AAnomalous Topological Active Matter
Kazuki Sone ∗ and Yuto Ashida
1, 2, † Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan (Dated: October 14, 2019)Active systems exhibit spontaneous flows induced by self-propulsion of microscopic constituents and canreach a nonequilibrium steady state without an external drive. Constructing the analogy between the quantumanomalous Hall insulators and active matter with spontaneous flows, we show that topologically protected soundmodes can arise in a steady-state active system in continuum space. We point out that the net vorticity of thesteady-state flow, which acts as a counterpart of the gauge field in condensed-matter settings, must vanish underrealistic conditions for active systems. The quantum anomalous Hall effect thus provides design principles forrealizing topological metamaterials. We propose and analyze the concrete minimal model and numerically cal-culate its band structure and eigenvectors, demonstrating the emergence of nonzero bulk topological invariantswith the corresponding edge sound modes. This new type of topological active systems can potentially expandpossibilities for their experimental realizations and may have broad applications to practical active metamateri-als. Possible realization of non-Hermitian topological phenomena in active systems is also discussed.
Topologically nontrivial bands, which have been at theforefront of condensed matter physics [1–6], can also ap-pear in various classical systems such as photonic [7–10] andphononic systems [11–17]. Such topologically nontrivial sys-tems exhibit unidirectional modes that propagate along theedge of a sample and are immune to disorder. The existenceof edge modes originates from the nontrivial topology char-acterized by bulk topological invariants of underlying pho-tonic or acoustic band structures. The topological edge modesgive rise to novel functionalities potentially applicable to, e.g.,sonar detection and heat diodes [11, 15]. Furthermore, theyare argued to be closely related to the mechanism of robust-ness in biological systems [18, 19].On another front, active matter, a collection of self-drivenparticles, has attracted much interest as an ideal platform tostudy biological physics [20–22] and out-of-equilibrium sta-tistical physics [23–28]. While a prototype of active matterhas been originally introduced to understand animal flockingbehavior [29, 30], recent experimental developments have al-lowed one to manipulate and observe artificial active systemsin a controlled manner by utilizing Janus particles [31], cat-alytic colloids [32] and external feedback control [33].The aim of this Letter is to show that a topologically non-trivial feature can ubiquitously emerge in a nonequilibriumsteady state of active matter and demonstrate it by analyzingthe concrete minimal model, which can be realized with cur-rent experimental techniques. Specifically, we first point outthat the net vorticity of the steady-state flow must vanish underrealistic conditions for active systems in the continuum space.Since the vorticity in active matter can act as a counterpart ofthe magnetic field in condensed matter systems, this fact indi-cates that the quantum anomalous Hall effect (QAHE) natu-rally provides design principles for realizing topological meta-materials. We propose and analyze the active matter model in-spired by the flat-band ferromagnet featuring the QAHE [34].We numerically calculate its band structure and eigenvectors,and demonstrate that they exhibit nonzero topological invari-ants with the corresponding edge modes. Possible relation to non-Hermitian topological phenomena is also discussed.Topological edge modes of active matter have been recentlydiscussed by several authors [19, 35–37]. There, the presenceof nonzero net vorticity of the active flows, which can act asan effective magnetic field, was crucial to support topologicaledge modes reminiscent of the quantum Hall effect [1, 38].Yet, this required the introduction of rather intricate structuresin active systems such as large defects [35], curved surface[36], and rotational forces [19, 37]. One of the novel aspectsintroduced by this Letter is to eliminate these bottlenecks byconstructing the analogy to the QAHE, significantly expand-ing possibilities for realizing topological metamaterials. Ourproposal is based on the simplest setup on a flat continuumspace with assuming no internal degrees of freedom of activeparticles. This class of systems is directly relevant to manyrealistic setups of active systems [39–43] and our design prin-ciple is applicable beyond the minimal model proposed here.
Emergent effective Hamiltonian for active matter.—
To de-scribe collective dynamics of active matter, we use the Toner-Tu equations [44–47], which are the hydrodynamic equationsfor active matter with a polar-type interaction: ∂ t ρ + ∇ · ( ρ v ) = 0 , (1) ∂ t v + λ ( v · ∇ ) v + λ ( ∇ · v ) v + λ ∇| v | = ( α − β | v | ) v − ∇ P + D B ∇ ( ∇ · v ) + D T ∇ v + D ( v · ∇ ) v + f , (2)where ρ ( r , t ) is the density field of active matter and v ( r , t ) 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 fora nonzero constant speed | v | = (cid:112) α/β if α is positive whilenegative α results in the nonordered state | v | = 0 . The coef-ficients in these equations can be obtained from microscopicmodels [48–53]. To simplify the problem, we ignore the termsincluding λ , and also the diffusive terms that contain thesecond-order derivative. This condition can be met in a vari-ety of active systems [35, 36, 50]. Effects of λ , terms can a r X i v : . [ c ond - m a t . s o f t ] O c t be taken into account by renormalizing λ if necessary [44].We also assume that the pressure P is proportional to ρ asappropriate for an ideal gas.Linearizing the Toner-Tu equations around a steady-statesolution, we derive an eigenvalue equation for the fluctua-tions of density and velocity fields, δρ ( r , t ) = ρ ( r , t ) − ρ ss ( r ) and δ v ( r , t ) = v ( r , t ) − v ss ( r ) , respectively, where ρ ss and v ss represent their steady-state values. We also assumethat the steady-state speed | v ss | is much smaller than thesound velocity c = (cid:112) P/ρ . To clarify the argument, wedefine the following dimensionless variables: r (cid:48) = r /a , t (cid:48) = ct/a , δρ (cid:48) ( r (cid:48) , t (cid:48) ) = δρ ( a r (cid:48) , at (cid:48) /c ) /ρ ss ( a r (cid:48) ) , δ v (cid:48) ( r (cid:48) , t (cid:48) ) = δ v ( a r (cid:48) , at (cid:48) /c ) /c and v (cid:48) ss ( r (cid:48) ) = v ss ( a r (cid:48) ) /c , where a is a char-acteristic length of a system that we specify as a lattice con-stant later. For the sake of notational simplicity, hereafter weexpress the dimensionless variables r (cid:48) , t (cid:48) , v (cid:48) ss as r , t , v ss . Theresulting linearized equation in the frequency domain is H δ ˜ ρδ ˜ v x δ ˜ v y = ω δ ˜ ρδ ˜ v x δ ˜ v y (3)with H being the effective Hamiltonian defined as H = − i v ss · ∇ − i∂ x − i∂ y − i∂ x − iλ v ss · ∇ − i∂ y − iλ v ss · ∇ , (4)where δ ˜ ρ ( r , ω ) , δ ˜ v x,y ( r , ω ) are the Fourier components in thefrequency domain. We here omit the spatial variation of ρ ss and the divergence of v ss as justified in the Supplemental Ma-terials. We note that, while the coefficient matrix H can beregarded as the effective Hamiltonian, it can in general be non-Hermitian when the diffusive terms in Eq. (2) are included. Absence of net vorticity in active matter.—
The linearizedequation (3) can be deformed into the Schr¨odinger-like equa-tion [35] ( − i ∇ − V ss ) δ ˜ ρ = ω δ ˜ ρ, (5)where V ss = ω ( λ + 1) v ss / (see the Supplemental Materialsfor the derivation). Equation (5) demonstrates that V ss acts asthe effective vector potential and its vorticity ∇ × V ss can beinterpreted as the effective magnetic field.We consider active particles without internal degrees offreedom, which reside on a two-dimensional plane with a pe-riodic structure of a unit cell Ω . 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 canbe neglected with respect to that of the unit-cell boundary ∂ Ω . We note that this condition does not preclude possibil-ities of minuscule defects created by, e.g., thin rods as real-ized in Ref. [43] or the presence of inhomogeneous potentialsrelevant to chemotactic bacteria subject to a nonuniform con-centration of chemical compounds [54].The net vorticity is then obtained by the integration over theunit cell and can be expressed via the Stokes’ theorem as (cid:90) Ω ( ∇ × V ss ) · d S = (cid:73) ∂ Ω V ss · d r . (6) (a)(b) FIG. 1. Active particles on the two-dimensional flat space moveunder the influence of periodically aligned pillars. The red andgreen curved arrows represent the steady-state flows. (a) Trivial sys-tem with a triangular-lattice geometry. The line integral along eachboundary of the unit cell (blue and red solid lines) cancels each otherbecause of the periodicity, leading to the vanishing net vorticity. (b)The proposed setup for topological active matter with a kagome-lattice geometry. Blue solid lines indicate the boundary of the unitcell. We set the side length of the unit cell as a = 1 . Due to a periodic structure of the steady flow along the unit-cell boundary ∂ Ω , one can show that the line integration in theright-hand side of Eq. (6) adds up to zero (see Fig. 1 for typicalexamples), resulting in the vanishing net vorticity. Thus, inthe active hydrodynamics of interest here, it is prohibited torealize an analog of the quantum Hall effect, which requires anexternal magnetic field indicating nonzero net vorticity. Underthe above conditions, we naturally arrive at the conclusion that a topological active matter must be realized as a counterpartof the QAHE , where the need for the external magnetic fieldcan be mitigated.We mention the reasons why the counterparts of the quan-tum Hall effect can be constructed in the previous setups[19, 35–37] despite the above argument. The active particlesin Refs. [19, 37] exhibit self-rotations and thus their internalangular momenta violate our assumption on the absence ofinternal degrees of freedom. The model in Ref. [37] includesexternal rotational force, where the Coriolis force acts as aneffective Lorentz force. In Ref. [35], the model includes largedefects around which additional line integrals contribute toextra vorticity in Eq. (6). The curved space is discussed inRef. [36]; it violates our assumption on flatness of the space.Altogether, the above effects lead to the emergence of the neteffective magnetic field and thus permit realizing the counter-parts of the conventional quantum Hall effect. The minimal model of topological active matter.—
To com-plete 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 underthe influence of small pillars located at each site of a kagomelattice [55]. As illustrated in Fig. 1(b), the steady-state ve-locity field v ss ( r ) aligns on each boundary of triangular andhexagonal subcells separated by blue solid and dashed lines.Thus, particles in the triangular subcells circulate in the coun-terclockwise direction (green arrows) that is opposite to thedirection of the particle circulation in the hexagonal subcells(red arrows) [56], resulting in the vanishing net vorticity. Weconfirm the emergence of this steady-state flow by perform-ing the particle-based numerical simulation (see Supplemen-tary Materials and Supplementary Movie published with themanuscript).It is noteworthy that such an “anticorrelated” velocity pro-file has been observed in bacterial experiments [43, 57] andalso in numerical simulations [35, 58]. In the cases oftriangular- and square-lattice structures (cf. Ref. [43] andFig. 1(a)), however, only topologically trivial bands can ap-pear due to the absence of a sublattice structure; this is whythe kagome-lattice structure (as considered here) is crucial forrealizing topological active matter. To gain physical insights,we point out the analogy between the present system and theQAHE [3, 34]; the collective density fluctuations are influ-enced by the local vorticity in such a way that electrons prop-agating through the crystal feel the Berry phase. One can thusexpect that sound modes can exhibit topologically nontrivialbands, as the electronic bands feature the QAHE [34]. Topological band structure.—
We obtain the bulk disper-sion by numerically diagonalizing the effective Hamiltonian H . To obtain accurate results, we add the redundant degreeof freedom without affecting the band structure by transform-ing H via a unitary matrix (see the Supplementary Materials).This transformation allows for calculations in the basis reflect-ing the centrosymmetry of the present system. If we calculatethe bulk band structure without this redundant degree of free-dom, we obtain unphysical k y -independent bands. While thisprescription generates redundant eigenstates with eigenvaluesof , it does not change any physical properties including thetopological feature. We thus analyze the eigenequation H (cid:48) δ ˜ ρδ ˜ v δ ˜ v δ ˜ v = ω δ ˜ ρδ ˜ v δ ˜ v δ ˜ v (7)with H (cid:48) being the effective Hamiltonian in the transformed k y wavenumber f r equen cy ω Γ k x k y k y wavenumber f r equen cy ω (a)(b) -1-2-1 FIG. 2. (a) The band structure of the nonordered system, i.e., v ss = 0 . The dispersion is plotted along the line at k x = 2 π/ with varying k y as indicated by the red arrow in the inset. We im-pose the twisted boundary conditions, ψ ( x + a ) = e i k · a ψ ( x ) , where ψ is an eigenfunction and a is a lattice vector. The parameters usedare c = 1 , ρ ss = 1 and λ = 0 . . (b) The enlarged view of theband structure in the green dashed box in (a). The orange solid (bluedashed) curves show the results with (without) the steady-state flows.The integer number at each band represents the Chern number. frame H (cid:48) = − i v ss · ∇ √ ∂ √ ∂ √ ∂ √ ∂ λ v ss · ∇ − λ v ss · ∇ − λ v ss · ∇ √ ∂ − λ v ss · ∇ λ v ss · ∇ − λ v ss · ∇ √ ∂ − λ v ss · ∇ − λ v ss · ∇ λ v ss · ∇ , (8)where we define the variables as δ ˜ v = 2 δ ˜ v x / √ δ ˜ v r / √ ,δ ˜ v = − δ ˜ v x / √ δ ˜ v y / √ δ ˜ v r / √ and δ ˜ v = − δ ˜ v x / √ − δ ˜ v y / √ δ ˜ v r / √ . Here, δ ˜ v r is the redundant degree of free-dom. The derivatives denote ∂ = ∂ x , ∂ = − ∂ x / √ ∂ y / and ∂ = − ∂ x / −√ ∂ y / , which correspond to the directionsalong the grid lines of the kagome lattice (cf. the blue dashedand solid lines in Fig. 1(b)). Figure 2 shows the band structureof the effective Hamiltonian H (cid:48) calculated by the differencemethod [59]. Since the effective Hamiltonian satisfies theparticle-hole symmetry, there is a counterpart for each eigen-vector whose eigenenergy has the same absolute value and theopposite sign. For the nonordered case | v ss | = 0 , there aredegeneracies at the edges of the first Brillouin zone. Nonzero | v ss | lifts those degeneracies and opens band gaps, character-istics of topological materials [3, 7, 9, 10, 12, 15, 16, 34–36]. Γ k x k y Wave number k x W a v e n u m b e r k y B e rr y c u r v a t u r e B FIG. 3. Berry curvature of a topologically nontrivial band (the mid-dle band with the Chern number C = − in Fig. 2). For the sakeof visibility, the Berry curvature is plotted by inverting its sign. Theparameters used are the same as in Fig. 2. We note that our effective Hamiltonian does not contain non-derivative terms that are necessary for the original proposal ofthe quantum anomalous Hall effect [3]. The kagome-latticestructure mitigates this requirement [34] as it can realize thelocal flux without next-nearest hoppings.We confirm that the proposed model exhibits a topologi-cally nontrivial band by calculating the bulk topological in-variant, i.e., the Chern number [2]. Specifically, the Chernnumber of the n -th band is defined as C n = 12 π (cid:90) BZ B n ( q ) · d S , (9)where B n ( q ) = ∇ q × A n ( q ) is the Berry curvature with A n ( q ) = i u n ( q ) · ( ∇ q u n ( q )) being the Berry connectionand u n ( q ) being the n -th eigenvector at wavenumber q . Wecalculate the Berry curvature and the Chern number for eachband following the numerical method proposed in Ref. [60].The calculation shows that many of the bands have nonzeroChern numbers (see e.g., Fig. 2(b)). Figure 3 shows the Berrycurvature of the topologically nontrivial acoustic band (cf. themiddle band with C = − in Fig. 2(b)), which exhibits sharppeaks at the edges of the first Brillouin zone.The bulk-edge correspondence predicts that the nonzeroChern number accompanies a unidirectional edge mode un-der open boundary conditions. While the correspondence hasbeen well established in tight-binding lattice models, it hasbeen recently argued to hold also in the continuum space [61].To test the existence of edge modes, we calculate the soundmodes for a supercell structure; many identical unit cellsare aligned with open boundary conditions in the x -directionwhile the periodic boundary conditions are imposed in the y -direction. Figure 4 shows the band structure in this setup andthe real-space profile of the sound mode at the gap between thetopologically distinct bands. The density fluctuation rapidlydecreases as we depart from the right end, indicating the pres-ence of the edge mode (Fig. 4a). The edge band connects thelower and upper bulk bands (Fig. 4b). There is one edge modeat the bulk gap as consistent with the sum of the Chern num- - - - k y wavenumber f r equen cy ω (a)(b) FIG. 4. (a) Spatial profile of the magnitude of the density fluctuationin an edge mode. We align 10 unit cells with open boundary condi-tions in the x -direction. Periodic boundary conditions are imposedin the y -direction. The wavenumber and the frequency are set to be k y = − (4 √ / π and ω = 7 . , respectively. (b) The correspond-ing band structure. The red curves show the dispersion of the edgemode in (a). The gray dashed line indicates the presence of the bulkband gap. bers of the bands below the energy gap, (cid:80) n < C n < = − . Summary and Discussions.—
We showed that topologicallynontrivial bands can arise in active systems without imple-menting intricate structures, which have been considered asprerequisites for realizing topological sound modes. Due tothe vanishing net vorticity of steady-state flows, we pointedout that the quantum anomalous Hall effect provides a nat-ural pathway to realize topological active materials. Thesefindings are supported by numerical calculations of the bandstructure of the simple model, which is inspired by the flat-band ferromagnet in solid-state systems.The present study opens several research directions. Firstly,our results expand possibilities for experimental realizationsof topological active systems. Recent experimental develop-ments have enabled one to measure and manipulate polar ac-tive matter by using, for example, bacteria. In particular, theexperimental setup realized by Ref. [43] is directly relevant toour model except for the lattice structure and thus, our the-oretical results can be tested with current experimental tech-niques. Secondly, our work suggests a simple and generalway to construct topological active matter by designing its pe-riodic structure (steady-state flow) with making the analogyto a profile of a tight-binding lattice (gauge field) relevant toelectronic topological materials.Thirdly, besides technological applications, one major mo-tivation in the field of active matter is to advance our under-standing of emergent nonequilibrium phenomena in biologi-cal systems. Biological systems modeled as active matter in-clude, for example, cells, molecular motors, and cytoskeletons[30, 47]. Topological edge modes may play an important rolein various biological functionalities, which are often robust todisorder.Finally, while we neglect the diffusive terms in the Toner-Tu equation (2), they can in general make the effective Hamil-tonian non-Hermitian and suppress the high-wavenumbermodes. It is worthwhile to explore non-Hermitian topolog-ical phenomena in active systems; of particular interest is anexotic topological feature that has no counterpart in Hermitiansystems [62–66]. In particular, asymmetrical flows in activematter may allow one to realize the non-Hermitian skin effect[64, 65, 67] and the quasiedge modes [66]. Such a featurecould lead to an emergence of novel functionalities unique toactive matter. We hope that our work stimulates further stud-ies in these directions.We thank Shunsuke Furukawa, Ryusuke Hamazaki,Takahiro Sagawa, Masahito Ueda, Daiki Nishiguchi, andKazumasa Takeuchi for useful discussions. 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The spatial variation of the density field and the steady-state flow
The equations for a steady state read as ρ ss ∇ · v ss + ( ∇ ρ ss ) · v ss = 0 , (S1) λ ( v ss · ∇ ) v ss = − c ρ ss ∇ ρ ss . (S2)Equation (S2) implies that a ∇ ρ ss /ρ ss = O (( | v ss | /c ) ) , where a is the length of each side of the unit cell. Therefore, when | v ss | /c = | v (cid:48) ss | is small, one can ignore the spatial variation of the density field in a steady state. We thus obtain a ∇ · v ss /c = O (( | v ss | /c ) ) from Eq. (S1). This implies that the divergence of the steady-state flow can also be ignored and the Toner-Tuequations permit the steady-state flow whose divergence is zero.To confirm the validity of the steady-state flow considered (cf. Fig. 1(b) in the main text), we perform the particle-basedsimulation for the bulk system. We use the following particle model [35], m ˙ v i = − γ v i + F ˆ v i + (cid:88) (cid:104) x i , x j (cid:105) ˆ v i /N + (cid:88) k ∇ i U ( | x i − x k | ) + (cid:112) γk B T ˆ ζ i ( t ) , (S3)where x i and v i are the position and velocity of each particle and ˆ v i is the unit vector parallel to v i , (cid:104) x i , x j (cid:105) denotes theneighbors of the i -th particle and N is the number of neighboring particles, U ( | x i − x k | ) represents repulsive potential betweenparticles introduced to avoid the condensation, and √ γk B T ˆ ζ i ( t ) is the white Gaussian noise term. Here, we use Yukawa po-tential U ( r ) = b/ ( re κr ) . We implement the pillars by using a steep one-sided harmonic repulsive potential k p x / . Performingthe numerical simulation of this model in our periodic kagome lattice starting from the random initial state, we observe theemergence of the steady-state flow discussed in the main text (see the Supplementary Movie published with the manuscript).Supplementary Figure S1 shows a typical snapshot of the particle-based simulation. These results justify the presence of thesteady-state flow in Fig. 1(b) in the main text as the present particle-based model should be described by the Toner-Tu equationsin the hydrodynamic limit.We can also make a qualitative discussion about the emergence of the steady-state flow. Since the actual system must be finiteand have walls around the system, the flow cannot come in and out of the edge of the entire system. Assuming the periodicityof the steady-state flow imposed by the periodic structure of the bulk, one can conclude that flows in and out of each unit cellare also forbidden in the bulk of the system (cf. the red arrows in the center of Supplementary Figure S2). Then, around the FIG. S1. Typical snapshot from the particle-based simulation for our periodic kagome lattice. Each circle represents an active particle and abar extending from it indicates the velocity of each particle. The total number of particles is 200 and the parameters are m = γ = 1 , F = 0 . , a = 60 , b = 5 , κ = 0 . , k p = 100 , and k B T = 2 × − . FIG. S2. Schematic diagram of the qualitative discussions for the steady-state flow. The flow depicted by the red arrows is prohibited by theboundary conditions. boundary of a unit cell, the steady-state flow must be parallel to the edge of a unit cell. For these reasons, the flow in Fig. 1(b)in the main text can naturally emerge as a steady state.We note that the linear stability analysis is a standard method to confirm the validity of the steady state. Our analysis basedon the linearized Toner-Tu equations corresponds to the linear stability analysis around the steady state of interest here. In oursetup, since the effective Hamiltonian is found to be Hermitian and thus exhibits real eigenvalues, unstable modes are absent atthe level of the current analysis. We note that, while the above arguments can justify the presence and stability of the steady-stateflow considered in the main text, the uniqueness of the steady state still remains as an open question and should be addressed byperforming full nonlinear calculations of the Toner-Tu equations.
Linearization of the Toner-Tu equations and the derivation of the Schr¨odinger-like equation
Here we show the details about the derivation of the linearized equation. The full Toner-Tu equations read ∂ t ρ + ∇ · ( ρ v ) = 0 , (S4) ∂ t v + λ ( v · ∇ ) v + λ ( ∇ · v ) v + λ ∇| v | = ( α − β | v | ) v − ∇ P + D B ∇ ( ∇ · v ) + D T ∇ v + D ( v · ∇ ) v + f . (S5)Since we neglect the diffusive terms and λ , λ terms, we obtain the simplified Toner-Tu equations ∂ t ρ + ∇ · ( ρ v ) = 0 , (S6) ∂ t v + λ ( v · ∇ ) v = ( α − β | v | ) v − c ρ ss ∇ ρ, (S7)where we assume the equation of state P = c ρ/ρ ss with c being the sound speed and that the external force f is absent. Weconsider the fluctuations of density and velocity fields from a steady state, δρ ( r , t ) = ρ ( r , t ) − ρ ss ( r ) , δ v ( r , t ) = v ( r , t ) − v ss ( r ) and neglect their second-order contributions in Eq. (S7). Then, we obtain ∂ t δρ + ( v ss · ∇ ) δρ = − ρ ss ∇ · δ v , (S8) ∂ t δ v + λ ( v ss · ∇ ) δ v = − β ( v ss · δ v ) v ss − c ∇ δρρ ss . (S9)Finally, we neglect the second-order contributions of v ss and arrive at the linearized equation (3) in the main text.We can also derive the Schr¨odinger-like equation from Eqs. (S8), (S9) with a little algebra. Applying the convective derivative ∂ t + λ ( v ss · ∇ ) to Eq. (S8), we obtain ( ∂ t + λ ( v ss · ∇ ))( ∂ t + ( v ss · ∇ )) δρ = c ∇ δρ. (S10)Assuming an oscillating solution δρ ( r , t ) = δ ˜ ρ ( r ) e iωt , the equation takes the form as [ c ∇ + ω − iω ( λ + 1) v ss · ∇ ] δ ˜ ρ = 0 , (S11)where we assume that | v ss | is small and neglect its second-order terms. This equation is equivalent to the Schr¨odinger-likeequation (5) in the main text aside the second-order contributions of | v ss | . 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 withrespect to spatial degrees of freedom. Specifically, we approximate the continuum space by a discretized triangular lattice andconsider quantities on each lattice point as values of density and velocity fields. Also, the derivatives ∂ x,y must be converted intothe difference between neighboring sites [59]. The way of this conversion is not unique and can lead to numerical errors; a naivediscretization of the effective Hamiltonian can fail to provide the correct band structure. Since our system is symmetric under π/ rotation, the band structure should also reflect that symmetry. However, one cannot symmetrize a pair of linear combinationsof the derivatives ∂ x,y under π/ rotation. This is the main reason why the calculated band structures can in practice break thesymmetry 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 theredundant degree of freedom δ ˜ v r and formally rewrite the linearized equation (3) in the main text as follows: H (cid:48)(cid:48) δ ˜ ρδ ˜ v x δ ˜ v y δ ˜ v r = ω δ ˜ ρδ ˜ v x δ ˜ v y δ ˜ v r , (S12)where we denote H (cid:48)(cid:48) as H (cid:48)(cid:48) = − i v ss · ∇ − i∂ x − i∂ y − i∂ x − iλ v ss · ∇ − i∂ y − iλ v ss · ∇
00 0 0 0 . (S13)This deformation only adds trivial eigenstates with zero eigenvalues. We next consider the unitary matrix U = √ √ − √ √ √ − √ − √ √ , (S14)and use it to transform H (cid:48)(cid:48) into the following form: H (cid:48) = U H (cid:48)(cid:48) U − = − i v ss · ∇ √ ∂ √ ∂ √ ∂ √ ∂ λ v ss · ∇ − λ v ss · ∇ − λ v ss · ∇ √ ∂ − λ v ss · ∇ λ v ss · ∇ − λ v ss · ∇ √ ∂ − λ v ss · ∇ − λ v ss · ∇ λ v ss · ∇ . (S15)Note that the eigenvalues are unchanged and the eigenvectors are only globally transformed by U . Thus, these transformations donot alter the topology of the system. However, this new basis now naturally reflects the underlying symmetry of kagome latticeand, in practice, allows one to accurately calculate the band structure with much less numerical errors than the naive discretizationin the basis of x, y directions. To further improve the numerical accuracy, we also implement a hybrid discretization of thederivatives. More specifically, to convert the derivatives into the discretized form, we use a forward difference, a backwarddifference and a central difference for the derivatives ∂ , , in the first row of the effective Hamiltonian H (cid:48)(cid:48) , the derivatives ∂ , , in the first column of the effective Hamiltonian H (cid:48)(cid:48) , and the derivative ∇ , respectively. This conversion sustains theHermiticity of the effective Hamiltonian and can remove substantial numerical errors that can contribute from large wavenumbercomponents.We compare the bulk band structures for the unordered state, | v ss | = 0 , obtained by the calculations with and withoutthe redundant degree of freedom. As shown in Supplementary Figure S3(a), with the proper discretization of the effectiveHamiltonian with the redundant degree of freedom, we can obtain the linear dispersion symmetric under π/ rotation. Onthe contrary, Supplementary Figure S3(b) demonstrates that the bulk band without the redundant degree of freedom shows theincorrect dispersion that is independent of k y . We can also obtain the analytical expressions of the dispersion relation for theunordered state. The correct dispersion is ω = 0 , ±| k | while we obtain the following dispersion from the calculation without the0 FIG. S3. (a) The correct band structure obtained for the continuum system discretized with the redundant degrees of freedom. (b) Unphysicalband structure obtained by the numerical calculations without the redundant degrees of freedom. redundant degree of freedom ω = 0 , M (cid:118)(cid:117)(cid:117)(cid:116) sin (cid:18) k x + 2 πnM (cid:19) + 13 (cid:34) sin (cid:32) k x / √ / k y + 2 πmM (cid:33) + sin (cid:32) − k x / √ / k y + 2 π ( m − n ) M (cid:33)(cid:35) , (S16)where M is the mesh number for the numerical calculation and n, m = 0 , , · · · , M − . Considering m = 0 , n = M/ , k x = 0 ,the frequency is independent of k y and always . With the other values of k x , the frequency can be also almost independentof k y if we choose a certain integer n . Even if we consider nonzero | v ss | , these unphysical bands still remain and affect thetopology of the system. Therefore we have to introduce the redundant degree of freedom. Bulk bands and edge dispersions
In our model, we find that some edge bands exist deep inside the bulk bands. In practice, it is difficult to graphically demon-strate that those edge bands actually connect gapped bulk bands. This is because the bulk bands are gapped but not “fullygapped” in the sense that bulk bands and an edge mode can have the same frequency but at different k x . As a consequence, inthe open-boundary spectrum E ( k y ) , the bands appear to overlap even if the corresponding bulk bands are actually gapped.To explain this point more explicitly, we calculate the bulk dispersion (under the periodic boundary conditions) for the param-eters we use in the main text. Supplementary Figure S4(a) shows the bulk dispersion relation around the frequency ω = 4 . for | ˜ v ss | = 0 . , which corresponds to the gap in Fig. 2(b) in the main text. We can confirm that the three bands are separated fromeach other and thus are gapped. However, they are not “fully gapped” in the above sense, i.e., no flat planes parallel to the k x - k y plane exist without crossing the three bands. On the other hand, the bulk bands around the frequency ω = 7 . for | ˜ v ss | = 0 . (corresponding to Fig. 4(b) in the main text) are “fully gapped” as shown in Supplementary Figure S4(b). Indeed, there exists aflat plane across the whole Brillouin zone without touching to the bulk bands (cf. the gray dashed horizontal line in Fig. 4(b) inthe main text). This helps us to graphically demonstrate the existence of an edge dispersion connecting the bulk bands.1 FIG. S4. Bulk band structures under the periodic boundary conditions (a) around ω = 4 . for | ˜ v ss | = 0 . and (b) around ω = 7 . for | ˜ v ss | = 0 .5