Exceptional non-Hermitian topological edge mode and its application to active matter

Topological materials, exhibiting robust edge-localized, scattering-free modes due to their nontrivial bulk topology (bulk-edge correspondence), have garnered significant interest. This correspondence, initially established in the integer quantum Hall effect, holds true in systems with symmetries like time-reversal symmetry in topological insulators. However, effective Hamiltonians in non-conservative systems (photonics, ultracold atoms, optomechanics, etc.) can become non-Hermitian, introducing complexities to the bulk-edge correspondence. While non-Hermitian topological materials have been classified based on bulk band topology, the bulk-edge correspondence is less clear-cut in these systems. This paper investigates a novel mechanism for robust gapless edge modes in non-Hermitian systems, distinct from the conventional bulk topology-based mechanisms and demonstrating a breakdown of the bulk-edge correspondence in such systems.

The existing literature extensively explores topological materials within the framework of Hermitian Hamiltonians, focusing on the bulk-edge correspondence and the role of symmetries like time-reversal symmetry. The discovery of the integer quantum Hall effect and subsequent research on topological insulators solidified the understanding of this correspondence. Recent studies have expanded this understanding to classify non-Hermitian topological materials based on bulk band topology. However, these studies often fail to adequately address the intricacies of the bulk-edge correspondence in non-Hermitian systems, where the interplay of dissipation and gain introduces new phenomena. This paper directly confronts this gap in the literature by examining the presence of robust edge modes in the absence of conventional topological protection.

The study employs several methodological approaches. First, a minimal tight-binding model is constructed using a two-layered Qi-Wu-Zhang (QWZ) model coupled with its time-reversal counterpart via a non-Hermitian term. This model, resembling the Bernevig-Hughes-Zhang model but with crucial non-Hermitian additions, allows the investigation of edge mode behavior in topologically trivial bulks. Numerical calculations of the edge band structure, under open and periodic boundary conditions, are performed to observe the emergence of gapless edge modes and exceptional points (EPs). The robustness of these modes against various perturbations and disorders (real on-site potentials, imaginary noise in coupling, imaginary on-site potentials) is analyzed numerically. An effective one-dimensional Hamiltonian is derived to characterize the low-energy dispersion of edge modes and investigate the conditions for maintaining gapless modes, tying the robustness to symmetry properties such as PT symmetry, CP symmetry, pseudo-Hermiticity, and chiral symmetry. Furthermore, a continuum toy model is developed and its edge band structure calculated to demonstrate the applicability of the exceptional edge mode mechanism to continuum systems. Finally, a continuum active matter model, based on chiral active matter with chirality flipping, is constructed to demonstrate the experimental feasibility of the proposed mechanism. The hydrodynamic equations are linearized to obtain an effective Hamiltonian, and numerical analysis of the edge band structure is performed to identify exceptional edge modes.

The research identifies a novel mechanism for robust gapless edge modes in non-Hermitian systems that are independent of the bulk topology. These exceptional edge modes, characterized by their association with exceptional points (EPs), are shown to exist even in topologically trivial bulks. The EPs act as a 'glue' connecting edge dispersions, ensuring robustness against disorder and perturbations. Numerical calculations using a two-layered non-Hermitian Bernevig-Hughes-Zhang model demonstrate the emergence of these modes in the bulk energy gap, even when the bulk topology is trivial. The strength of the non-Hermitian coupling is varied to demonstrate the independence of these modes from bulk topology, confirming the breakdown of the bulk-edge correspondence. Further analysis shows the robustness of these modes against different types of disorder. A condition for the robustness of gapless edge modes (Im(β² + γ²) ≤ |Im α|) is derived from an effective Hamiltonian analysis, linking the stability to the underlying symmetries (PT, CP, pseudo-Hermiticity, chiral). The study further constructs a model for a topological insulator laser utilizing these exceptional edge modes, demonstrating amplified wave packets propagating along the edge with nonzero group velocity. Real-space simulations illustrate the amplification and unidirectional propagation of edge modes even in the presence of edge disorder. Finally, a chiral active matter model confirms the presence of exceptional edge modes in a more realistic physical system, supporting the broad applicability of the findings.

The discovery of exceptional edge modes challenges the conventional understanding of topological phases in non-Hermitian systems. The bulk-edge correspondence, a cornerstone of Hermitian topological systems, is shown to be insufficient for predicting the existence of these robust edge modes. The findings highlight the importance of considering the role of EPs and their associated topological structures in understanding non-Hermitian systems. The proposed mechanism provides an alternative route for designing devices with scattering-free edge currents, going beyond the existing classifications of non-Hermitian topological phases. The application to topological insulator lasers provides a potential pathway for technological advancements. The active matter model strengthens the significance of these findings by showing the feasibility of experimental realization.

This research reveals a novel mechanism for robust gapless edge modes in non-Hermitian systems, independent of bulk topology, demonstrating a breakdown of the bulk-edge correspondence. These exceptional edge modes, tied to exceptional points, show robustness against disorder, are applicable to topological insulator lasers, and are shown to exist in both tight-binding and continuum models, including a model of chiral active matter. This offers a new design principle for scattering-free edge currents, transcending existing classifications of non-Hermitian topological phases. Future work could explore exceptional edge modes in higher dimensions and further refine the understanding of their interaction with different types of disorder and symmetries.

The study primarily focuses on two-dimensional systems. While the effective Hamiltonian analysis provides insights into the robustness of the exceptional edge modes, extending these findings to higher-dimensional systems requires further investigation. The active matter model, while demonstrating experimental feasibility, focuses on a specific type of chiral active matter. The generality of the findings for different active matter systems requires additional research. Furthermore, the experimental realization of the proposed topological insulator laser remains a challenge.