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# Topological localization in out-of-equilibrium dissipative systems

Edited by Tom C. Lubensky, University of Pennsylvania, Philadelphia, PA, and approved August 1, 2018 (received for review December 7, 2017)

## Significance

Topological insulators and their analogs in mechanical materials support conducting states only on their surface. We show that such topologically protected edge modes can also occur as the steady states of classical systems driven out of equilibrium. As proof of principle of the generic applicability of such notions, we show the existence of topologically localized states in a collection of interacting particles described by a hydrodynamic theory and discuss a general procedure to establish them in stochastic networks. In both cases, dissipative processes that break time-reversal symmetry are key to topological protection. Our results provide design principles for robust edge modes in synthetic systems as well as for the localization of flow of matter and information in biology.

## Abstract

In this paper, we report that notions of topological protection can be applied to stationary configurations that are driven far from equilibrium by active, dissipative processes. We consider two physically disparate systems: stochastic networks governed by microscopic single-particle dynamics, and collections of driven interacting particles described by coarse-grained hydrodynamic theory. We derive our results by mapping to well-known electronic models and exploiting the resulting correspondence between a bulk topological number and the spectrum of dissipative modes localized at the boundary. For the Markov networks, we report a general procedure to uncover the topological properties in terms of the transition rates. For the active fluid on a substrate, we introduce a topological interpretation of fluid dissipative modes at the edge. In both cases, the presence of dissipative couplings to the environment that break time-reversal symmetry are crucial to ensuring topological protection. These examples constitute proof of principle that notions of topological protection do indeed extend to dissipative processes operating out of equilibrium. Such topologically robust boundary modes have implications for both biological and synthetic systems.

Theoretical and experimental studies of biophysical mechanisms, such as error correction in DNA replication (1), adaptation in molecular motors controlling flagellar dynamics (2⇓–4), and timing of events in the cell cycle (5), are beginning to show the close connection between robust functioning and nonequilibrium forces. Theoretical results (6, 7) have also elucidated the connection between energy dissipation and fluctuations in a wide class of nonequilibrium systems and have shown how dissipation can be used to tune steady states of many-body nonequilibrium systems (8). However, unlike the behavior and characteristics of equilibrium systems, where no energy is dissipated, general principles governing fluctuations about a steady state or the steady state itself in far-from-equilibrium conditions are just being discovered. As such, strategies that allow for the engineering of specific robust steady states in nonequilibrium biological and soft matter systems are highly desirable. In this paper, we take an approach that is motivated by the physics of topological insulators to construct such states in nonequilibrium systems.

The discovery of robust localized edge states in mechanical systems (9) that resemble those found in topologically nontrivial electronic (10) and photonic (11) systems has enabled the development of new design principles. For instance, it has been shown that assemblies of coupled mechanical oscillators can support topologically protected directed modes at their boundary (12, 13). Such assemblies can function as waveguides robust to backscattering. Localized edge modes have also been discovered in mechanical lattices (9, 14). These edge modes can be set up either at the edge of a lattice (9) or at topological defects in the interior of the lattice (14). They are formally zero-energy “free” modes but unlike the commonly found long-wavelength, zero-energy modes in the bulk of marginally stable lattices, these topologically protected boundary modes are highly resistant to perturbations due to disorder and environmental fluctuations. Like their electronic and photonic counterparts, topological modes in the above mentioned mechanical systems have been characterized by topological indices, such as winding or Chern numbers (15⇓–17).

In this paper, we propose that topologically protected modes can be encoded in a variety of biological and soft matter contexts at the cost of energy dissipation. Topological protection can occur in non-Hermitian quantum models with dissipation (18, 19) or through time-periodic driving (20). We focus here on a very different class of driven dissipative classical systems that have no obvious topological properties. Like in the electronic and mechanical analogs, a topological index can be associated with the bulk of the system, which predicts protected zero or soft modes localized at the edge or an interface. The modes are highly robust and insensitive to perturbations. In all of the instances considered in this paper, topological protection and the presence of an associated tunable spectral gap require that the fundamental equations of motion contain dissipative couplings.

We derive our results in two broad and apparently dissimilar contexts. Both systems that we consider can be mapped to noninteracting model quantum Hamiltonians with topologically protected states using a procedure formally similar to that in ref. 9. In the first, we consider biochemical networks with connectivity motivated by those of networks commonly encountered in biophysical information processing and control (21). We show that the spectrum of the master equation rate matrix can support localized edge modes that are separated from the bulk via a band gap. In the second, we consider a hydrodynamic description of active matter, specifically that of collections of driven rotating particles in a confined geometry on a frictional substrate (22, 23), that exhibit large-scale flow localized at the boundaries. The spectrum of dissipative modes in this context implies a rapid relaxation of fluctuations of the flow in the bulk, while edge states, including the stationary state, when allowed by boundary conditions, can be longer lived. This implication of topological protection in dissipative systems is distinct from the directed propagating edge modes described recently in fluids (24⇓–26) that are more directly analogous to single-particle quantum models (10).

The hydrodynamic equations have a structure very different from that of master equations considered in the first example. The two address phenomena at very different length scales and timescales: one being a coarse-grained phenomenological description of disordered matter and the other being a microscopic approach involving a network of discrete states. Nonetheless, they both describe dissipative phenomena characterized by the production of entropy and lack time-reversal symmetry. Our results elucidate the design principles required for robust steady states in various biophysical and synthetic systems.

## Topological Protection in Markov Networks

In this section, we consider the out-of-equilibrium statistical dynamics of stochastic processes described by a Markov network (27). Such descriptions are routinely used in statistical mechanics as reduced models of chemical and biophysical processes (28). For equilibrium systems, the steady states and dynamics of fluctuations about it can be described in terms of energy landscapes, but such a simple description is not available for out-of-equilibrium processes. Hence, any insight into the existence and robustness of steady states of systems driven out of equilibrium by energy-consuming processes, such as those involved in biological functions, is potentially valuable.

Here, we focus on the steady state of lower-dimensional networks characterized by uniform or nearly uniform transition rates between various mesoscopic states. This is in analogy with periodically ordered lattices in electronic and metamaterials that host localized topological states at their edges (10). We show in the following that the steady states of certain Markov networks can indeed be mapped onto the ground states of Su–Schrieffer–Heeger (SSH)-like models for 1D periodic systems with topological properties—the “forward” and “backward” transition rates in the master equation play the role of the hopping rates between the two sites of a unit cell in the SSH model (29).

A simple illustration of this is depicted in Fig. 1: the probability flow in a 1D Markov chain with constant rate of site-to-site transition rates in the bulk regions on the left and right of an interface separating them is shown in Fig. 1 *A* and *C*. By mapping the steady state of the 1D Markov chain to the ground state of a Hamiltonian of a topologically nontrivial tight binding model (Fig. 1 *B* and *D*), we show that a possibly “disordered” interface connecting two “bulk” regions of the network with different transition rates can host localized topological modes depending on the transition rates in the bulk. Our results follow from the bulk–boundary correspondence inherent in topological systems, where the existence of localized zero modes at an edge can be predicted by studying the properties of the system in the bulk (9).

The procedure for establishing the mapping between the stochastic process and the Hamiltonian of a topological tight binding model is distinct from and more direct than the previous work, in which we suggested that the properties of certain out-of-equilibrium Markov states can be understood in terms of topological winding numbers (30). Indeed, we explicitly provide forms of tight binding Hamiltonians with ground states that are the steady states of the out-of-equilibrium stochastic processes that we consider. While the 1D network in Fig. 1 is fairly trivial, the procedure outlined below can be used to construct effective tight binding Hamiltonians for more complex biophysical networks with many cycles.

We begin by recalling that the dynamics of Markovian systems can be modeled using a master equation (28)**2** counts the number of zero modes of W that are localized at the interfacial region (9). This is exactly equal to one if the unique steady-state solution,

We now show that the condition in Eq. **2** can be related to a topological quantity calculated from the master equation in the bulk network. Note that the state to state transition matrix, W, does not itself possess the symmetries usually associated with topological protection in electronic or mechanical materials (10, 31). The eigenvalue spectrum of the master equation necessarily has one zero eigenvalue, with the rest of the eigenvalues being less than zero (28). Furthermore, W is usually nonhermitian and can have complex eigenvalues. In this form, Eq. **1** does not possess any obvious topological properties.

To uncover the topological properties of the master equation, we first note that the rate of change of probability can be expressed as (32)

We are interested in cases when the trace count as described by Eq. **2** predicts the existence of localized zero modes. We will show below with concrete examples that Eq. **2** can be expressed as a topological invariant by using the fact that the information about the interface between two homogeneous bulk subregions is contained in

As an illustration of these steps, we first consider the 1D Markov chain in Fig. 1. The dynamics of the random walker can be described by the master equation in Eq. **1**. Using the above-mentioned decomposition into *z* matrix. The trace in Eq. **5**, denoted by **5** can only take values *A*, whereas it is localized at the opposite ends when *C*. The decomposition into *A* and *C* for the 1D random walker are shown in detail in *SI Appendix*.

That **2** can be expressed as a difference of integer topological winding numbers characteristic of the left and right bulks,

The topological winding numbers for the 1D random walker can be derived by considering the transition matrix in one of the bulk regions **6**, **7**, one can see that H is isomorphic to the Hamiltonian of the quantum SSH model with hopping rates

We have mapped the zero modes of the 1D Markov chain directly to those of a corresponding SSH model, with the backward and forward transition rates playing the role of the two hopping parameters in the tight binding model of the Hamiltonian. The polarization generated in a topologically nontrivial SSH model is simply related to current generated by the bias in the random walk model. In this sense, dissipation in the bulk of the random walk model plays a crucial role in the generation of localized states. In analogy with the SSH model, two connected chains with opposite polarizations of probability flux will naturally lead to an accumulation of probability at the interface as the system approaches steady state.

The Markov-state model in Fig. 1 does not possess multiple cycles. Such cycles can allow for feedback at the cost of energy dissipation and are features common to Markov-state representations of many out-of-equilibrium biophysical processes (3). To derive our results in the context of out-of-equilibrium stochastic models relevant for biological processes, we consider the minimal Markov-state model shown in Fig. 2*A* that was introduced in ref. 30. This ladder-like Markov network possesses two horizontal rails with transition rates

The Markov-state model is composed of two translationally invariant bulk-like regions with an interface connecting them. Specifically, the rates of transitions in the bulk regions do not depend on the position along the horizontal axis. The rates in the interfacial region interpolate between the two bulk regions. The transition rates in the right bulk region are denoted by the (^{∼}) symbol to distinguish them from those on the left. As discussed in ref. 30, the spatial connectivity and structure of this Markov-state network resemble those of networks routinely used to study adaptation (4), kinetic proofreading (33, 34), and cell signal sensing (35). These and other Markov-state representations of biophysical processes can often be decomposed into bulk-like subgraphs stitched together by interfaces as indicated in Fig. 2*A*. The subgraphs are formed by finite periodic replication of a particular module or motif.

Since we are mainly interested in networks of the form in Fig. 2, which possess translational symmetry along one (horizontal) axis and the interface spans the other (vertical) axis, we decompose the rate matrix of this system as**10**, because the bare *SI Appendix*. Again, the crucial point in this decomposition is that **W** has topologically protected modes whenever the following Hermitian operator constructed with *SI Appendix*.

The polarization implicit in the Hamiltonian H reflects the currents generated by the master equation rate matrix W. For instance, the effective Hamiltonian that we have depicted in Fig. 2*B* is composed of two horizontal rails, each with a lattice structure that resembles that of the SSH model. The two rails can potentially have polarizations with the same magnitude but opposite signs. This choice corresponds to the case *C*, we show the numerically computed count of localized zero eigenmodes of W using Eq. **2** and compare it with that of the effective matrix H. We also show the gaps in the eigenvalue spectrum for W and H in Fig. 2 *C* and *D*, respectively, by numerically computing the first two eigenvalues as the hopping rates are varied. Such a gap between the steady state and subsequent eigenvalues in the eigenvalue spectrum makes the steady state robust against random fluctuations. We indeed find that, as long as the Hamiltonian H has a winding number mismatch that supports a localized zero mode at the interface, W is also guaranteed to have a zero mode localized to the interface.

This correspondence between a master equation operator, W, and a topologically nontrivial Hamiltonian, H, is the main result of the first part of the paper. It establishes that the topological properties of W can be computed by simply constructing winding numbers for the matrix

## Topological Protection in Many-Body Systems

There is broad interest in dissipative, steady-state structures (such as ordered phases and topological defects therein) that emerge in collections of particles driven away from equilibrium in both synthetic (39, 40) and biological contexts (41, 42). Such steady states, if topologically protected, are likely to be robust against disorder and can be categorized into distinct topological classes—thus guiding applications that involve organization away from equilibrium (8, 43).

The results of the previous sections are, however, specific to effectively single-particle Markov-state processes. A natural question is whether similar statements about topological modes can be made for collections of many interacting particles out of equilibrium. Indeed, an effective Markov-state representation of the dynamics of a many-particle interacting system cannot be simplified in the same way as the models in Figs. 1 and 2. While a full microscopic description of a many-particle interacting system is in general intractable without performing detailed simulations, the relaxation of such a system toward equilibrium or a steady state can be described in terms of a few slowly decaying collective modes with long wavelengths. Such a hydrodynamic description in terms of long-wavelength collective modes differs fundamentally from the electronic or mechanical materials with periodic order in which topologically protected edge states typically occur (9, 10).

As a nontrivial application of this notion of topological protection to a many-particle system, we focus on collections of actively rotating particles (rotors) with an intrinsic “spin” angular momentum degree of freedom (Fig. 3*A*). The collective dynamics of such particles are more complex and relatively less understood than those that involve linear self-propulsion (39), although there is now a wide range of experiments where such active angular momentum injection in the bulk can be realized in a controlled manner (23, 44). Examples of such chiral active flows include shaken chiral grains (22), liquid crystals in a rotating magnetic field (45), and light-powered colloids (46). Instances of naturally occurring flowing matter with actively generated rotation by molecular motors range from the rotational beating of flagella of swimming bacteria to active torque generation in the cellular cytoskeleton (47). Boundary flows where the rotors circulate around the edge of a container have been seen in ref. 22 and more recently in simulations in ref. 23. In a biological context, such flows have been implicated in the breaking of left–right symmetry during morphogenesis (48). However, connections to topological protection, if any, have not been explored. We now show that driven rotors in a finite 2D geometry on a dissipative substrate can indeed support topologically protected localized edge states. These states allow the system to robustly localize long-lived flows and fluctuations to its edge, while fluctuations in the bulk dissipatively decay over a finite lifetime set by friction. This topological interpretation of the hydrodynamic modes is the main result of this part of the manuscript.

### Chiral Active Hydrodynamics: Flow and Fluctuations.

The dynamics of a collection of actively spinning rotors can be described by a coarse-grained hydrodynamic theory formulated in terms of conservation laws and constitutive relations derived using the general principles of irreversible thermodynamics formulated in the seminal work of de Groot and Mazur (49). The key hydrodynamic variables in the theory are intrinsic rotation rate (or spin angular velocity) **12** together with the uniform active torque, τ, driving each rotor determine the dynamics. We do not consider the theoretically allowed “odd viscosity,” which is nondissipative and is important at high drive (53).

The coupled dynamical equations for the spin angular velocity, Ω, and vorticity, ω, are derived by extremizing the above dissipation functional (49), R, as

In a confined geometry, the rotors are prevented from rotating freely at the walls, which induces a spatial profile in the angular velocity and therefore, in the vorticity (22, 23). In fact, a finite cluster of uniformly spinning unconfined rotors also exhibits edge vorticity. These steady-state solutions of the hydrodynamic theory are detailed in *SI Appendix*. Taken together, these hydrodynamic solutions illustrate for different boundary conditions that a steady-state flow can occur only at the edge of a finite system with substrate friction and not in the bulk, which has zero vorticity.

The dissipation-dominated dynamics (the convective term, being quadratic in velocity, is negligible for small flows or small-amplitude flow fluctuations) about the uniform steady state of chiral active fluids, **13**. The bulk diffusive modes of the spin and vorticity fluctuations about the steady-state flow,

### Topological Interpretation of Dissipative Hydrodynamic Modes.

While M in itself has no apparent topological properties, its positive semidefinite character for purely dissipative dynamics allows us to construct a suitable square root H, which may possess a nontrivial topological index (9), as we now show. In doing so, we are inspired by the example of Dirac (54) as well as of mechanical lattices (9, 31). Importantly, however, our objective in taking this square root is not to go from second-order (in time) to first-order dynamics but to construct topologically nontrivial operators.

In the regime of weak spin–vorticity coupling, **14** reduce to nearly independent spin and velocity fluctuations, with timescales of decay that are given by the eigenvalues of Eq. **14**. In the small q limit, these inverse decay times are proportional to **15**, is like the magnetic field of a monopole (10, 55): *SI Appendix* has details).

We note that the integrated Berry curvature (“Berry flux”) in the bulk here is a half-integer, since the wavenumbers **15**. In the bulk, either of **13** have to decay away from the interface to their bulk values. This is easy to see for the simple 1D differential operator, which allows exponentially localized zero eigenstates, *SI Appendix*, this representation is the square of a topologically nontrivial finite matrix (with a bulk topological index) only for specific choices of boundary conditions. The boundary conditions effectively specify the sign of m in the square root matrix H.

We emphasize that, although for a quantum model and many classical models, the Chern number is associated with the direction of propagation of waves (10, 11, 31) (specifically the difference in the number of right- and left-moving edge modes), such an interpretation is absent in the purely dissipative dynamics that we consider here. The fictitious frequencies (i.e., eigenvalues of H) that we construct here are just the square roots of the inverse relaxation timescales associated with purely diffusive modes (and not propagating waves):

### Topological Properties of Chiral Active Flows.

The conclusions outlined above can be easily generalized to the chiral active dynamical operator in Eq. **13**. The very long-wavelength normal modes (**14** are obtained by diagonalizing M in the small q limit**17** has the form **15**) with corresponding effective Dirac masses: **16**. The principle of bulk edge correspondence when applied to these Dirac Hamiltonians for a chiral edge flow setup ensures the presence of

### Bulk Vs. Edge Spectrum of Dissipative Modes.

Bulk diffusive modes with dynamics governed by **17**, relax with timescales **13**: *y* axis (as shown in Fig. 3*A*) of the form **14**. This analysis implies that any long-lived fluctuations including the steady-state flow (corresponding to

The distinction in the relaxation of bulk and edge modes is a consequence of topological protection, distinct from the propagating wave-like edge modes in quantum and classical dynamics that are not purely dissipative. We again note that, while the dissipative edge modes are topologically protected as described above, the chirality of the edge flow of rotors results directly from the sense of the driving torque and is not a consequence of the topology.

A representative spectrum of inverse timescales of dissipative modes is plotted in Fig. 3*B*. In the weak spin–vorticity coupling *B*) and mainly vorticity (blue lines in Fig. 3*B*) fluctuations. The solid curves in Fig. 3*B* indicate the bulk spectra, whereas the dashed lines in Fig. 3*B* correspond to diffusive long-lived fluctuation modes of spin and vorticity that are localized at the edge and are compatible with stress-free boundary conditions. At *SI Appendix*.

Although we ignored the convective term in our discussion so far, we point out now that its presence does not preclude the topological properties of the diffusive hydrodynamic operator. When perturbed about a uniform flow state, **14**, making it non-Hermitian. However, this does not affect the spectral gap at

Finally, we note the role played by the coupling of spin and vorticity in our analysis. The spin–vorticity coupling, Γ, “mixes” fluctuations in Ω and ω to order

### Molecular Simulations of Driven Rotors.

As another test of these ideas, we set up molecular dynamics simulations of a system composed of driven rotors confined in a box. It has been well-established that the hydrodynamic equations in Eq. **14** are good descriptors of the dynamics of such rotor systems (22, 23). The equations of motion of individual rotors in our molecular dynamics simulations are*Materials and Methods*.

To check for the existence of localized vorticity predicted by the theory, we computed the time-averaged velocities of the center of mass of the rotors. As is clear from Fig. 4*A*, this system supports localized edge flows. This flow is immune to backscattering in the presence of obstacles and persists even if the edges of the box used in the simulation are jagged (as depicted in Fig. 4*C*). When the rotors are actively driven only in a partial region of the simulation box (Fig. 4*B*), the flow is localized at the boundary of this region of driving. This may be experimentally realized, for example, in the light-powered micromotors studied in ref. 46 by selectively shining light in the region of interest.

According to a solution of the hydrodynamic equations with a specific choice of boundary conditions (23) (*SI Appendix*), one expects the inverse effective localization length κ to scale like *D*). The loss of topological protection in a finite system at nonzero friction can be interpreted as a signature of the narrowing of the associated gap in the eigenvalue spectrum of flow solutions in relation to fluctuations as friction is decreased, resulting eventually in an unstable steady state. Similar features are apparent as topological protection is lost in finite-sized electronic and mechanical topological insulator models (57). These observations taken together constitute a demonstration of the robustness of edge flows to boundary perturbations.

### Topological Protection of Propagating Sound Modes.

In this paper, we focus on dissipation-dominated processes and topologically interpret the bulk and edge fluctuation spectrum of chiral active hydrodynamics. Even if inertial terms were present in these hydrodynamic equations, the gap in the imaginary (dissipative) part of the spectrum of fluctuation relaxation times inherent in Eq. **14** is preserved. However, the presence of density fluctuations coupled to the longitudinal component of velocity can, in principle, allow propagating sound waves with gapped frequency spectrum (26). By relaxing the incompressibility constraint and perturbing around a uniform edge velocity *SI Appendix* that sound modes can propagate in the regime of wavenumbers: *SI Appendix* are in fact isomorphic to those found for sound in polar active flows derived from Toner-Tu active hydrodynamics (26). This implies the existence of topologically protected sound modes for chiral active flows when placed in the microfluidic lattice setup proposed in ref. 26 if the rotors are driven in opposite senses in neighboring cells.

## Discussion

Using two disparate examples of complex dynamical systems that are out of equilibrium, we have shown in this paper that topologically protected states can arise in principle in a variety of dissipative systems: both stochastic networks and active flows. Unlike other systems for which topological protection has been previously explored, such as mechanical lattices (9) or propagating sound modes in hydrodynamic equations (25, 26), dissipation is key to the phenomena considered here. In both cases considered, the topological properties are not readily apparent from the structure of the relevant operators that govern their dynamics. We reveal their topological properties by decomposing them suitably to map the properties of their steady-state solutions to the zero-energy states of Hamiltonian models that can be characterized by a topological index. Our work indicates that both single- and many-particle dynamics with interactions at both microscopic and macroscopic scales can result in topological modes that are localized at boundaries. Furthermore, our results also suggest that the chiral edge flows ubiquitously seen in synthetic and biological matter are potentially robust, topologically protected modes. This has applications for self-assembly of metamaterials and can also guarantee robust localized flows of both information and matter in biology.

## Materials and Methods

The molecular dynamics simulations were performed by evolving the Langevin equation with Euler dynamics. The rotors in our simulations were composite particles composed of three equally spaced point particles arranged along a line. The particles interact according to a Yukawa potential with a decay length *A*, *C*, and *D* were performed with *B* were performed with

## Acknowledgments

We acknowledge very useful discussions with William Irvine, Sid Nagel, Tom Witten, Jayson Paulose, Anton Souslov, and Zhenghan Liao. K.D. and S.V. were funded by National Science Foundation Grant DMR-MRSEC 1420709. K.K.M. acknowledges support from NIH Grant R01-GM110066. K.K.M. is also supported by the Director, Office of Science, Office of Basic Energy Sciences, Chemical Sciences Division of the US Department of Energy under Contract DE-AC02-05CH11231. S.V. acknowledges funding from the University of Chicago and Army Research Office Grant W911NF-16-1-0415.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: svaikunt{at}uchicago.edu.

Author contributions: K.D., K.K.M., and S.V. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1721096115/-/DCSupplemental.

Published under the PNAS license.

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