Non-Hermitian Topological Band Theory

• Non-Hermitian topological band theory is a framework that generalizes conventional band topology to systems with non-Hermitian Hamiltonians, characterized by complex eigenvalues and exceptional points.
• It introduces distinct gap notions such as line-gaps and point-gaps, along with novel invariants like winding numbers, braid group classifications, and biorthogonal Chern numbers.
• Applications span photonics, cold-atom systems, and metamaterials, underpinning phenomena such as the non-Hermitian skin effect and robust bulk–boundary correspondence.

Non-Hermitian topological band theory is a generalization of conventional band topology that encompasses complex eigenvalue spectra, exceptional point singularities, and non-orthogonal eigenstates arising in systems governed by non-Hermitian Hamiltonians. Unlike Hermitian systems, where band structures admit a real energy spectrum and unitary evolution, non-Hermitian physics naturally describes open, dissipative, gain/loss, or non-reciprocal systems, prominent in photonics, cold-atom settings, active metamaterials, and beyond. This framework necessitates new mathematical definitions of band gaps, novel topological invariants, and the inclusion of phenomena such as the non-Hermitian skin effect and exceptional degeneracies, leading to an enriched classification of topological phases and boundary states.

1. Gap Notions and Band Structure in the Complex Plane

Line-Gap vs. Point-Gap. In non-Hermitian systems, two non-equivalent notions of a spectral gap are central: a line-gap excludes a prescribed line (e.g., $\mathrm{Re}E=0$) from the spectrum at every $\mathbf{k}$, while a point-gap excludes a reference point $E_0\in\mathbb{C}$—i.e., $E_0\notin\mathrm{Spec}[H(\mathbf{k})]$ for all $\mathbf{k}$ (Shen et al., 2017, Wojcik et al., 2019, Nakamura et al., 21 Apr 2025). The point-gap notion is critical for capturing genuinely non-Hermitian phenomena (e.g., the skin effect), whereas conventional Hermitian invariants are associated with line-gap topology.

Separability and Band Gaps. A non-Hermitian band is separable if its eigenvalue locus does not cross those of any other band on the Brillouin zone (Shen et al., 2017). In the complex plane, isolated bands—i.e., those whose energy images are surrounded by gaps—generalize the concept of insulating bands. The most general setting involves separable bands with non-degenerate spectra across the Brillouin zone, or a prescribed separation between compact subsets of bands (separation gaps), which further refines possible invariants (Yang et al., 2023).

2. Topological Invariants: Winding, Chern, and Braid Group

1D Winding/Braid Group. In one dimension, the key invariant is the winding of the complex eigenvalues (or their differences) around a base point $E_0$. For $N$-band systems, the robust topological invariant is the conjugacy class of a braid in $B_N$ (the Artin braid group), reflecting how eigenvalues permute as $k$ winds over the Brillouin zone (Wojcik et al., 2019, Wojcik et al., 2021). For two bands, this reduces to a familiar integer winding number; for $N>2$, the invariants are intrinsically non-Abelian (Rui et al., 2022).

2D Chern Numbers and Fractional Winding. The non-Hermitian analogue of the Chern number employs the biorthogonal basis of left/right eigenvectors. The Chern number is constructed from the biorthogonal Berry curvature: $\mathbf{k}$0 which remains quantized as long as the relevant band remains isolated (Shen et al., 2017, Silveirinha, 2019). In sectors with band permutation (e.g., via nonseparable bands connected by PHLs), fractional winding numbers $\mathbf{k}$1 arise, reflecting $\mathbf{k}$2-cycle exchanges and associated with robust topological phase phenomena in the absence of exceptional points (Ryu et al., 2024).

Higher-Dimensional Fragile and Torsion Invariants. In 2D, the presence of non-trivial braid-group sectors can reduce the possible topological invariants to cyclic groups $\mathbf{k}$3 or $\mathbf{k}$4, making topological structure "fragile": the invariant can be trivialized by the addition of trivial bands (Li et al., 2019, Wojcik et al., 2019). In 3D, Hopf-type invariants protected by point gaps emerge for two-band systems, such as the non-Hermitian Hopf insulator with $\mathbf{k}$5– or $\mathbf{k}$6–valued topological charge depending on symmetry and dimensionality (Nakamura et al., 21 Apr 2025).

Green Function and Gauge-Free Approaches. In continuum models, the non-Hermitian Chern number can be cast as a frequency-momentum space integral involving the retarded Green's function,

$\mathbf{k}$7

where the analyticity of $\mathbf{k}$8 in a vertical strip of the complex-$\mathbf{k}$9 plane defines the generalized band gap (Silveirinha, 2019).

3. Exceptional Points, EP Braiding, and Pseudo-Hermitian Lines

Exceptional Points and Braiding. Non-Hermitian systems generally allow robust (codimension-2) exceptional point (EP) degeneracies, where eigenvalues and eigenvectors coalesce. Their presence is encoded by braid-group data in the momentum space: for example, in 2D, the fundamental group of the Brillouin zone punctured by $E_0\in\mathbb{C}$0 EPs leads to constraint equations relating the braids around cycles and the EPs themselves (Wojcik et al., 2021). Braiding of bands at EPs underlies nontrivial phase transitions and can generate non-Hermitian topological phase transitions not present in Hermitian settings (Rui et al., 2022).

Pseudo-Hermitian Lines (PHLs). Topological features can persist without EPs, mediated by PHLs along which the Hamiltonian is pseudo-Hermitian (i.e., $E_0\in\mathbb{C}$1 for a fixed invertible $E_0\in\mathbb{C}$2) (Ryu et al., 2024). In 2D, a PHL can be non-contractible on the torus, allowing nontrivial topology even in the absence of band-touching points. The permutation group $E_0\in\mathbb{C}$3 organizes the possible exchange classes of bands, and fractional winding numbers result from the action of PHLs.

Symmetry Engineering and Indicator Formulas. Non-Hermitian symmetries generalize the Hermitian Altland-Zirnbauer classes to a rich Bernard-LeClair symmetry taxonomy (Zhou et al., 2018). Generalized inversion and chiral symmetries lead to topological indicators expressible in terms of simple functions at inversion-invariant momenta, facilitating the diagnosis of skin effects and EP-related topological transitions (Okugawa et al., 2021).

4. Bulk–Boundary Correspondence and the Non-Hermitian Skin Effect

Non-Hermitian Skin Effect (NHSE). In contrast to Hermitian systems, the bulk spectrum under periodic (PBC) and open (OBC) boundary conditions can differ dramatically: a nonzero point-gap winding predicts the macroscopic accumulation of bulk states at the system boundary under OBC—the skin effect. The OBC spectrum collapses onto a region of the complex plane corresponding to zero point-gap winding (Longhi, 2021, Sun et al., 13 Jul 2025). In models with translational symmetry, this is encoded by non-Bloch band theory, where boundaries are incorporated by shifting momenta into the complex plane; in disordered or aperiodic systems, the Lyapunov exponent formulation generalizes this physics (Sun et al., 13 Jul 2025).

Bulk–Edge Correspondence. The number of protected edge states is controlled by the change in the appropriate topological invariant (e.g., winding number or Chern number) across a boundary or domain wall. For lossy systems described by Maxwell's equations, the non-Hermitian Chern number remains quantized and edge modes traverse the complex-frequency band gaps as in Hermitian systems, provided the analyticity strip is preserved (Silveirinha, 2019).

Flat Bands and Higher-Order Skin Modes. Real-space decimation methods elucidate the formation of flat bands and compact localized states, as well as higher-order skin effects, where not only edges but also corners accumulate exponentially localized states, signaled by corresponding topological invariants in the real-space framework (Banerjee et al., 2023).

5. Interplay of Symmetry, Fragility, and Topological Classification

Symmetry-Protected Phases and Fragility. Non-Hermitian Bernard-LeClair symmetries and point-gap/line-gap distinctions result in a "38-fold periodic table" for symmetry-protected topological phases (Zhou et al., 2018). However, certain invariants are only stable as long as the band number and symmetry structure are fixed; adding trivial bands can trivialize non-Hermitian “torsion” (fragile) invariants (Li et al., 2019). For two-band systems, non-Hermitian Hopf-type invariants (in 3D and 4D) exist, which do not extend to higher bands (Nakamura et al., 21 Apr 2025).

Homotopy and Braid-Frame Classifications. The homotopy-theoretic approach reveals a two-level hierarchy: eigenvalue braiding (non-Abelian and Abelian braid-group structure) classifies sectors, while eigenvector (frame) topology further distinguishes subclasses in each sector (Li et al., 2019, Wojcik et al., 2021, Yang et al., 2023). This framework naturally unifies and extends K-theoretic line-gap schemes and captures both stable and fragile topological phenomena unique to non-Hermitian band physics.

Pseudo-Hermitian Topologies. Pseudo-Hermitian operators support phases where the spectrum remains real for arbitrarily large non-Hermiticity, circumventing the limitations of conventional PT-symmetric models and giving rise to robust topological phases with well-defined bulk-boundary correspondence (Long et al., 2021, Zhu et al., 2021).

6. Physical Realizations, Experimental Platforms, and Applications

Optics, Photonics, Atomic, and Electronic Metamaterials. Non-Hermitian band topology is realized in systems with balanced gain/loss, radiative decay, nonreciprocal couplings, and engineered dissipation, such as photonic crystals, cold-atom arrays, acoustic devices, and topological lasers (Silveirinha, 2019, Xie et al., 1 Jan 2026).

Atomic Lattices and Dirac-Type Models. Long-range radiative coupling in bipartite atomic lattices leads to non-Hermitian Dirac equations with complex Fermi velocities and biorthogonal bulk and edge state topology (Xie et al., 1 Jan 2026). Analytic edge-state solutions verify bulk-edge correspondence under non-Hermitian deformations.

Twisted Graphene Systems. The combination of non-reciprocal hopping and sublattice-staggered mass in twisted bilayer graphene aligned with hBN produces non-Hermitian valley Hall phases, gapless nodal-line physics across extended parameter regions, and robustness of Dirac points against symmetry-breaking perturbations (Bera et al., 1 May 2025).

Measurement Protocols. Experimental access to non-Hermitian geometric tensors, Berry curvature, and quantum metric is feasible via dilated Hermitian embeddings and modulation spectroscopy in platforms such as superconducting qubits, NV centers, and trapped ions (Zhu et al., 2021).

7. Outlook and Open Directions

Non-Hermitian topological band theory has established new paradigms for classifying and understanding topological matter beyond Hermitian constraints. Open directions include the systematic classification of intrinsic/extrinsic point-gap invariants for higher-band and higher-dimensional systems, the full characterization of exceptional degeneracy braiding in real multi-band structures, robust detection protocols for fragile invariants, and the integration of interaction and Floquet effects. The interplay of symmetry, disorder, non-Hermitian topology, and dynamical phenomena continues to drive rapid developments at the intersection of condensed matter physics, optics, quantum engineering, and mathematical physics (Wojcik et al., 2019, Yang et al., 2023, Nakamura et al., 21 Apr 2025).