Topological localization in out-of-equilibrium dissipative systems

Obstacle-enhanced spontaneous oscillation of confined active granules

Spontaneous oscillation in particle numbers has been reported recently, in which two chambers connected by a narrow channel are alternately filled and emptied by self-propelled particles. The challenge in realizing the application of this oscillation lies in promotion of the oscillatory periodicity. By placing an asymmetric obstacle at an appropriate position near a channel opening, we can significantly improve the oscillation quality, which approaches the quality of an ideal oscillation. Additionally, we experimentally explore the relationship between the oscillation quality and various system parameters such as the obstacle position. Based on experimental observations, we incorporate a random noise into our previous model and properly reproduce the experimental results. The agreement between theory and experiment uncovers the mechanism of delicate competition between noise and unidirectional particle flow in influencing the oscillation quality. Our findings provide new insights for the optimization of the oscillation quality, expand the conventional rectification capability of the ratchet effect due to the obstacle, and make it possible for spontaneous oscillation to serve as a reliable source for rhythmic signals.

Diffusion of active particles driven by odd interactions

Odd systems do not conserve energy, violate time-reversal symmetry, and remain far from equilibrium. How odd interactions between particles affect their diffusion remains unknown. To investigate this issue, we studied the diffusion and glass transition of a two-dimensional Kob-Andersen mixture, where Brownian particles interact via the Lennard-Jones potential and nonconservative odd forces. Our findings indicate a significant influence of odd interactions on the system's diffusion dynamics. Odd interactions always promote diffusion. These interactions lead to a nonmonotonic relationship between the effective diffusion coefficient and particle number density. Specifically, in systems with low oddness, the diffusion coefficient decreases steadily with increasing particle number density. Conversely, in systems with moderate oddness, an optimal particle number density exists that maximizes the diffusion coefficient. For systems with high oddness, we observe two distinct peaks in the diffusion coefficient-particle number density relationship. Furthermore, our investigation into the glass transition under dense conditions reveals that adjusting the oddness at low temperatures can induce a transition from a glassy state to a liquid state. Our findings offer a deeper insight into the diffusion processes in systems with odd interactions from a critical perspective.

Reviving the Lieb-Schultz-Mattis theorem in open quantum systems

In closed systems, the celebrated Lieb-Schultz-Mattis (LSM) theorem states that a one-dimensional locally interacting half-integer spin chain with translation and spin rotation symmetries cannot have a non-degenerate gapped ground state. However, the applicability of this theorem is diminished when the system interacts with a bath and loses its energy conservation. In this letter, we propose that the LSM theorem can be revived in the entanglement Hamiltonian when the coupling to the bath renders the system short-range correlated. Specifically, we argue that the entanglement spectrum cannot have a non-degenerate minimum, isolated by a gap from other states. We further support the results with numerical examples where a spin-[Formula: see text] system is coupled to another spin-[Formula: see text] chain serving as the bath. Compared with the original LSM theorem that primarily addresses UV-IR correspondence, our findings reveal that the UV data and topological constraints also play a pivotal role in shaping the entanglement in open quantum many-body systems.

Chiral edge waves in a dance-based human topological insulator

Topological insulators are insulators in the bulk but feature chiral energy propagation along the boundary. This property is topological in nature and therefore robust to disorder. Originally discovered in electronic materials, topologically protected boundary transport has since been observed in many other physical systems. Thus, it is natural to ask whether this phenomenon finds relevance in a broader context. We choreograph a dance in which a group of humans, arranged on a square grid, behave as a topological insulator. The dance features unidirectional flow of movement through dancers on the lattice edge. This effect persists when people are removed from the dance floor. Our work extends the applicability of wave physics to dance.

A topological mechanism for robust and efficient global oscillations in biological networks

Long and stable timescales are often observed in complex biochemical networks, such as in emergent oscillations. How these robust dynamics persist remains unclear, given the many stochastic reactions and shorter time scales demonstrated by underlying components. We propose a topological model that produces long oscillations around the network boundary, reducing the system dynamics to a lower-dimensional current in a robust manner. Using this to model KaiC, which regulates the circadian rhythm in cyanobacteria, we compare the coherence of oscillations to that in other KaiC models. Our topological model localizes currents on the system edge, with an efficient regime of simultaneously increased precision and decreased cost. Further, we introduce a new predictor of coherence from the analysis of spectral gaps, and show that our model saturates a global thermodynamic bound. Our work presents a new mechanism and parsimonious description for robust emergent oscillations in complex biological networks.

Phase separation, edge currents, and Hall effect for active matter with Magnus dynamics

We examine run-and-tumble disks in two-dimensional systems where the particles also have a Magnus component to their dynamics. For increased activity, we find that the system forms a motility-induced phase-separated (MIPS) state with chiral edge flow around the clusters, where the direction of the current is correlated with the sign of the Magnus term. The stability of the MIPS state is non-monotonic as a function of increasing Magnus term amplitude, with the MIPS region first extending down to lower activities followed by a break up of MIPS at large Magnus amplitudes into a gel-like state. We examine the dynamics in the presence of quenched disorder and a uniform drive and find that the bulk flow exhibits a drive-dependent Hall angle. This is a result of the side jump effect produced by scattering from the pinning sites and is similar to the behavior found for skyrmions in chiral magnets with quenched disorder.

Nonequilibrium protection effect and spatial localization of noise-induced fluctuations: Quasi-one-dimensional driven lattice gas with partially penetrable obstacle

We consider a nonequilibrium transition that leads to the formation of nonlinear steady-state structures due to the gas flow scattering on a partially penetrable obstacle. The resulting nonequilibrium steady state (NESS) corresponds to a two-domain gas structure attained at certain critical parameters. We use a simple mean-field model of the driven lattice gas with ring topology to demonstrate that this transition is accompanied by the emergence of local invariants related to a complex composed of the obstacle and its nearest gas surrounding, which we refer to as obstacle edges. These invariants are independent of the main system parameters and behave as local first integrals, at least qualitatively. As a result, the complex becomes insensitive to the noise of external driving field within the overcritical domain. The emerged invariants describe the conservation of the number of particles inside the obstacle and strong temporal synchronization or correlation of gas states at obstacle edges. Such synchronization guarantees the equality to zero of the total edge current at any time. The robustness against external drive fluctuations is shown to be accompanied by strong spatial localization of induced gas fluctuations near the domain wall separating the depleted and dense gas phases. Such a behavior can be associated with nonequilibrium protection effect and synchronization of edges. The transition rates between different NESSs are shown to be different. The relaxation rates from one NESS to another take complex and real values in the sub- and overcritical regimes, respectively. The mechanism of these transitions is governed by the generation of shock waves at the back side of the obstacle. In the subcritical regime, these solitary waves are generated sequentially many times, while only a single excitation is sufficient to rearrange the system state in the overcritical regime.

Topological phases in the dynamics of the simple exclusion process

We study the dynamical large deviations of the classical stochastic symmetric simple exclusion process (SSEP) by means of numerical matrix product states. We show that for half filling, long-time trajectories with a large enough imbalance between the number hops in even and odd bonds of the lattice belong to distinct symmetry-protected topological (SPT) phases. Using tensor network techniques, we obtain the large deviation (LD) phase diagram in terms of counting fields conjugate to the dynamical activity and the total hop imbalance. We show the existence of high activity trivial and nontrivial SPT phases (classified according to string order parameters) separated by either a critical phase or a critical point. Using the leading eigenstate of the tilted generator, obtained from infinite-system density-matrix renormalization group simulations, we construct a near-optimal dynamics for sampling the LDs, and show that the SPT phases manifest at the level of rare stochastic trajectories. We also show how to extend these results to other filling fractions, and discuss generalizations to asymmetric SEPs.

Collective motion of chiral particles in complex noise environments

Collective motion of chiral particles in complex noise environments is investigated based on the Vicsek model. In the model, we added chirality, along with complex noise, affecting particles clustering motion. Particles can only avoid noise interference in a specific channel, and this consideration is more realistic due to the complexity of the environment. Via simulations, we find that the channel proportion, p, critically influences chiral particle synchronization. Specifically, we observe a disorder-order transition at critical [Formula: see text], only when [Formula: see text], the system can achieve global synchronization. Combined with our definition of spatial distribution parameter and observation of the model, the reason is that particles begin to escape from the noise region under the influence of complex noise. In addition, the value of [Formula: see text] increases linearly with velocity, while it decreases monotonically with the increase in chirality and interaction radius. Interestingly, an appropriate noise amplitude minimizes [Formula: see text]. Our findings may inspire novel strategies to manipulate self-propelled particles of distinct chirality to achieve desired spatial migration and global synchronization.

Role of Topology in Relaxation of One-Dimensional Stochastic Processes

Stochastic processes are commonly used models to describe dynamics of a wide variety of nonequilibrium phenomena ranging from electrical transport to biological motion. The transition matrix describing a stochastic process can be regarded as a non-Hermitian Hamiltonian. Unlike general non-Hermitian systems, the conservation of probability imposes additional constraints on the transition matrix, which can induce unique topological phenomena. Here, we reveal the role of topology in relaxation phenomena of classical stochastic processes. Specifically, we define a winding number that is related to topology of stochastic processes and show that it predicts the existence of a spectral gap that characterizes the relaxation time. Then, we numerically confirm that the winding number corresponds to the system-size dependence of the relaxation time and the characteristic transient behavior. One can experimentally realize such topological phenomena in magnetotactic bacteria and cell adhesions.

Density-mediated spin correlations drive edge-to-bulk flow transition in active chiral matter

We demonstrate that edge currents develop in active chiral matter due to boundary shielding over a wide range of densities corresponding to a gas, fluid, and crystal. The system is composed of spinning disk-shaped grains with chirally arranged tilted legs confined in a circular vibrating chamber. The edge currents are shown to increasingly drive circulating bulk flows with area fraction as percolating clusters develop due to increasing spin-coupling between neighbors mediated by frictional contacts. Edge currents are observed even in the dilute limit. While, at low area fraction, the average flux vanishes except within a distance that is of the order of a particle diameter of the boundary, the penetration depth grows with increasing area fraction until a solid-body rotation is achieved corresponding to the highest packing, where the particles are fully caged with hexagonal order and spin in phase with the entire packing. A coarse-grained model, based on the increased collisional interlocking of the particles with area fraction and the emergence of order, captures the observed flow fields.

Odd Mobility of a Passive Tracer in a Chiral Active Fluid

Chiral active fluids break both time-reversal and parity symmetry, leading to exotic transport phenomena unobservable in ordinary passive fluids. We develop a generalized Green-Kubo relation for the anomalous lift experienced by a passive tracer suspended in a two-dimensional chiral active fluid subjected to an applied force. This anomalous lift is characterized by a transport coefficient termed the odd mobility. We validate our generalized response theory using molecular dynamics simulations, and we show that the asymmetric tracer mobility may be understood mechanically in terms of asymmetric deformations of the tracer-fluid density distribution function. We show that the even and odd components of the mobility decay at different rates with tracer size, suggesting the possibility of size-based particle separation using a chiral active working fluid.

Odd viscosity-induced Hall-like transport of an active chiral fluid

Broken time-reversal and parity symmetries in active spinner fluids imply a nondissipative "odd viscosity," engendering phenomena unattainable in traditional passive or active fluids. Here we show that the odd viscosity itself can lead to a Hall-like transport when the active chiral fluid flows through a quenched matrix of obstacles, reminiscent of the anomalous Hall effect. The Hall-like velocity depends significantly on the spinner activity and longitudinal flow due to the interplay between odd viscosity and spinner-obstacle collisions. Our findings underscore the importance of odd viscosity in active chiral matter and elucidate its essential role in the anomalous Hall-like effect.

A topological fluctuation theorem

Fluctuation theorems specify the non-zero probability to observe negative entropy production, contrary to a naive expectation from the second law of thermodynamics. For closed particle trajectories in a fluid, Stokes theorem can be used to give a geometric characterization of the entropy production. Building on this picture, we formulate a topological fluctuation theorem that depends only by the winding number around each vortex core and is insensitive to other aspects of the force. The probability is robust to local deformations of the particle trajectory, reminiscent of topologically protected modes in various classical and quantum systems. We demonstrate that entropy production is quantized in these strongly fluctuating systems, and it is controlled by a topological invariant. We demonstrate that the theorem holds even when the probability distributions are non-Gaussian functions of the generated heat.

Emergent spiral vortex of confined biased active particles

Confinement is known to have profound effects on the collective dynamics of many active systems. Here, we investigate a modeled active system in circular confinement consisting of biased active particles, where the direction of active force deviates a biased angle from the principle orientation of the anisotropic interaction. We find that such particles can spontaneously form a spiral vortex with two concentric and counter-rotating regions near the boundary. The emerged vortex can be measured by the vortex order parameter which shows nonmonotonic dependencies on both the biased angle and the strength of the anisotropic interaction. Our work can provide an understanding of such dynamic behaviors and enable different strategies for designing ordered collective behaviors.

Subradiant-to-Subradiant Phase Transition in the Bad Cavity Laser

We show that the onset of steady-state superradiance in a bad cavity laser is preceded by a dissipative phase transition between two distinct phases of steady-state subradiance. The transition is marked by a nonanalytic behavior of the cavity output power and the mean atomic inversion, as well as a discontinuity in the variance of the collective atomic inversion. In particular, for repump rates below a critical value, the cavity output power is strongly suppressed and does not increase with the atom number, while it scales linearly with atom number above this value. Remarkably, we find that the atoms are in a macroscopically entangled steady state near the critical region with a vanishing fraction of unentangled atoms in the large atom number limit.

Complete absorption of topologically protected waves

Chiral edge states can transmit energy along imperfect interfaces in a topologically robust and unidirectional manner when protected by bulk-boundary correspondence. However, in continuum systems, the number of states at an interface can depend on boundary conditions. Here we design interfaces that host a net flux of the number of modes into a region, trapping incoming energy. As a realization, we present a model system of two topological fluids composed of counter-spinning particles, which are separated by a boundary that transitions from a fluid-fluid interface into a no-slip wall. In these fluids, chiral edge states disappear, which implies non-Hermiticity and leads to an interplay between topology and energy dissipation. Solving the fluid equations of motion, we find explicit expressions for the disappearing modes. We then conclude that energy dissipation is sped up by mode trapping. Instead of making efficient waveguides, our paper shows how topology can be exploited for applications towards acoustic absorption, shielding, and soundproofing.

Topologically Protected Transport of Cargo in a Chiral Active Fluid Aided by Odd-Viscosity-Enhanced Depletion Interactions

The discovery of topological edge states that unidirectionally propagate along the boundary of system without backscattering has enabled the development of new design principles for material or information transport. Here, we show that the topological edge flow supported by the chiral active fluid composed of spinners can even robustly transport an immersed intruder with the aid of the spinner-mediated depletion interaction between the intruder and boundary. Importantly, the effective interaction significantly depends on the dissipationless odd viscosity of the chiral active fluid, which originates from the spinning-induced breaking of time-reversal and parity symmetries, rendering the transport controllable. Our findings propose a novel avenue for robust cargo transport and could open a range of new possibilities throughout biological and microfluidic systems.

Entropy production fluctuations encode collective behavior in active matter

We derive a general lower bound on distributions of entropy production in interacting active matter systems. The bound is tight in the limit that interparticle correlations are small and short-ranged, which we explore in four canonical active matter models. In all models studied, the bound is weak where collective fluctuations result in long-ranged correlations, which subsequently links the locations of phase transitions to enhanced entropy production fluctuations. We develop a theory for the onset of enhanced fluctuations and relate it to specific phase transitions in active Brownian particles. We also derive optimal control forces that realize the dynamics necessary to tune dissipation and manipulate the system between phases. In so doing, we uncover a general relationship between entropy production and pattern formation in active matter, as well as ways of controlling it.

Topological Dynamic Matter

The principles of topology in condensed matter physics have expanded to areas such as photonics, acoustics, electronics, and mechanics. Their extension to dynamic (soft) matter could enable the control and design of topological thermodynamic (micro)states and nonreciprocal dynamics, potentially leading to paradigmatic applications in molecular and thermal waveguiding, logics, and energy management. This Perspective explores distinct topological concepts for dynamic matter and prospective function. Topological tools are exemplified and discussed for the study of nonlocal order parameters or invariants in dynamic molecular matter, toward the engineering of assemblies, reactions, and system chemistry with unconventional global properties-a scope which has the potential to push the frontiers of physical chemistry and transform chemical topology from form to function.

Topological Phase Transition in Coupled Rock-Paper-Scissors Cycles

A hallmark of topological phases is the occurrence of topologically protected modes at the system's boundary. Here, we find topological phases in the antisymmetric Lotka-Volterra equation (ALVE). The ALVE is a nonlinear dynamical system and describes, for example, the evolutionary dynamics of a rock-paper-scissors cycle. On a one-dimensional chain of rock-paper-scissor cycles, topological phases become manifest as robust polarization states. At the transition point between left and right polarization, solitary waves are observed. This topological phase transition lies in symmetry class D within the "tenfold way" classification as also realized by 1D topological superconductors.

Dynamical Crystallites of Active Chiral Particles

One of the intrinsic characteristics of far-from-equilibrium systems is the nonrelaxational nature of the system dynamics, which leads to novel properties that cannot be understood and described by conventional pathways based on thermodynamic potentials. Of particular interest are the formation and evolution of ordered patterns composed of active particles that exhibit collective behavior. Here we examine such a type of nonpotential active system, focusing on effects of coupling and competition between chiral particle self-propulsion and self-spinning. It leads to the transition between three bulk dynamical regimes dominated by collective translative motion, spinning-induced structural arrest, and dynamical frustration. In addition, a persistently dynamical state of self-rotating crystallites is identified as a result of a localized-delocalized transition induced by the crystal-melt interface. The mechanism for the breaking of localized bulk states can also be utilized to achieve self-shearing or self-flow of active crystalline layers.

Exceptional non-Hermitian topological edge mode and its application to active matter

Topological materials exhibit edge-localized scattering-free modes protected by their nontrivial bulk topology through the bulk-edge correspondence in Hermitian systems. While topological phenomena have recently been much investigated in non-Hermitian systems with dissipations and injections, the fundamental principle of their edge modes has not fully been established. Here, we reveal that, in non-Hermitian systems, robust gapless edge modes can ubiquitously appear owing to a mechanism that is distinct from bulk topology, thus indicating the breakdown of the bulk-edge correspondence. The robustness of these edge modes originates from yet another topological structure accompanying the branchpoint singularity around an exceptional point, at which eigenvectors coalesce and the Hamiltonian becomes nondiagonalizable. Their characteristic complex eigenenergy spectra are applicable to realize lasing wave packets that propagate along the edge of the sample. We numerically confirm the emergence and the robustness of the proposed edge modes in the prototypical lattice models. Furthermore, we show that these edge modes appear in a model of chiral active matter based on the hydrodynamic description, demonstrating that active matter can exhibit an inherently non-Hermitian topological feature. The proposed general mechanism would serve as an alternative designing principle to realize scattering-free edge current in non-Hermitian devices, going beyond the existing frameworks of non-Hermitian topological phases.

Entropic vibrational resonance

We demonstrate the existence of vibrational resonance associated with the presence of an uneven boundary. When the motion of a Brownian particle is confined in a region with an uneven boundary, constrained to a double cavity, a high-frequency signal may produce a peak in the spectral power amplification of the other low-frequency signal and therefore to the appearance of the vibrational resonance phenomenon. The mechanism of vibrational resonance in constrained boundaries is different from that in energetic potentials and is termed entropic vibrational resonance (EVR). The EVR can be observed even if the bias force is absent in any direction. Through careful analysis, we clarify two types of mechanisms of the EVR. The one mechanism is ascribed to the transition from a bistable system to a monostable system, and the other corresponds to the match between the escape rate and the natural frequency of the low-frequency signal. Our work merges the vibrational resonance with an uneven boundary, thus extending the scope of the vibrational resonance and shedding new light on the concept of resonance.

Pattern Engineering of Living Bacterial Colonies Using Meniscus-Driven Fluidic Channels

Creating adaptive, sustainable, and dynamic biomaterials is a forthcoming mission of synthetic biology. Engineering spatially organized bacterial communities has a potential to develop such bio-metamaterials. However, generating living patterns with precision, robustness, and a low technical barrier remains as a challenge. Here we present an easily implementable technique for patterning live bacterial populations using a controlled meniscus-driven fluidics system, named as MeniFluidics. We demonstrate multiscale patterning of biofilm colonies and swarms with submillimeter resolution. Utilizing the faster bacterial spreading in liquid channels, MeniFluidics allows controlled bacterial colonies both in space and time to organize fluorescently labeled Bacillus subtilis strains into a converged pattern and to form dynamic vortex patterns in confined bacterial swarms. The robustness, accuracy, and low technical barrier of MeniFluidics offer a tool for advancing and inventing new living materials that can be combined with genetically engineered systems, and adding to fundamental research into ecological, evolutional, and physical interactions between microbes.

Oscillating collective motion of active rotors in confinement

Due to its inherent out-of-equilibrium nature, active matter in confinement may exhibit collective behavior absent in unconfined systems. Extensive studies have indicated that hydrodynamic or steric interactions between active particles and boundary play an important role in the emergence of collective behavior. However, besides introducing external couplings at the single-particle level, the confinement also induces an inhomogeneous density distribution due to particle-position correlations, whose effect on collective behavior remains unclear. Here, we investigate this effect in a minimal chiral active matter composed of self-spinning rotors through simulation, experiment, and theory. We find that the density inhomogeneity leads to a position-dependent frictional stress that results from interrotor friction and couples the spin to the translation of the particles, which can then drive a striking spatially oscillating collective motion of the chiral active matter along the confinement boundary. Moreover, depending on the oscillation properties, the collective behavior has three different modes as the packing fraction varies. The structural origins of the transitions between the different modes are well identified by the percolation of solid-like regions or the occurrence of defect-induced particle rearrangement. Our results thus show that the confinement-induced inhomogeneity, dynamic structure, and compressibility have significant influences on collective behavior of active matter and should be properly taken into account.

Unsupervised Manifold Clustering of Topological Phononics

Classification of topological phononics is challenging due to the lack of universal topological invariants and the randomness of structure patterns. Here, we show the unsupervised manifold learning for clustering topological phononics without any a priori knowledge, neither topological invariants nor supervised trainings, even when systems are imperfect or disordered. This is achieved by exploiting the real-space projection operator about finite phononic lattices to describe the correlation between oscillators. We exemplify the efficient unsupervised manifold clustering in typical phononic systems, including a one-dimensional Su-Schrieffer-Heeger-type phononic chain with random couplings, amorphous phononic topological insulators, higher-order phononic topological states, and a non-Hermitian phononic chain with random dissipations. The results would inspire more efforts on applications of unsupervised machine learning for topological phononic devices and beyond.

Time-reversal symmetry breaking in two-dimensional nonequilibrium viscous fluids

We study the rheological signatures of departure from equilibrium in two-dimensional viscous fluids with and without internal spin. Under the assumption of isotropy, we provide the most general linear constitutive relations for stress and couple stress in terms of the velocity and spin fields. Invoking Onsager's regression hypothesis for fluctuations about steady states, we derive the Green-Kubo formulas relating the transport coefficients to time-correlation functions of the fluctuating stress. In doing so, we show that one of the nonequilibrium transport coefficients, the odd viscosity, requires time-reversal symmetry breaking in the case of systems without internal spin. However, the Green-Kubo relations for systems with internal spin also show that there is a possibility for nonvanishing odd viscosity even when time-reversal symmetry is preserved. Furthermore, we find that breakdown of equipartition in nonequilibrium steady states results in the decoupling of the two rotational viscosities relating the vorticity and the internal spin.

Robust boundary flow in chiral active fluid

We perform experiments on an active chiral fluid system of self-spinning rotors in a confining boundary. Along the boundary, actively rotating rotors collectively drive a unidirectional material flow. We systematically vary rotor density and boundary shape; boundary flow robustly emerges under all conditions. Flow strength initially increases then decreases with rotor density (quantified by area fraction ϕ); peak strength appears around a density ϕ=0.65. Boundary curvature plays an important role: flow near a concave boundary is stronger than that near a flat or convex boundary in the same confinements. Our experimental results in all cases can be reproduced by a continuum theory with single free fitting parameter, which describes the frictional property of the boundary. Our results support the idea that boundary flow in active chiral fluid is topologically protected; such robust flow can be used to develop materials with novel functions.

Anomalous Topological Active Matter

Active systems exhibit spontaneous flows induced by self-propulsion of microscopic constituents and can reach a nonequilibrium steady state without an external drive. Constructing the analogy between the quantum anomalous Hall insulators and active matter with spontaneous flows, we show that topologically protected sound modes can arise in a steady-state active system in continuum space. We point out that the net vorticity of the steady-state flow, which acts as a counterpart of the gauge field in condensed-matter settings, must vanish under realistic conditions for active systems. The quantum anomalous Hall effect thus provides design principles for realizing topological metamaterials. We propose and analyze the concrete minimal model and numerically calculate its band structure and eigenvectors, demonstrating the emergence of nonzero bulk topological invariants with the corresponding edge sound modes. This new type of topological active systems can potentially expand possibilities for their experimental realizations and may have broad applications to practical active metamaterials. Possible realization of non-Hermitian topological phenomena in active systems is also discussed.

Active microrheology, Hall effect, and jamming in chiral fluids

We examine the motion of a probe particle driven through a chiral fluid composed of circularly swimming disks. We find that the probe particle travels in both the longitudinal direction, parallel to the driving force, and in the transverse direction, perpendicular to the driving force, giving rise to a Hall angle. Under constant driving force, we show that the probe particle velocity in both the longitudinal and transverse directions exhibits nonmonotonic behavior as a function of the activity of the circle swimmers. The Hall angle is maximized when a resonance occurs between the frequency of the chiral disks and the motion of the probe particle. As the density of the chiral fluid increases, the Hall angle gradually decreases before reaching zero when the system enters a jammed state. We show that the onset of jamming depends on the chiral particle swimming frequency, with a fluid state appearing at low frequencies and a jammed solid occurring at high frequencies.

Topological Waves in Fluids with Odd Viscosity

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

Energy dissipation and fluctuations in a driven liquid

Minimal models of active and driven particles have recently been used to elucidate many properties of nonequilibrium systems. However, the relation between energy consumption and changes in the structure and transport properties of these nonequilibrium materials remains to be explored. We explore this relation in a minimal model of a driven liquid that settles into a time periodic steady state. Using concepts from stochastic thermodynamics and liquid state theories, we show how the work performed on the system by various nonconservative, time-dependent forces-this quantifies a violation of time reversal symmetry-modifies the structural, transport, and phase transition properties of the driven liquid.