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

The study addresses how robust edge modes arise in non-Hermitian systems, where gain/loss and dissipation make effective Hamiltonians non-Hermitian. In Hermitian systems, robust edge modes are predicted by bulk topology via bulk-edge correspondence (e.g., quantum Hall and time-reversal-invariant topological insulators). However, in non-Hermitian systems, bulk-edge correspondence can fail. The authors hypothesize and demonstrate that robust gapless edge modes can emerge independently of bulk band topology, protected instead by exceptional points (EPs), where eigenvalues and eigenvectors coalesce and the Hamiltonian becomes defective. This mechanism indicates a fundamental breakdown of bulk-edge correspondence in non-Hermitian systems and suggests new routes to scattering-free edge transport and device applications such as topological insulator lasers.

The paper situates itself within work on non-Hermitian topological phases and classifications (including periodic tables for non-Hermitian symmetries) and reports that edge physics is subtler than Hermitian counterparts. Prior studies showed non-Hermitian effects (PT symmetry, pseudo-Hermiticity, skin effects) and attempts to generalize bulk invariants to predict edge states. Several works explored topological edge modes in active matter and chiral fluids, often protected by bulk topology or symmetries. The authors emphasize that existing bulk-based classification schemes (e.g., Z, Z2 invariants under non-Hermitian symmetries) may not predict the edge modes they find, which are instead stabilized by EP-induced branch point structures. They also note prior demonstrations of EP physics in optics, optomechanics, and electronics, and topological lasers where gain engineering along edges was required.

The authors develop and analyze multiple models and numerical studies: - Two-layer non-Hermitian QWZ/BHZ-like lattice model: Construct a minimal tight-binding model by coupling a Qi-Wu-Zhang (QWZ) Chern insulator (with two chiral edge modes per edge due to layer coupling) and its time-reversal counterpart via a non-Hermitian interlayer term iΣ with tunable real parameters β, β'. The full Hamiltonian H is a 4×4 block matrix with H0 and H1 (time-reversed) on the diagonal and iΣ off-diagonal. To break pseudo-Hermiticity and avoid alternative bulk classifications, they add a Hermitian coupling γ σ_x ⊗ σ_x, forming H' with time-reversal symmetry requiring Z2 bulk characterization. - Edge band calculations: Use ribbon geometry (open in x, periodic in y) to compute edge dispersions numerically for super-ribbons (1×50). They show gapless edge bands with EPs within the bulk energy gap, despite a trivial bulk gap (even number of edge modes per edge in constituent layers). - Parameter sweeps: Vary β, β' to induce emergence of EP pairs in edge dispersions without closing the bulk gap (demonstrating independence from bulk topology). Provide examples from Fig. 2: Hermitian case β=β'=0 has gapped edges; increasing non-Hermitian coupling brings edge bands together; at critical coupling, bands coalesce; beyond, EP pairs appear while the bulk remains gapped. - Effective 1D edge Hamiltonian: Propose a generic 2×2 effective edge Hamiltonian H(k_y) with parameters α, β, γ capturing two counter-propagating edge modes coupled non-Hermitically. Derive eigenvalues E(k_y)=E0 ± sqrt((k_y+α)^2 − β^2 − γ^2), EPs at k_y = −α ± sqrt(β^2+γ^2), and derive conditions for (i) gapless edge modes and (ii) exceptional edge modes. Show robustness criteria linked to PT/CP/pseudo-Hermitian/chiral symmetries and parameter constraints (e.g., Im(β^2+γ^2) ≤ |Im α| for gapless; Im α=0, Im(β^2+γ^2)=0, Re(β^2+γ^2)>0 for exceptional). - Disorder robustness: Introduce various disorder types (random real on-site potentials; imaginary and real noise in couplings; onsite imaginary potential ig σ_z I) and compute edge spectra. Connect robustness to a modified PT symmetry of the edge problem and to avoided crossings in the imaginary part preventing real-gap opening. - Lasing edge mode construction: Provide a general effective edge Hamiltonian for lasing modes with nonzero group velocity and nonzero imaginary parts localized to edges: H_edge = [[E0 + i α ∂_y, i β], [i β', E0 + i δ]] with α ≠ 1 and ββ' > 0. Derive dispersion E(k_y)=E0 + ((α−1)k_y ± sqrt((α+1)^2 k_y^2 − 4 β β'))/2 with EPs at finite k_y and finite group velocity. Build a concrete lattice model H = [[2 H_QWZ, i β σ_x], [i β' σ_x, H_QWZ]] and compute edge bands. - Real-space dynamics: Simulate finite 20×20 samples with open boundaries in both directions using RK4 time integration (dt=0.001). Demonstrate edge-selective amplification and propagation when non-Hermitian coupling is present, including robustness to edge defects. - Continuum model: Construct a 4×4 continuum Hamiltonian by coupling a C=2 model and its time-reversal via non-Hermitian couplings (parameters M, α, β, b, b'), discretize x with central differences (50 sites), periodic in y (k_y as Bloch momentum), and compute edge bands showing EP-protected edge modes with topologically trivial bulk. - Active matter model: Start from a Vicsek-type chiral active particle model with chirality flipping at rate γ, polar interactions, and momentum coupling between clockwise/counter-clockwise species. Derive hydrodynamic equations via Boltzmann-Ginzburg-Landau approach, linearize around the disordered state, obtain a block matrix effective Hamiltonian with non-Hermitian inter-species couplings A and dissipative terms C. Discretize as above to compute edge spectra in a cylinder, showing EPs and robust edge modes in the linearized hydrodynamics.

- Discovery of exceptional edge modes: Robust gapless edge modes arise from exceptional points (EPs) in non-Hermitian systems, independent of bulk topological invariants. EPs act as branch-point singularities that glue edge dispersions, preventing gap opening under perturbations and certain disorders. - Breakdown of bulk-edge correspondence: Edge EPs can appear and persist while the bulk gap remains open and topologically trivial (e.g., Z2 trivial), hence bulk band topology fails to predict their presence. Parameter sweeps (e.g., Fig. 2) show EP emergence without bulk-gap closing. - Effective theory conditions: For H(k_y)= [[E0 + k_y + α, iβ + γ], [iβ − γ, E0 − k_y − α]], gapless edge modes exist iff |Im sqrt(β^2+γ^2)| ≤ |Im α|; exceptional edge modes require Im α = 0, Im(β^2+γ^2)=0, and Re(β^2+γ^2) > 0. These conditions tie robustness to PT/CP/pseudo-Hermiticity/chiral symmetry constraints. - Disorder robustness: Exceptional edge modes persist with random real on-site potentials and imaginary noise in couplings due to a modified PT symmetry and avoidance in the imaginary energy preventing real-gap opening. Real noise in couplings can gap edges, but adding onsite imaginary potential restores robustness by lifting degeneracy in Im(E). - Lasing edge modes with nonzero group velocity: Constructed models exhibit edge-only positive imaginary parts and finite group velocity between EPs, enabling amplified unidirectional propagation without engineered edge gain. Real-space simulations show edge amplification from random initial states and robust propagation around defects. - Continuum and active matter realization: A continuum C=2-based model and a linearized hydrodynamic model of chiral active matter both exhibit EP-protected edge modes with trivial bulk. In the active matter case, EPs appear at ω=0 and edge modes avoid real crossings via imaginary part splitting, implying robustness; all imaginary parts are nonpositive (no lasing) in the presented parameters. - Symmetry insights: EP protection in 1D edge problems is supported by PT, CP, pseudo-Hermiticity, or chiral symmetries; increasing certain symmetry-breaking components (e.g., |Im γ| in the effective Hamiltonian) can annihilate EPs at a critical value.

The findings demonstrate that in non-Hermitian systems, edge robustness can stem from EP topology rather than bulk band topology, fundamentally challenging the conventional bulk-edge correspondence. By identifying EPs as the protective mechanism, the work explains how gapless edge states can persist with a trivial bulk invariant and clarifies which symmetries and parameter regimes ensure robustness under disorder. This EP-based edge protection provides a practical route to realizing scattering-free edge transport and lasing: edge modes can be selectively amplified and guided without fine-tuned gain distribution, enabling potential device implementations in photonics, acoustics, or electronic circuits. In active matter, the hydrodynamic model reveals that edge-propagating oscillations could naturally arise from non-Hermitian dynamics with chirality flipping and momentum coupling, highlighting active media as platforms for non-Hermitian topology. Overall, the results recast the role of open boundaries and suggest that edge-focused exceptional topology must complement bulk classifications in non-Hermitian settings.

The paper establishes a general mechanism for robust, gapless non-Hermitian edge modes protected by exceptional points, independent of bulk topological invariants. It develops effective theories and concrete lattice/continuum models demonstrating EP-induced edge robustness and shows applications to lasing edge transport with nonzero group velocity. It further extends the concept to active matter via linearized hydrodynamics of chiral particles with chirality flipping, showing EP-protected edge modes in realistic settings. Future work includes: experimental realization in photonics, mechanics, electronics, and active matter; extending to higher-dimensional systems and complex geometries; integrating EP-protected edges with non-Hermitian skin effects; engineering active systems with controlled gain to achieve lasing in hydrodynamic realizations; and developing refined classifications incorporating boundary EP topology beyond bulk Bloch Hamiltonians.

- Active matter model exhibits nonpositive imaginary parts of eigenfrequencies in the presented parameter regime, precluding lasing; additional gain mechanisms would be needed for active lasing. - Exceptional edge modes can disappear when symmetry-breaking non-Hermitian parameters (e.g., Im γ in the effective edge model) exceed critical values, indicating sensitivity to certain perturbations. - Robustness depends on disorder types; some Hermitian coupling disorder can gap edge modes unless compensated by suitable non-Hermitian onsite terms. - The study is largely theoretical and numerical; experimental validation remains to be demonstrated. - Effective models consider specific coupling forms and symmetries; generality across all non-Hermitian platforms may require case-by-case verification.