Topological chiral random walker

Abstract

Understanding how biological and synthetic systems achieve robust function in noisy environments remains a fundamental challenge across the physical and life sciences. To connect robust behavior with non-trivial topological features present already in the dynamics of individual units, here we introduce the topological chiral random walker (TCRW) model. While exploring the system, a TCRW locates edges and boundaries in the system and develops topologically protected edge currents even in the presence of defects and disorder. Drawing on the bulk-boundary correspondence found in hard condensed matter systems allows us to rationalize the emergence of robust edge currents through topological features of the dynamic spectrum. We show that chiral motion and rotational noise with opposite chirality are two crucial components in our inherently non-Hermitian model. As proofs of principle, we first show that a topological walker outperforms diffusive motion to efficiently solve complex mazes due to its property of remaining on the edge with some rare detachments. Second, we use this model to design building blocks that can perform efficient self-assembly overcoming the timescale bottlenecks of diffusion-limited growth and reducing self-assembly times by approximately 80%.

Figures (12)

• Figure 1: Chiral random walker.(a) Schematic showing the discrete dynamics of a chiral random walker combining a chiral move (translation and rotation) and rotational noise with opposite chirality. Notice the dynamics of the TCRW for specific values of $D_r$ and $\omega$. (b) Sample trajectories of such a walker for different values of chirality $\omega$ and $D_r=10^{-3}$. (c) MSD of walkers for different values of $\omega$ and $D_r=10^{-3}$, confirms that walker experiences normal diffusion. (d) The diffusion coefficient decreases linearly with chirality $\omega$ independent of the value of $D_r$.
• Figure 2: Chiral edge current.(a) A CRW is placed on a 2D grid with hard walls, where $P(X,Y)$ is the probability of finding a walker on the grid point $(X,Y)$. (b) Sample trajectory of the walker for the first $10^6$ steps of the simulation. (c)-(e) Total current $\bm J$, the chiral current $\bm J_{\omega}$, and the noise current $\bm J_{D_r}$ for $\omega = 1$. (f)-(j) Corresponding plots for an achiral walker with $\omega=0.5$ and (k)-(o) for a CRW with $\omega=0$ on a grid with defects on the edge and in the bulk of the system. (l) The trajectory of the walker develops edge currents along external and internal boundaries. (o) The internal and external boundary edge currents have opposite chiralities. Note that introducing defects and observing the edge current in (o) confirms that the system is sensitive to edges due to the topological properties of the dynamics. For simulations we used $D_r=10^{-3}$ and total number of steps $T=10^{10}$.
• Figure 3: Impact of rotational noise $D_r$ and chirality $\omega$. The black lines show the results obtained from the exact steady-state solution of the transition matrix. (a) The ratio $P_{\text{edge}}$/$P_{\text{bulk}}$ becomes independent of system size and confirms the decay of edge localization in the system by increasing $D_r$ for $\omega=1$. (b) The ratio of the total values of $\bm J_{D_r}$ and $\bm J_\omega$ are depicted for different $D_r$. The current scatters to the bulk for values of $D_r$ around one. (c)-(d) shows the orientation of the currents along the left edge for different $D_r$. Note, that the chiral current $\bm J_{\omega}$ is consistently directed towards bulk with $\pi/4$. (e) The angle of the current $J_{D_r}$ along the left edge goes from $-\pi/2$ to zero as we vary $D_r$ from zero to one. The angle is measured with respect to the horizontal axis. Note that for all plots in the second row (f)-(j) there is a symmetry with respect to $\omega = 0.5$. (f) The ratio $P_{\text{edge}}$/$P_{\text{bulk}}$ is independent of $\omega$. (g)-(i) The same as (b)-(d) but for varying $\omega$. (j) The angle of the currents along the left edge with respect to $\omega$.
• Figure 4: Topological origin of the edge localization and edge current.(a) Labeling of states for $\omega=1$ where CCW edge current is expected. Edge states are union of "to edge" and "along edge" states. (b) Spectrum of the model in PBC. The gap closes at $\omega=0.5$, where a topological transition occurs. (c) Decomposition into real and imaginary part of the spectrum in the presence of OBC for a small lattice of $L=2$. Note that the spectrum is colored based on the localization of corresponding eigenvectors on the edge states. (d) Real part of spectrum for a fully chiral case $\omega=1$. Note the coalescence of eigenvalues at $D_r \to 0$. (e) Same as (d) but for fixed $D_r$. (f) Spectrum of the model on a complex plane for a fully chiral case. Coloring differentiates localization on edge and current along edge. (g) Same as (f) but for changing $\omega$. (h) Spectrum of a model with hybrid boundary conditions (periodic along the $y$-direction and open along the $x$-direction). Note the bands localized on the edge. (i) Band structure of the HPBC on a closed circle defined by $(\cos k_x, \cos k_y)$. Different coloring is used to show localization on different states.
• Figure 5: Maze solving.(a) Trajectories of chiral walkers (top and bottom) and an achiral walker (middle). Orange shading indicates the walkers' paths. The blue circle marks the maze entrance, and the green star marks the exit of the maze. (b) Mean first-passage time $\tau_M$ to solve the maze, depending on the starting position and initial director. (c) Maze-solving time $\tau_M$ as a function of the chirality $\omega$, for different $D_r$. Top: unscaled. Bottom: rescaled by $D_r$. Lines show the average solving time measured from the transition matrix of a maze, and the markers show the average of times measured directly from simulations of a chiral walker. (d) Scaling of $\tau_M$ with maze size $L$. The upper dashed line shows $\tau_M\propto L^3$ and the lower dashed line shows $\tau_M\propto L^2$. (e) Solving disconnected mazes. (f) Solving mazes with wider passage ways ($D_r=10^{-3}$).