[Paper Review] Guided accumulation of active particles by topological design of a second-order skin effect

Lucas S. Palacios, S. Tchoumakov+4 more|arXiv (Cornell University)|Dec 28, 2020

Metamaterials and Metasurfaces Applications70 references99 citations

TL;DR

This paper proposes a topological design using a second-order non-Hermitian skin effect to guide and accumulate self-propelled active particles in 2D microfabricated devices without external fields. By engineering a periodic topographical structure with detailed balance at the unit cell level, the system induces spontaneous edge and corner localization of particles due to a non-zero topological invariant ν, which governs the robustness of particle accumulation. The stochastic circuit model predicts and confirms that devices with ν = 1 guide particles more efficiently than trivial ν = 0 devices, offering a blueprint for active matter applications in sensing and drug delivery.

ABSTRACT

Collective guidance of out-of-equilibrium systems without using external fields is a challenge of paramount importance in active matter, ranging from bacterial colonies to swarms of self-propelled particles. Designing strategies to guide active matter and exploiting enhanced diffusion associated to its motion will provide insights for application from sensing, drug delivery to water remediation. However, achieving directed motion without breaking detailed balance, for example by asymmetric topographical patterning, is challenging. Here we engineer a two-dimensional periodic topographical design with detailed balance in its unit cell where we observe spontaneous particle edge guidance and corner accumulation of self-propelled particles. This emergent behaviour is guaranteed by a second-order non-Hermitian skin effect, a topologically robust non-equilibrium phenomenon, that we use to dynamically break detailed balance. Our stochastic circuit model predicts, without fitting parameters, how guidance and accumulation can be controlled and enhanced by design: a device guides particles more efficiently if the topological invariant characterizing it is non-zero. Our work establishes a fruitful bridge between active and topological matter, and our design principles offer a blueprint to design devices that display spontaneous, robust and predictable guided motion and accumulation, guaranteed by out-of-equilibrium topology.

Motivation & Objective

• To achieve directed, robust, and predictable guidance of self-propelled active particles without external fields such as magnetic or electric fields.
• To explore the emergence of a second-order non-Hermitian skin effect in two-dimensional active matter systems, a phenomenon previously unobserved experimentally.
• To establish a design principle for microfluidic devices that exploit out-of-equilibrium topology to control particle motion and accumulation.
• To validate the role of the topological invariant ν in determining the efficiency of particle guidance and corner accumulation.
• To bridge active matter and topological physics by demonstrating a non-Hermitian topological phenomenon in a classical, soft-matter system.

Proposed method

• Designing microfabricated 2D periodic topographical devices with alternating left/right-oriented ratchet channels and vertically coupled microchannels to create a lattice with detailed balance at the unit cell level.
• Engineering the system so that the non-Hermitian skin effect dynamically breaks detailed balance at the edges, leading to particle accumulation at corners.
• Using a stochastic circuit model based on non-Hermitian Hamiltonians to describe particle dynamics without fitting parameters, with the model's predictions matching experimental data.
• Computing the topological invariant ν = wH × wnH, where wH is the Hermitian winding number and wnH is the non-Hermitian winding number, to classify the system as topological (ν = 1) or trivial (ν = 0).
• Employing particle tracking via a convolutional neural network to monitor trajectories and compute probability distributions and Shannon entropy for quantifying localization.
• Applying Fourier analysis to estimate spatial fluctuations and perform local averaging over 2–3 cells to reduce noise in entropy measurements, enabling quantitative comparison between theory and experiment.

Experimental results

Research questions

• RQ1 Can a second-order non-Hermitian skin effect be engineered and observed in a classical, active matter system without external fields?
• RQ2 How does the topological invariant ν = wH × wnH govern the efficiency of particle guidance and corner accumulation in 2D active particle systems?
• RQ3 To what extent can a stochastic circuit model predict particle dynamics in such systems without fitting parameters?
• RQ4 Does the presence of inversion symmetry allow for a second-order skin effect, and how does it differ from the first-order effect?
• RQ5 Can topological design principles be used to create robust, predictable, and efficient particle accumulation in active matter devices?

Key findings

• The second-order non-Hermitian skin effect leads to spontaneous accumulation of self-propelled particles at the corners of the 2D device, with no external fields applied.
• Devices with a non-zero topological invariant ν = 1 exhibit significantly enhanced particle accumulation at corners compared to trivial ν = 0 devices, as confirmed by both experiment and theory.
• The stochastic circuit model accurately predicts particle guidance and accumulation without fitting parameters, validating the theoretical framework.
• Shannon entropy analysis shows a clear difference in localization between topological (ν = 1) and trivial (ν = 0) devices, especially at high particle densities, confirming the non-Hermitian skin effect.
• The system maintains detailed balance at the unit cell level, yet the non-Hermitian skin effect breaks it dynamically at the edges, enabling robust, topologically protected particle transport.
• The topological invariant ν is found to be a product of 1D Hermitian (wH) and non-Hermitian (wnH) winding numbers, confirming its role as a higher-order topological invariant in 2D.