Anomalous Topological Active Matter
Active systems exhibit spontaneous flows induced by self-propulsion of microscopic constituents and can reach to 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 the 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. As a consequence, the quantum anomalous Hall effect naturally 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.
Topologically nontrivial bands, which have been at the forefront of condensed matter physics Thouless et al. (1982); Simon (1983); F. D. M. Haldane (1988); Kane and Mele (2005); Hasan and Kane (2010); Qi and Zhang (2011), can also appear in various classical systems such as photonic Haldane and Raghu (2008); Lu et al. (2014); Rechtsman et al. (2013); Khanikaev et al. (2013) and phononic systems Huber (2016); Nash et al. (2015); Süsstrunk and Huber (2015); Kane and Lubensky (2014); Yang et al. (2015); Fleury et al. (2016); Delplace et al. (2017). Such topologically nontrivial systems exhibit unidirectional modes that propagate along the edge of a sample and are immune to disorder. The existence of edge modes originates from the nontrivial topology characterized by bulk topological invariants of underlying photonic or acoustic band structures. This property gives rise to novel functionalities potentially applicable to, e.g., sonar detection and heat diodes Huber (2016); Yang et al. (2015). Furthermore, topologically protected edge modes are argued to be closely related to the mechanism of robustness in biological systems Murugan and Vaikuntanathan (2017); Dasbiswas et al. (2017).
On another front, active matter, a collection of self-driven particles, has attracted much interest as an ideal platform to study biological physics Brugués and Needleman (2014); Saw et al. (2017); Kawaguchi et al. (2017) and out-of-equilibrium statistical physics Ganguly and Chaudhuri (2013); Fodor et al. (2016); Krishnamurthy et al. (2016); Mandal et al. (2017); Pietzonka and Seifert (); Martin et al. (2018). While a prototype of active matter has been originally introduced to understand animal flocking behavior Vicsek et al. (1995); Vicsek and Zafeiris (2012), recent experimental developments have allowed one to manipulate and observe artificial active systems in a controlled manner by utilizing Janus particles Jiang et al. (2010), catalytic colloids Palacci et al. (2013) and external feedback control Khadka et al. (2018).
The aim of this Letter is to show that a topologically nontrivial feature can ubiquitously emerge in a nonequilibrium steady state of active matter and demonstrate it by analyzing the concrete minimal model, which can be realized with current experimental techniques. Specifically, we first point out that the net vorticity of the steady-state flow must vanish under realistic conditions for active systems in the continuum space. Since the vorticity in active matter can act as a counterpart of the magnetic field in condensed matter systems, this fact indicates that the quantum anomalous Hall effect (QAHE) naturally provides design principles for realizing topological metamaterials. We propose and analyze the active matter model inspired by the flat-band ferromagnet featuring the QAHE Ohgushi et al. (2000). We numerically calculate its band structure and eigenvectors, and demonstrate that they exhibit nonzero topological invariants with the corresponding edge modes. Possible relation to non-Hermitian topological phenomena is also discussed.
Topological edge modes of active matter have been recently discussed by several authors Souslov et al. (2017); Shankar et al. (2017); Souslov et al. (2018); Dasbiswas et al. (2017). There, the presence of nonzero net vorticity of the active flows, which can act as an effective magnetic field, was crucial to support topological edge modes reminiscent of the quantum Hall effect Klitzing et al. (1980); Thouless et al. (1982). Yet, this required the introduction of rather intricate structures in active systems such as large defects Souslov et al. (2017), curved surface Shankar et al. (2017), and rotational forces Dasbiswas et al. (2017); Souslov et al. (2018). One of the novel aspects introduced by this Letter is to eliminate these bottlenecks by constructing the analogy to the QAHE, significantly expanding possibilities for realizing topological metamaterials. Our proposal is based on the simplest setup on a flat continuum space with assuming no internal degrees of freedom of active particles. This class of systems is directly relevant to many realistic setups of active systems Sokolov et al. (2007); Cavagna et al. (2010); Deseigne et al. (2010); Schaller et al. (2010); Nishiguchi et al. (2018) and our design principle is applicable beyond the minimal model proposed here.
Emergent effective Hamiltonian for active matter.— To describe collective dynamics of active matter, we use the Toner-Tu equations Toner and Tu (1995, 1998); Toner et al. (2005); Marchetti et al. (2013), which are the hydrodynamic equations for active matter with a polar-type interaction:
where is the density field of active matter and is the local average of velocities of self-propelled particles. Equation (1) presents the equation of continuity. In Eq. (2), the first term of its right-hand side suggests a preference for a nonzero constant speed if is positive while negative results in the nonordered state . The coefficients in these equations can be obtained from microscopic models Bertin et al. (2006); Peshkov et al. (2014); Farrell et al. (2012); Solon and Tailleur (2013); Suzuki et al. (2015); Bricard et al. (2013). To simplify the problem, we ignore the terms including and also the diffusive terms that contain the second-order derivative. This condition can be met in a variety of active systems Souslov et al. (2017); Shankar et al. (2017); Farrell et al. (2012). We also assume that the pressure is proportional to as appropriate for an ideal gas.
Linearizing the Toner-Tu equations around a steady-state solution, we derive an eigenvalue equation for the fluctuations of density and velocity fields, and , respectively, where and represent their steady-state values. We also assume that the steady-state speed is much smaller than the sound velocity . To clarify the argument, we define the following dimensionless variables: , , , and , where is a characteristic length of a system that we specify as a lattice constant later. For the sake of notational simplicity, hereafter we express the dimensionless variables , , as , , . The resulting linearized equation in the frequency domain is
with being the effective Hamiltonian defined as
where , are the Fourier components in the frequency domain. We here omit the spatial variation of and the divergence of as justified in the Supplemental Materials. We note that, while the coefficient matrix can be regarded as the effective Hamiltonian, it can in general be non-Hermitian when the diffusive terms or terms in Eq. (2) are included.
where (see the Supplemental Materials for the derivation). Equation (5) demonstrates that (i.e., the steady-state flow aside from a constant factor) acts as the effective vector potential and its vorticity can be interpreted as the effective magnetic field. Below we point out that the net vorticity integrated over the whole two-dimensional space must vanish under realistic conditions relevant to a variety of active systems. Since the net vorticity in active systems plays the role of the net magnetic field, this argument has a profound consequence when one considers topological aspects of active matter as detailed later.
We consider active particles without internal degrees of freedom, which reside on a two-dimensional plane with a periodic structure of a unit cell . To avoid intricate structures, we assume that non-negligibly large defects are absent, i.e., the length of the perimeter of a defect in each unit cell can be neglected with respect to that of the unit-cell boundary . We note that this condition does not preclude possibilities of minuscule defects created by, e.g., thin rods as realized in Ref. Nishiguchi et al. (2018) or the presence of inhomogeneous potentials relevant to chemotactic bacteria subject to a nonuniform concentration of chemical compounds Saragosti et al. (2011).
The net vorticity is then obtained by the integration over the unit cell and can be expressed via the Stokes’ theorem as
Due to a periodic structure of the steady flow along the unit-cell boundary , one can show that the line integration in the right-hand side of Eq. (6) adds up to zero (see Fig. 1 for typical examples), resulting in the vanishing net vorticity. Thus, in the active hydrodynamics of interest here, it is prohibited to realize an analog of the quantum Hall effect, which requires an external magnetic field indicating nonzero net vorticity. Under the above conditions, we naturally arrive at the conclusion that a topological active matter must be realized as a counterpart of the QAHE, where the need for the external magnetic field can be mitigated.
The minimal model of topological active matter.— To complete the analogy between the active matter and the QAHE, we propose the minimal model illustrated in Fig. 1(b). There, active particles obeying the Toner-Tu equations move under the influence of small pillars located at each site of a kagome lattice 111We assume that, while these pillars can influence the flow of particles, their sizes are small enough such that the conditions discussed above Eq. (6) are satisfied.. As illustrated in Fig. 1(b), the steady-state velocity field aligns on each boundary of triangular and hexagonal subcells separated by blue solid and dashed lines. Thus, particles in the triangular subcells circulate in the counterclockwise direction (green arrows) that is opposite to the direction of the particle circulation in the hexagonal subcells (red arrows) 222The reversed flow is also allowed as a steady-state solution; the initial condition determines which flow can be realized in a system., resulting in the vanishing net vorticity.
It is noteworthy that such an “anticorrelated” velocity profile has been observed in bacterial experiments Wioland et al. (2016); Nishiguchi et al. (2018) and also in particle-based numerical simulations Souslov et al. (2017); Pearce and Turner (2015). In the cases of triangular- and square-lattice structures (c.f. Ref. Nishiguchi et al. (2018) and Fig. 1(a)), however, only topologically trivial bands can appear due to the absence of a sublattice structure; this is why the kagome-lattice structure (as considered here) is crucial for realizing topological active matter.
The proposed system exhibits topologically nontrivial bands despite the absence of net vorticity. This makes a sharp contrast to the previous setups Souslov et al. (2017); Shankar et al. (2017); Souslov et al. (2018); Dasbiswas et al. (2017), which require rather intricate structures to realize nonvanishing net vorticity that is prerequisite for constructing a counterpart of the quantum Hall effect. To gain physical insights, we point out the analogy between the present system and the QAHE originally introduced in the context of condensed matter physics F. D. M. Haldane (1988); Ohgushi et al. (2000). As inferred from Eq. (5), the steady-state flow of the active system acts as an analog of the electromagnetic vector potential. Then, the collective density fluctuations are influenced by the local vorticity in such a way that electrons propagating through the crystal feel the Berry phase. One can thus expect that sound modes can exhibit topologically nontrivial bands, as the electronic bands feature the QAHE Ohgushi et al. (2000).
Topological band structure.— Diagonalizing the effective Hamiltonian numerically, we calculate the bulk-band dispersion. In practice, we find that there is a subtlety in obtaining a precise hydrodynamic dispersion of the present continuum system. To get an accurate result, we add the redundant degree of freedom without affecting the band structure by transforming via a unitary matrix (see the Supplemental Materials for details). This transformation allows for calculations in the basis naturally reflecting the centrosymmetry of the present system. While this prescription generates redundant eigenstates with eigenvalues of , it does not change any physical properties including the topological feature. We thus analyze the eigenequation
with being the effective Hamiltonian in the transformed frame
where we define the variables as and Here, is the redundant degree of freedom. The derivatives denote and , which correspond to the directions along the grid lines of the kagome lattice (c.f. the blue dashed and solid lines in Fig. 1(b)). Figure 2 shows the band structure of the effective Hamiltonian calculated by the difference method Smith (1985). Since the effective Hamiltonian satisfies the particle-hole symmetry, there is a counterpart for each eigenvector whose eigenenergy has the same absolute value and the opposite sign. For the nonordered case , there are degeneracies at the edges of the first Brillouin zone. Nonzero lifts those degeneracies and opens band gaps, characteristics of topological materials F. D. M. Haldane (1988); Ohgushi et al. (2000); Yang et al. (2015); Souslov et al. (2017); Fleury et al. (2016); Rechtsman et al. (2013); Khanikaev et al. (2013); Shankar et al. (2017); Nash et al. (2015); Haldane and Raghu (2008).
We confirm that the proposed model exhibits a topologically nontrivial band by calculating the bulk topological invariant, i.e., the Chern number Simon (1983). Specifically, the Chern number of the n-th band is defined as
where is the Berry curvature with being the Berry connection and being the n-th eigenvector at wavenumber . We calculate the Berry curvature and the Chern number for each band following the numerical method proposed in Ref. Fukui et al. (2005). The calculation shows that many of the bands have nonzero Chern numbers (see e.g., Fig. 2(b)). Figure 3 shows the Berry curvature of the topologically nontrivial acoustic band (c.f. the middle band with in Fig. 2(b)), which exhibits sharp peaks at the edges of the first Brillouin zone. We note that the values of the Berry curvature are inverted in this plot for the sake of visibility.
The bulk-edge correspondence predicts that the nonzero Chern number accompanies a unidirectional edge mode under open boundary conditions. While the correspondence has been well established in tight-binding lattice models, it has been recently argued to hold also in the continuum space Silveirinha (2019). To test the existence of such edge modes in our model, we calculate the sound modes by considering a supercell structure; many identical unit cells are aligned with open boundary conditions in the -direction while the periodic boundary conditions are imposed in the -direction. Figure 4 shows the real-space profile of the sound mode at the gap between the topologically distinct bands. The density fluctuation rapidly decreases as we depart from the right end, indicating the presence of the edge mode.
Summary and Discussions.— We showed that topologically nontrivial bands can arise in active systems without implementing intricate structures, which have been considered as prerequisites for realizing topological sound modes. Due to the vanishing net vorticity of steady-state flows, we pointed out that the quantum anomalous Hall effect provides a natural pathway to realize topological active materials. These findings are supported by numerical calculations of the band structure of the simple model, which is inspired by the flat-band ferromagnet in solid-state systems. Our construction of the analogy to the quantum anomalous Hall effect should provide a useful design principle applicable to a multitude of practical metamaterials.
The present study opens several research directions. Firstly, our results expand possibilities for experimental realizations of topological active systems. Recent experimental developments have enabled one to measure and manipulate polar active matter by using, for example, bacteria. In particular, the experimental setup realized by Ref. Nishiguchi et al. (2018) is directly relevant to our model except for the lattice structure and thus, our theoretical results can be tested with current experimental techniques. Secondly, our work suggests a simple and general way to construct topological active matter by designing its periodic structure (steady-state flow) with making the analogy to a profile of a tight-binding lattice (gauge field) relevant to electronic topological materials. In this manner, one can construct topological active matter with various types of symmetries on demand. Edge modes in such a system can behave differently from the existing models and potentially enrich applications to metamaterials.
Thirdly, besides technological applications, one major motivation in the field of active matter is to advance our understanding of emergent nonequilibrium phenomena in biological systems. Biological systems modeled as active matter include, for example, cells, molecular motors, and cytoskeletons Marchetti et al. (2013); Vicsek and Zafeiris (2012). Topological edge modes may play an important role in various biological functionalities, which are often robust to disorder. If the collective response of active constituents turns out to be topologically protected, the net vorticity should be zero as discussed above. In this respect, our model should serve as the prototype for the search of topological phenomena in active biological systems.
Finally, while we neglect the diffusive terms and terms in the Toner-Tu equation (2), they can in general make the effective Hamiltonian non-Hermitian. It is worthwhile to explore non-Hermitian topological phenomena in active systems; of particular interest is an exotic topological feature that has no counterpart in Hermitian systems Lee (2016); Leykam et al. (2017); Kunst et al. (2018); Yao and Wang (2018); Gong et al. (2018). Such a feature may lead to an emergence of novel functionalities unique to active matter. We hope that our work stimulates further studies in these directions.
We thank Shunsuke Furukawa, Ryusuke Hamazaki, Takahiro Sagawa and Masahito Ueda for useful discussions. Y.A. acknowledges support from the Japan Society for the Promotion of Science through Program for Leading Graduate Schools (ALPS) and Grant No. JP16J03613.
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.1 Linearization of the Toner-Tu equations and the derivation of the Schrödinger-like equation
Here we show the details about the derivation of the linearized equation. The full Toner-Tu equations read
Since we neglect the diffusive terms and terms, we obtain the simplified Toner-Tu equations
where we assume the equation of state with being the sound speed and that the external force is absent. We consider the fluctuation of density and velocity fields from a steady state, , and neglect the second-order terms in Eq. (S4). Then, we obtain
Finally, we neglect the second-order contributions of and arrive at the linearized equation (3) in the main text.
Assuming an oscillating solution , the equation takes the form as
where we assume that is small and neglect its second-order terms. This equation is equivalent to the Schrödinger-like equation (5) in the main text except for the second-order contributions of .
.2 Evaluation of the spatial variation of the density field and justification for considering the steady-state flow
The equations for a steady state read
Equation (S10) shows , where is the length of each side of the unit cell. Therefore, when is small, we can ignore the spatial variation of the density field in a steady state. Then, we can obtain from Eq. (S9). This implies that the divergence of the steady-state flow is also small enough to be ignored and the Toner-Tu equations permit considering the steady-state flow, whose divergence is zero. We note that the divergence of the steady-state flow in Fig. 1(b) in the main text becomes zero even if is large.
.3 Deformation of the effective Hamiltonian in the basis appropriate for the band calculations
To numerically obtain the band structure in the continuum space, we must in practice discretize the effective Hamiltonian with respect to spatial degrees of freedom. Specifically, we approximate the continuum space by a discretized triangular lattice and consider quantities on each lattice point as values of density and velocity fields. Also, the derivatives must be converted into the difference between neighboring sites. The way of this conversion is not unique and can lead to numerical errors; a naive discretization of the effective Hamiltonian can fail to provide the proper band structure. Since our system is symmetric under rotation, the band structure should also reflect that symmetry. However, one cannot symmetrize a pair of linear combinations of the derivatives under rotation. This is the main reason why the calculated band structures can in practice break the symmetry or have substantial numerical errors due to large wavenumber components.
To solve this problem, we deform the effective Hamiltonian without affecting the topology of the system. First, we add the redundant degree of freedom and formally rewrite the linearized equation (3) in the main text as follows:
where we denote as
This deformation only adds trivial eigenstates with zero eigenvalues. We next consider the unitary matrix
and use it to transform into the following form:
Note that the eigenvalues are unchanged and the eigenvectors are only globally transformed by . Thus, these transformations do not alter the topology of the system. However, this new basis now naturally reflects the underlying symmetry of kagome lattice and, in practice, allows one to accurately calculate the band structure with much less numerical errors than the naive discretization in the basis of directions. To further improve the numerical accuracy, we also implement a hybrid discretization of the derivatives. More specifically, to convert the derivatives into the discretized form, we use a forward difference, a backward difference and a central difference for the derivatives in the first row of the effective Hamiltonian , the derivatives in the first column of the effective Hamiltonian , and the derivative , respectively. This conversion sustains the Hermiticity of the effective Hamiltonian and can remove substantial numerical errors that can contribute from large wavenumber components.