physical Society. (B, C) Reprinted with permission from [105]. Copyright©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©2021 the American Physical Society. 
The connection between topological insulators and edge flows in chiral active fluids can be drawn more rigorously [107, 71]. In a weakly compressible chiral active fluid in a circular container with odd viscosity and without substrate friction, a stationary linear solid bodylike 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 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 [26]. It is calculated by an integration over qspace, and a nonzero odd viscosity is necessary for a regularisation such that the Chern numbers are well defined^{5)} [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 nonzero 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 finiteelement 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 nonHermitian, 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 nonreciprocal 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 nonHermitian system can be modified by the boundary conditions [109]. Skin modes thus might serve as an alternative design for scatteringfree 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 selforganisation 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 counterclockwise, 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 [114117]. Interestingly, the behaviour of these rotors can be controlled by introducing active soft boundaries, which consist of interconnected particles with both clockwise and counterclockwise rotation [118] (Figure 10B). Additionally, the interconnection of these particles in different geometries leads to fascinating selforganising behaviours, reminiscent of amphiphilic behaviour seen in surfactants, such as doublestranded soft asymmetric boundaries (confer Figure 10C) show affinity to clockwise/counterclockwise interfaces [114] which could be employed for segregation or ordering in rotor systems.
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©2010 IOP Publishing Ltd. (B) Rotor binary mixture with soft active boundaries for varying boundary composition. Reprinted with permission from [118]. Copyright©2015 National Academy of Sciences. (C) Binary mixture of vibrational granular rotors with a doublestranded rotor chain in the evolution of time. Reprinted with permission from [114]. Copyright©2021 The Author(s). (D) A chiral active fluid forced through a grid of fixed obstacles. Reprinted with permission from [69]. Copyright©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©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©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©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 nonzero chirality. Reprinted with permission from [121]. Copyright©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©2019 the American Physical Society. 
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 steadystate 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 steadystate flow can still serve as an analogue to the anomalous Hall effect, where spinorbit 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 [124126]. 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 [5, 4, 128] is further subject to torque, the linear selfdriven 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 selfphoretic 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 noslip 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 counterrotating flagella bundle acting into opposite directions resulting in a torque from the hydrodynamic interactions between the rotating bacterium and a noslip wall [130, 3]. 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 selfpropelling 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].
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 brightfield image of V. cholerae. (A) Reprinted with permission from [130]. Copyright©2006 Elsevier. (B) Reprinted with permission from [132]. Copyright©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©2017 National Academy of Sciences. (D) Asymmetric Zn/Au rods (inset) show selfelectrophoresis 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©2020 American Chemical Society. (E) Asymmetric Lshaped colloids exhibit selfphoretic 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©2013 the American Physical Society. 
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 selfpropelled polymers with intrinsic curvature and chiral circular dynamics can selfassemble into vortex structures such as closed rings, arising from only steric interactions [142, 143]. Moreover, when the chiral active swimmers are Lshaped, 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 nonreciprocal 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 longwavelength density fluctuations akin to crystalline structures [30, 146]. Crystals exhibit longrange order and the structure factor S_{q} and density fluctuations behave like and , 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 and λ=d [146, 147]. When a system shows density fluctuations with λ>d and a structure factor , 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, 149151]. However, recently, chiral active fluids have been shown to exhibit hyperuniformity [146, 30, 148, 152], leading to the suppression of largescale 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 longrange 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 noninteracting 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 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 longranged 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 longranged interactions [30], as has also been observed for an ensemble of point vortices [152]. Without longranged 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 fluidlike behaviour of a noninteracting spinner fluid [148].
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 Rdependent 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©2019 The Author(s). 
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, antisymmetric stresses, odd diffusion, topologically protected edge modes, and nondissipative 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 [72, 58, 8991]. 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 [105, 26]. 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 [35, 34], 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.
Greek indices are used for the spacial dimensions and summation over repeated indices is implied, in other words, is the divergence of the velocity field v, where . Furthermore, δ_{αβ} is the Kronecker delta which equals unity if α=β and equals zero otherwise, and ε_{αβ} is the twodimensional LeviCivita symbol with ε_{xy}=1, ε_{yx}=1, and ε_{aa}=0.
Funding
J.M. gratefully acknowledges the National Natural Sience Foundation of China for supporting this project within the Research Fund for International Young Scientists (12350410368). Y.G. acknowledges financial support from the Natural Science Foundation of Guangdong Province (2024A1515011343), and the Key Project of Guangdong Provincial Department of Education (2023ZDZX3021).
Author contributions
J.M. and Y.G. conceived the work; J.M., J.O.N., R.L. and Y.G. conducted the literature review; J.M. took a lead in drafting the manuscript; all authors have participated in critical reading and revising the manuscript, and approved the final version; Y.G. supervised the entire work.
Conflict of interest
The authors declare no conflict of interest.
References
 Marchetti MC, Joanny JF, Ramaswamy S, et al. Hydrodynamics of soft active matter. Rev Mod Phys 2013; 85: 1143–1189.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Elgeti J, Winkler RG, Gompper G. Physics of microswimmerssingle particle motion and collective behavior: A review. Rep Prog Phys 2015; 78: 056601.[Article] [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
 Gompper G, Winkler RG, Speck T, et al. The 2020 motile active matter roadmap. J PhysCondens Matter 2020; 32: 193001.[Article] [CrossRef] [PubMed] [Google Scholar]
 Paxton WF, Kistler KC, Olmeda CC, et al. Catalytic nanomotors: Autonomous movement of striped nanorods. J Am Chem Soc 2004; 126: 13424–13431.[Article] [CrossRef] [PubMed] [Google Scholar]
 Howse JR, Jones RAL, Ryan AJ, et al. Selfmotile colloidal particles: From directed propulsion to random walk. Phys Rev Lett 2007; 99: 048102.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Jiang HR, Yoshinaga N, Sano M. Active motion of a Janus particle by selfthermophoresis in a defocused laser beam. Phys Rev Lett 2010; 105: 268302.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Gangwal S, Cayre OJ, Bazant MZ, et al. Inducedcharge electrophoresis of metallodielectric particles. Phys Rev Lett 2008; 100: 058302.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Ghosh A, Fischer P. Controlled propulsion of artificial magnetic nanostructured propellers. Nano Lett 2009; 9: 2243–2245.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Zhang J, Luijten E, Grzybowski BA, et al. Active colloids with collective mobility status and research opportunities. Chem Soc Rev 2017; 46: 5551–5569.[Article] [CrossRef] [PubMed] [Google Scholar]
 Xie H, Sun M, Fan X, et al. Reconfigurable magnetic microrobot swarm: Multimode transformation, locomotion, and manipulation. Sci Robot 2019; 4: eaav8006.[Article] [Google Scholar]
 Yang M, Zhang Y, Mou F, et al. Swarming magnetic nanorobots biointerfaced by heparinoidpolymer brushes for in vivo safe synergistic thrombolysis. Sci Adv 2023; 9: eadk7251.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Zheng J, Chen J, Jin Y, et al. Photochromism from wavelengthselective colloidal phase segregation. Nature 2023; 617: 499–506.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Urso M, Ussia M, Peng X, et al. Reconfigurable selfassembly of photocatalytic magnetic microrobots for water purification. Nat Commun 2023; 14: 6969.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Schmidt CK, MedinaSánchez M, Edmondson RJ, et al. Engineering microrobots for targeted cancer therapies from a medical perspective. Nat Commun 2020; 11: 5618.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Grosser S, Lippoldt J, Oswald L, et al. Cell and nucleus shape as an indicator of tissue fluidity in carcinoma. Phys Rev X 2021; 11: 011033.[Article] [NASA ADS] [Google Scholar]
 Ilina O, Gritsenko PG, Syga S, et al. Cellcell adhesion and 3D matrix confinement determine jamming transitions in breast cancer invasion. Nat Cell Biol 2020; 22: 1103–1115.[Article] [CrossRef] [PubMed] [Google Scholar]
 Gottheil P, Lippoldt J, Grosser S, et al. State of cell unjamming correlates with distant metastasis in cancer patients. Phys Rev X 2023; 13: 031003.[Article] [NASA ADS] [Google Scholar]
 MedinaSánchez M, Schwarz L, Meyer AK, et al. Cellular cargo delivery: Toward assisted fertilization by spermcarrying micromotors. Nano Lett 2016; 16: 555–561.[Article] [CrossRef] [PubMed] [Google Scholar]
 Nagel AM, Greenberg M, Shendruk TN, et al. Collective dynamics of model pilibased twitchermode bacilliforms. Sci Rep 2020; 10: 10747.[Article] [Google Scholar]
 Worlitzer VM, Jose A, Grinberg I, et al. Biophysical aspects underlying the swarm to biofilm transition. Sci Adv 2022; 8: eabn8152.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Palagi S, Fischer P. Bioinspired microrobots. Nat Rev Mater 2018; 3: 113–124.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Langer R. New methods of drug delivery. Science 1990; 249: 1527–1533.[Article] [CrossRef] [PubMed] [Google Scholar]
 Tran S, DeGiovanni P Piel B, et al. Cancer nanomedicine: A review of recent success in drug delivery. Clin Transl Med 2017; 6: e44.[Article] [CrossRef] [Google Scholar]
 Kim DK, Dobson J. Nanomedicine for targeted drug delivery. J Mater Chem 2009; 19: 6294.[Article] [CrossRef] [Google Scholar]
 Liebchen B, Levis D. Chiral active matter. EPL 2022; 139: 67001.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Shankar S, Souslov A, Bowick MJ, et al. Topological active matter. Nat Rev Phys 2022; 4: 380–398.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Fürthauer S, Strempel M, Grill SW, et al. Active chiral fluids. Eur Phys J E 2012; 35: 89.[Article] [CrossRef] [PubMed] [Google Scholar]
 Lenz P, Joanny JF, Jülicher F, et al. Membranes with rotating motors. Phys Rev Lett 2003; 91: 108104.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Noji H, Yoshida M. The rotary machine in the cell, ATP synthase. J Biol Chem 2001; 276: 1665–1668.[Article] [CrossRef] [PubMed] [Google Scholar]
 Huang M, Hu W, Yang S, et al. Circular swimming motility and disordered hyperuniform state in an algae system. Proc Natl Acad Sci USA 2021; 118: e2100493118.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Petroff AP, Wu XL, Libchaber A. Fastmoving bacteria selforganize into active twodimensional crystals of rotating cells. Phys Rev Lett 2015; 114: 158102.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Tan TH, Mietke A, Li J, et al. Odd dynamics of living chiral crystals. Nature 2022; 607: 287–293.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Drescher K, Leptos KC, Tuval I, et al. Dancing Volvox: Hydrodynamic bound states of swimming algae. Phys Rev Lett 2009; 102: 168101.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Mecke J, Gao Y, Ramírez Medina CA, et al. Simultaneous emergence of active turbulence and odd viscosity in a colloidal chiral active system. Commun Phys 2023; 6: 324.[Article] [Google Scholar]
 Soni V, Bililign ES, Magkiriadou S, et al. The odd free surface flows of a colloidal chiral fluid. Nat Phys 2019; 15: 1188–1194.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Yan J, Bae SC, Granick S. Rotating crystals of magnetic Janus colloids. Soft Matter 2015; 11: 147–153.[Article] [Google Scholar]
 Han K, Kokot G, Das S, et al. Reconfigurable structure and tunable transport in synchronized active spinner materials. Sci Adv 2020; 6: eaaz8535.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Grzybowski BA, Stone HA, Whitesides GM. Dynamic selfassembly of magnetized, millimetresized objects rotating at a liquidair interface. Nature 2000; 405: 1033–1036.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Modin A, Ben Zion MY, Chaikin PM. Hydrodynamic spinorbit coupling in asynchronous optically driven microrotors. Nat Commun 2023; 14: 4114.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Liu P, Zhu H, Zeng Y, et al. Oscillating collective motion of active rotors in confinement. Proc Natl Acad Sci USA 2020; 117: 11901–11907.[Article] [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
 Yang X, Ren C, Cheng K, et al. Robust boundary flow in chiral active fluid. Phys Rev E 2020; 101: 022603.[Article] [Google Scholar]
 Gao Y, Balin AK, Dullens RPA, et al. Thermal analog of gimbal lock in a colloidal ferromagnetic Janus rod. Phys Rev Lett 2015; 115: 248301.[Article] [Google Scholar]
 Bililign ES, Balboa Usabiaga F, Ganan YA, et al. Motile dislocations knead odd crystals into whorls. Nat Phys 2022; 18: 212–218.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Mun J, Kim M, Yang Y, et al. Electromagnetic chirality: From fundamentals to nontraditional chiroptical phenomena. Light Sci Appl 2020; 9: 139.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Liu Y, Xiao J, Koo J, et al. Chiralitydriven topological electronic structure of DNAlike materials. Nat Mater 2021; 20: 638–644.[Article] [CrossRef] [PubMed] [Google Scholar]
 Xu L, Wang X, Wang W, et al. Enantiomerdependent immunological response to chiral nanoparticles. Nature 2022; 601: 366–373.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Kim DS, Kim M, Seo S, et al. Natureinspired chiral structures: Fabrication methods and multifaceted applications. Biomimetics 2023; 8: 527.[Article] [CrossRef] [PubMed] [Google Scholar]
 Michaeli K, KantorUriel N, Naaman R, et al. The electron’s spin and molecular chirality  How are they related and how do they affect life processes? Chem Soc Rev 2016; 45: 6478–6487.[Article] [Google Scholar]
 Arora P, Sood AK, Ganapathy R. Emergent stereoselective interactions and selfrecognition in polar chiral active ellipsoids. Sci Adv 2021; 7: eabd0331.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Zhang B, Sokolov A, Snezhko A. Reconfigurable emergent patterns in active chiral fluids. Nat Commun 2020; 11: 4401.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Shah ZH, Wang S, Xian L, et al. Highly efficient chemicallydriven micromotors with controlled snowmanlike morphology. Chem Commun 2020; 56: 15301–15304.[Article] [CrossRef] [PubMed] [Google Scholar]
 Zhou X, Wang S, Xian L, et al. Ionic effects in ionic diffusiophoresis in chemically driven active colloids. Phys Rev Lett 2021; 127: 168001.[Article] [Google Scholar]
 Banerjee D, Souslov A, Abanov AG, et al. Odd viscosity in chiral active fluids. Nat Commun 2017; 8: 1573.[Article] [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
 Fruchart M, Scheibner C, Vitelli V. Odd viscosity and odd elasticity. Annu Rev Condens Matter Phys 2023; 14: 471–510.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Scheibner C, Souslov A, Banerjee D, et al. Odd elasticity. Nat Phys 2020; 16: 475–480.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Avron JE. Odd viscosity. J Statistical Phys 1998; 92: 543–557.[Article] [CrossRef] [MathSciNet] [Google Scholar]
 Khain T, Scheibner C, Fruchart M, et al. Stokes flows in threedimensional fluids with odd and parityviolating viscosities. J Fluid Mech 2022; 934: A23.[Article] [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Hosaka Y, Komura S, Andelman D. Nonreciprocal response of a twodimensional fluid with odd viscosity. Phys Rev E 2021; 103: 042610.[Article] [Google Scholar]
 Han M, Fruchart M, Scheibner C, et al. Fluctuating hydrodynamics of chiral active fluids. Nat Phys 2021; 17: 1260–1269.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Souslov A, Gromov A, Vitelli V. Anisotropic odd viscosity via a timemodulated drive. Phys Rev E 2020; 101: 052606.[Article] [Google Scholar]
 Yang Q, Zhu H, Liu P, et al. Topologically protected transport of cargo in a chiral active fluid aided by oddviscosityenhanced depletion interactions. Phys Rev Lett 2021; 126: 198001.[Article] [Google Scholar]
 Hargus C, Epstein JM, Mandadapu KK. Odd diffusivity of chiral random motion. Phys Rev Lett 2021; 127: 178001.[Article] [Google Scholar]
 Kalz E, Vuijk HD, Abdoli I, et al. Collisions enhance selfdiffusion in odddiffusive systems. Phys Rev Lett 2022; 129: 090601.[Article] [Google Scholar]
 Braverman L, Scheibner C, VanSaders B, et al. Topological defects in solids with odd elasticity. Phys Rev Lett 2021; 127: 268001.[Article] [Google Scholar]
 Chen Y, Li X, Scheibner C, et al. Realization of active metamaterials with odd micropolar elasticity. Nat Commun 2021; 12: 5935.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Lapa MF, Hughes TL. Swimming at low Reynolds number in fluids with odd, or Hall, viscosity. Phys Rev E 2014; 89: 043019.[Article] [Google Scholar]
 Yasuda K, Ishimoto K, Kobayashi A, et al. Timecorrelation functions for odd Langevin systems. J Chem Phys 2022; 157: 095101.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Reichhardt CJO, Reichhardt C. Active rheology in oddviscosity systems. EPL 2022; 137: 66004.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Lou X, Yang Q, Ding Y, et al. Odd viscosityinduced Halllike transport of an active chiral fluid. Proc Natl Acad Sci USA 2022; 119: e2201279119.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Banerjee D, Vitelli V, Jülicher F, et al. Active viscoelasticity of odd materials. Phys Rev Lett 2021; 126: 138001.[Article] [Google Scholar]
 Souslov A, Dasbiswas K, Fruchart M, et al. Topological waves in fluids with odd viscosity. Phys Rev Lett 2019; 122: 128001.[Article] [Google Scholar]
 Reichhardt C, Reichhardt CJO. Active microrheology, Hall effect, and jamming in chiral fluids. Phys Rev E 2019; 100: 012604.[Article] [Google Scholar]
 Banerjee D, Souslov A, Vitelli V. Hydrodynamic correlation functions of chiral active fluids. Phys Rev Fluids 2022; 7: 043301.[Article] [Google Scholar]
 Rao P, Bradlyn B. Resolving Hall and dissipative viscosity ambiguities via boundary effects. Phys Rev B 2023; 107: 075148.[Article] [Google Scholar]
 Toner J, Tu Y. Flocks, herds, and schools: A quantitative theory of flocking. Phys Rev E 1998; 58: 4828–4858.[Article] [Google Scholar]
 Baskaran A, Marchetti MC. Statistical mechanics and hydrodynamics of bacterial suspensions. Proc Natl Acad Sci USA 2009; 106: 15567–15572.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Doostmohammadi A, IgnésMullol J, Yeomans JM, et al. Active nematics. Nat Commun 2018; 9: 3246.[Article] [CrossRef] [PubMed] [Google Scholar]
 Reinken H, Klapp SHL, Bär M, et al. Derivation of a hydrodynamic theory for mesoscale dynamics in microswimmer suspensions. Phys Rev E 2018; 97: 022613.[Article] [Google Scholar]
 Steffenoni S, Falasco G, Kroy K. Microscopic derivation of the hydrodynamics of activeBrownianparticle suspensions. Phys Rev E 2017; 95: 052142.[Article] [Google Scholar]
 Landau L, Lifshitz E. Fluid mechanics, Course of Theoretical Physics. 2nd Ed. Oxford: Pergamon Press, 1987 [Google Scholar]
 Markovich T, Lubensky TC. Odd viscosity in active matter: Microscopic origin and 3D effects. Phys Rev Lett 2021; 127: 048001.[Article] [Google Scholar]
 Dahler JS, Scriven LE. Angular momentum of continua. Nature 1961; 192: 36–37.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Mecke J, Ripoll M. Birotor hydrodynamic microswimmers: From single to collective behaviour. EPL 2023; 142: 27001.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Hosaka Y, Komura S, Andelman D. Hydrodynamic lift of a twodimensional liquid domain with odd viscosity. Phys Rev E 2021; 104: 064613.[Article] [Google Scholar]
 Ganeshan S, Abanov AG. Odd viscosity in twodimensional incompressible fluids. Phys Rev Fluids 2017; 2: 094101.[Article] [Google Scholar]
 Purcell EM. Life at low Reynolds number. Am J Phys 1977; 45: 3–11.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Yuan H, Olvera de la Cruz M. Stokesian dynamics with odd viscosity. Phys Rev Fluids 2023; 8: 054101.[Article] [Google Scholar]
 Abanov A, Can T, Ganeshan S. Odd surface waves in twodimensional incompressible fluids. SciPost Phys 2018; 5: 010.[Article] [CrossRef] [Google Scholar]
 Hosaka Y, Golestanian R, Vilfan A. Lorentz reciprocal theorem in fluids with odd viscosity. Phys Rev Lett 2023; 131: 178303.[Article] [Google Scholar]
 Hosaka Y, Andelman D, Komura S. Pair dynamics of active force dipoles in an oddviscous fluid. Eur Phys J E 2023; 46: 18.[Article] [CrossRef] [PubMed] [Google Scholar]
 Hosaka Y, Golestanian R, DaddiMoussaIder A. Hydrodynamics of an odd active surfer in a chiral fluid. New J Phys 2023; 25: 083046.[Article] [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Lier R, Duclut C, Bo S, et al. Lift force in odd compressible fluids. Phys Rev E 2023; 108: L023101.[Article] [Google Scholar]
 Tsai JC, Ye F, Rodriguez J, et al. A chiral granular gas. Phys Rev Lett 2005; 94: 214301.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 MassanaCid H, Levis D, Hernández RJH, et al. Arrested phase separation in chiral fluids of colloidal spinners. Phys Rev Res 2021; 3: L042021.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Fily Y, Baskaran A, Marchetti MC. Cooperative selfpropulsion of active and passive rotors. Soft Matter 2012; 8: 3002.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Llopis I, Pagonabarraga I. Hydrodynamic regimes of active rotators at fluid interfaces. Eur Phys J E 2008; 26: 103–113.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Alert R, Casademunt J, Joanny JF. Active turbulence. Annu Rev Condens Matter Phys 2022; 13: 143–170.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Reeves CJ, Aranson IS, Vlahovska PM. Emergence of lanes and turbulentlike motion in active spinner fluid. Commun Phys 2021,4: 92. [Google Scholar]
 Hanna S, Hess W, Klein R. Selfdiffusion of spherical Brownian particles with hardcore interaction. Physica AStatistical Mech its Appl 1982; 111: 181–199.[Article] [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 van Roon DM, Volpe G, Telo da Gama MM, et al. The role of disorder in the motion of chiral active particles in the presence of obstacles. Soft Matter 2022; 18: 6899–6906.[Article] [Google Scholar]
 Vega Reyes F, LópezCastaño MA, RodríguezRivas Á. Diffusive regimes in a twodimensional chiral fluid. Commun Phys 2022; 5: 256.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Petroff AP, Libchaber A. Nucleation of rotating crystals by Thiovulummajus bacteria. New J Phys 2018; 20: 015007.[Article] [Google Scholar]
 Surówka P, Souslov A, Jülicher F, et al. Odd Cosserat elasticity in active materials. Phys Rev E 2023; 108: 064609.[Article] [Google Scholar]
 Poncet A, Bartolo D. When soft crystals defy Newton’s Third Law: Nonreciprocal mechanics and dislocation motility. Phys Rev Lett 2022; 128: 048002.[Article] [Google Scholar]
 Dasbiswas K, Mandadapu KK, Vaikuntanathan S. Topological localization in outofequilibrium dissipative systems. Proc Natl Acad Sci USA 2018; 115: E9031–E9040.[Article] [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
 Jia LL, Irvine WTM, Shelley MJ. Incompressible active phases at an interface. Part 1. Formulation and axisymmetric odd flows. J Fluid Mech 2022; 951: A36.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Sone K, Ashida Y. Anomalous topological active matter. Phys Rev Lett 2019; 123: 205502.[Article] [Google Scholar]
 Scheibner C, Irvine WTM, Vitelli V. Nonhermitian band topology and skin modes in active elastic media. Phys Rev Lett 2020; 125: 118001.[Article] [Google Scholar]
 Lee CH, Thomale R. Anatomy of skin modes and topology in nonHermitian systems. Phys Rev B 2019; 99: 201103.[Article] [Google Scholar]
 Sone K, Ashida Y, Sagawa T. Exceptional nonHermitian topological edge mode and its application to active matter. Nat Commun 2020; 11: 5745.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 van Zuiden BC, Paulose J, Irvine WTM, et al. Spatiotemporal order and emergent edge currents in active spinner materials. Proc Natl Acad Sci USA 2016; 113: 12919–12924.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Götze IO, Gompper G. Dynamic selfassembly and directed flow of rotating colloids in microchannels. Phys Rev E 2011; 84: 031404.[Article] [Google Scholar]
 Götze IO, Gompper G. Flow generation by rotating colloids in planar microchannels. EPL 2010; 92: 64003.[Article] [CrossRef] [Google Scholar]
 Scholz C, Ldov A, Pöschel T, et al. Surfactants and rotelles in active chiral fluids. Sci Adv 2021; 7: eabf8998.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Scholz C, Engel M, Pöschel T. Rotating robots move collectively and selforganize. Nat Commun 2018; 9: 931.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Yeo K, Lushi E, Vlahovska PM. Collective dynamics in a binary mixture of hydrodynamically coupled microrotors. Phys Rev Lett 2015; 114: 188301.[Article] [CrossRef] [PubMed] [Google Scholar]
 Yeo K, Lushi E, Vlahovska PM. Dynamics of inert spheres in active suspensions of microrotors. Soft Matter 2016; 12: 5645–5652.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Spellings M, Engel M, Klotsa D, et al. Shape control and compartmentalization in active colloidal cells. Proc Natl Acad Sci USA 2015; 112: E4642–E4650.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Mijalkov M, Volpe G. Sorting of chiral microswimmers. Soft Matter 2013; 9: 6376.[Article] [Google Scholar]
 Nishiguchi D, Aranson IS, Snezhko A, et al. Engineering bacterial vortex lattice via direct laser lithography. Nat Commun 2018; 9: 4486.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Souslov A, van Zuiden BC, Bartolo D, et al. Topological sound in activeliquid metamaterials. Nat Phys 2017; 13: 1091–1094.[Article] [Google Scholar]
 Li W, Li L, Shi Q, et al. Chiral separation of rotating robots through obstacle arrays. Powder Tech 2022; 407: 117671.[Article] [CrossRef] [Google Scholar]
 Toner J, Tu Y. Longrange order in a twodimensional dynamical XY model: How birds fly together. Phys Rev Lett 1995; 75: 4326–4329.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Farhadi S, Machaca S, Aird J, et al. Dynamics and thermodynamics of airdriven active spinners. Soft Matter 2018; 14: 5588–5594.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Workamp M, Ramirez G, Daniels KE, et al. Symmetryreversals in chiral active matter. Soft Matter 2018; 14: 5572–5580.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 LópezCastaño MA, Márquez Seco A, Márquez Seco A, et al. Chirality transitions in a system of active flat spinners. Phys Rev Res 2022; 4: 033230.[Article] [CrossRef] [Google Scholar]
 Petroff AP, Whittington C, Kudrolli A. Densitymediated spin correlations drive edgetobulk flow transition in active chiral matter. Phys Rev E 2023; 108: 014609.[Article] [Google Scholar]
 Yan J, Han M, Zhang J, et al. Reconfiguring active particles by electrostatic imbalance. Nat Mater 2016; 15: 1095–1099.[Article] [Google Scholar]
 Kümmel F, ten Hagen B, Wittkowski R, et al. Circular motion of asymmetric selfpropelling particles. Phys Rev Lett 2013; 110: 198302.[Article] [CrossRef] [PubMed] [Google Scholar]
 Lauga E, DiLuzio WR, Whitesides GM, et al. Swimming in circles: Motion of bacteria near solid boundaries. Biophys J 2006; 90: 400–412.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Riedel IH, Kruse K, Howard J. A selforganized vortex array of hydrodynamically entrained sperm cells. Science 2005; 309: 300–303.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Utada AS, Bennett RR, Fong JCN, et al. Vibrio cholerae use pili and flagella synergistically to effect motility switching and conditional surface attachment. Nat Commun 2014; 5: 4913.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Schmidt F, Liebchen B, Löwen H, et al. Lightcontrolled assembly of active colloidal molecules. J Chem Phys 2019; 150: 094905.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Lauga E, Powers TR. The hydrodynamics of swimming microorganisms. Rep Prog Phys 2009; 72: 096601.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Han M, Yan J, Granick S, et al. Effective temperature concept evaluated in an active colloid mixture. Proc Natl Acad Sci USA 2017; 114: 7513–7518.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 FernandezRodriguez MA, Grillo F, Alvarez L, et al. Feedbackcontrolled active brownian colloids with spacedependent rotational dynamics. Nat Commun 2020; 11: 4223.[Article] [CrossRef] [PubMed] [Google Scholar]
 Erglis K, Wen Q, Ose V, et al. Dynamics of magnetotactic bacteria in a rotating magnetic field. Biophys J 2007; 93: 1402–1412.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Caprini L, Löwen H, Marini Bettolo Marconi U. Chiral active matter in external potentials. Soft Matter 2023; 19: 6234–6246.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Du S, Wang H, Zhou C, et al. Motor and rotor in one: Lightactive ZnO/Au twinned rods of tunable motion modes. J Am Chem Soc 2020; 142: 2213–2217.[Article] [CrossRef] [PubMed] [Google Scholar]
 Liebchen B, Levis D. Collective behavior of chiral active matter: Pattern formation and enhanced flocking. Phys Rev Lett 2017; 119: 058002.[Article] [Google Scholar]
 Li JR, Zhu W, Li JJ, et al. Chiralityinduced directional rotation of a symmetric gear in a bath of chiral active particles. New J Phys 2023; 25: 043031.[Article] [CrossRef] [MathSciNet] [Google Scholar]
 Denk J, Huber L, Reithmann E, et al. Active curved polymers form vortex patterns on membranes. Phys Rev Lett 2016; 116: 178301.[Article] [Google Scholar]
 Dunajova Z, Mateu BP, Radler P, et al. Chiral and nematic phases of flexible active filaments. Nat Phys 2023; 19: 1916–1926.[Article] [Google Scholar]
 Liu Y, Yang Y, Li B, et al. Collective oscillation in dense suspension of selfpropelled chiral rods. Soft Matter 2019; 15: 2999–3007.[Article] [Google Scholar]
 Kreienkamp KL, Klapp SHL. Clustering and flocking of repulsive chiral active particles with nonreciprocal couplings. New J Phys 2022; 24: 123009.[Article] [Google Scholar]
 Lei QL, Ciamarra MP, Ni R. Nonequilibrium strongly hyperuniform fluids of circle active particles with large local density fluctuations. Sci Adv 2019; 5: eaau7423.[Article] [Google Scholar]
 Torquato S. Hyperuniform states of matter. Phys Rep 2018; 745: 1–95.[Article] [Google Scholar]
 Zhang B, Snezhko A. Hyperuniform active chiral fluids with tunable internal structure. Phys Rev Lett 2022; 128: 218002.[Article] [Google Scholar]
 Wagner M, RocaBonet S, Ripoll M. Collective behavior of thermophoretic dimeric active colloids in threedimensional bulk. Eur Phys J E 2021; 44: 43.[Article] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
 Stenhammar J, Tiribocchi A, Allen RJ, et al. Continuum theory of phase separation kinetics for active brownian particles. Phys Rev Lett 2013; 111: 145702.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Qi K, Westphal E, Gompper G, et al. Emergence of active turbulence in microswimmer suspensions due to active hydrodynamic stress and volume exclusion. Commun Phys 2022; 5: 49.[Article] [NASA ADS] [CrossRef] [Google Scholar]
 Oppenheimer N, Stein DB, Zion MYB, et al. Hyperuniformity and phase enrichment in vortex and rotor assemblies. Nat Commun 2022; 13: 804.[Article] [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Lei QL, Ni R. Hydrodynamics of randomorganizing hyperuniform fluids. Proc Natl Acad Sci USA 2019; 116: 22983–22989.[Article] [NASA ADS] [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
 Liu R, Gong J, Yang M, et al. Local rotational jamming and multistage hyperuniformities in an active spinner system. Chin Phys Lett 2023; 40: 126402.[Article] [Google Scholar]
 Aggarwal A, Kirkinis E, Olvera de la Cruz M. Thermocapillary migrating odd viscous droplets. Phys Rev Lett 2023; 131: 198201.[Article] [Google Scholar]
 McNeill JM, Choi YC, Cai YY, et al. Threedimensionally complex phase behavior and collective phenomena in mixtures of acoustically powered chiral microspinners. ACS Nano 2023; 17: 7911–7919.[Article] [CrossRef] [PubMed] [Google Scholar]
All Figures
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©2001 The Author(s). (B) Marine algae Effrenium voratum (σ≈10 m) with superimposed trajectory showing chiral circular swimming behaviour at the airliquid interface. Reprinted with permission from [30]. Copyright©2021 National Academy of Sciences. (C) Bacteria Thiovulum majus (σ≈10 m) on a surface induce a chiral tornadolike flow that leads to an attraction and mutual orbital rotation of neighbouring cells. Reprinted with permission from [31]. Copyright©2015 American Physical Society. (D) Actively spinning starfish embryos (σ≈200 m) form a corotating pair by flow generated by each other. Reprinted with permission from [32]. Copyright©2022 Springer Nature. (E) Volvox colonies (σ≈500 m) have a ciliated surface of beating flagella pairs on each somatic cell (small dots), leading to directed motion and chiral rotation. Reprinted with permission from [33]. Copyright©2009 the American Physical Society. (F) Silica rodlike colloids (σ≈1 m) with an adhered magnetic tip perpendicular to the symmetry axis [42] orient perpendicular to the substrate and rotate in sync with an externally applied rotating magnetic field. The colloids excite a rotational flow field advecting nearby colloids. Shown streamlines are obtained from simulations. Reprinted with permission from [34]. Copyright©2023 The Author(s). (G) Chiral magnetic colloidal spinners (σ≈2 m) drag the surrounding fluid and exert hydrodynamic transverse and magnetic attractive forces. Upper image and lower image are reprinted with permission from [43, 35], respectively. Copyright©2019 and 2022 Springer Nature. (H) Vaterite colloidal particles (σ≈212 m) asynchronously rotate in circularly polarised light resulting from birefringence, leading to hydrodynamic spinorbit coupling. Reprinted with permission from [39]. Copyright©2023 The Author(s). (I) 3Dprinted granular gearlike rotors (D1=16 mm, D2=21 mm) with tilted bristles at the bottom can be brought into a state of active rotation powered by vertical vibration. Reprinted with permission from [40]. Copyright©2020 National Academy of Sciences. (J) Two 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©2020 the American Physical Society. 

In the text 
Figure 2 Collective behaviours in chiral active systems. (A) The circle swimming algae E. voratum generates a periodaveraged 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©2021 National Academy of Sciences. (B) Anisotropic pearshaped 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©2020 The Author(s). (C) Hydrodynamic interactions in an ensemble of isotropic rotors leads to a cascade of transverse dynamics and the formation of multiscale clockwise and counterclockwise vortices. Reprinted with permission from [34]. Copyright©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©2019 Springer 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©2022 Springer Nature. (F) Spontaneous assembly of swimming starfish embryos (σ≈200 m) into a chiral active crystal featuring sustained overdamped odd elastic waves. Reprinted with permission from [32]. Copyright©2022 Springer Nature. 

In the text 
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 steadystate velocity profile decays exponentially from the boundaries. 

In the text 
Figure 4 Vorticity (top) and density (bottom) correlations resulting from odd viscosity in particle based hydrodynamic simulations (left) and experiments (right). The weak compressibility and the presence of a radial effective pressure resulting from odd viscosity amounts to density inhomogeneities , where v^{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 v^{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 . Accordingly, density inhomogeneities resulting from odd shear stresses can only be perceived in longlived vortical flows, since viscous stresses are transported much faster then odd shear stresses. Reprinted with permission from [34]. Copyright©2023 The Author(s). 

In the text 
Figure 5 Streamlines of the flow created by a point force into the xdirection in a quasitwodimensional 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©2021 the American Physical Society. (C) Trajectory (red) of a particle (red) subject to a constant force into the xdirection 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©2019 the American Physical Society. 

In the text 
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. 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 corotates with the rotor and the azimuthal flow profile decays like r^{1} for disks or r^{2} for spheres. 

In the text 
Figure 7 (A) Linear density gradient (colour) leading to flux (arrows) with transverse component arising from . (B, C) Diffusion coefficients and 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 GreenKubo relations and the relation between flux and concentration by maintaining a constant density gradient (boundary flux). Reprinted with permission from [62]. Copyright©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©2022 the American Physical Society. 

In the text 
Figure 8 (A) Nonpotential force between two masses acting transverse () and radial () to the conncting spring. (B) Compressing the spring results in a radial force, while extension results into an opposite radial force. The closed cycle () 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 selfsustaining 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©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©2022 Springer Nature. 

In the text 
Figure 9 (A) Band structure 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©2019 the American Physical Society. (B, C) Reprinted with permission from [105]. Copyright©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©2021 the American Physical Society. 

In the text 
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©2010 IOP Publishing Ltd. (B) Rotor binary mixture with soft active boundaries for varying boundary composition. Reprinted with permission from [118]. Copyright©2015 National Academy of Sciences. (C) Binary mixture of vibrational granular rotors with a doublestranded rotor chain in the evolution of time. Reprinted with permission from [114]. Copyright©2021 The Author(s). (D) A chiral active fluid forced through a grid of fixed obstacles. Reprinted with permission from [69]. Copyright©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©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©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©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 nonzero chirality. Reprinted with permission from [121]. Copyright©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©2019 the American Physical Society. 

In the text 
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 brightfield image of V. cholerae. (A) Reprinted with permission from [130]. Copyright©2006 Elsevier. (B) Reprinted with permission from [132]. Copyright©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©2017 National Academy of Sciences. (D) Asymmetric Zn/Au rods (inset) show selfelectrophoresis 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©2020 American Chemical Society. (E) Asymmetric Lshaped colloids exhibit selfphoretic 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©2013 the American Physical Society. 

In the text 
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 Rdependent 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©2019 The Author(s). 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.