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.

Figure 1

Active particles on the two-dimensional flat space move under the influence of periodically aligned pillars. The red and green curved arrows represent the steady-state flows. (a) Triangular lattice geometry accompanying topologically trivial bands. The line integral along each boundary of the unit cell (blue and red solid lines) cancels each other because of the periodicity, leading to the vanishing net vorticity. (b) The proposed setup for topological active matter with kagome lattice geometry. Blue solid lines indicate the boundary of the unit cell. We set the side length of the unit cell as $a=1$.

Figure 2

(a) The band structure of the nonordered system, i.e., ${\mathbf{v}}_{\mathrm{ss}}=0$. The dispersion is plotted along the line at $k_x=2\pi /3$ with varying $k_y$ as indicated by the red arrow in the inset. We impose the twisted boundary conditions, $\psi (\mathbf{x}+\mathbf{a})=e^{i\mathbf{k}·\mathbf{a}}\psi (\mathbf{x})$, where $\psi$ is an eigenfunction and $\mathbf{a}$ is a lattice vector. The parameters used are $c=1$, ${\rho }_{\mathrm{ss}}=1$ and $\lambda =0.8$. (b) The enlarged view of the band structure in the green dashed box in (a). The orange solid (blue dashed) curves show the results with (without) the steady-state flows. The integer number at each band represents the Chern number.

Figure 3

Berry curvature of a topologically nontrivial band [the middle band with the Chern number $C=-2$ in Fig. 2(b)]. For the sake of visibility, the Berry curvature is plotted by inverting its sign. The parameters used are the same as in Fig. 2.

Figure 4

(a) Spatial profile of the magnitude of the density fluctuation in an edge mode. The wave number and the frequency are set to be $k_y=-(4\sqrt{3}/15)\pi$ and $\omega =7.6$, respectively. (b) The corresponding band structure under the open (periodic) boundary conditions in the $x$ ($y$) direction. The red curves show the dispersions associated with the edge mode in (a). The gray horizontal dashed line indicates the presence of the bulk band gap.