Non-Hermitian Topology

• Non-Hermitian topology is the study of topological properties in systems governed by non-Hermitian operators with complex spectra and biorthogonal eigenstates.
• It reveals phenomena such as exceptional points, non-Hermitian skin effects, and novel gap classifications that challenge conventional band theory.
• Its applications span photonic, acoustic, and electronic systems, providing insights into robust mode localization and phase transitions.

Non-Hermitian topology is the branch of topological band theory that studies the robust, symmetry-constrained properties of physical systems described by non-Hermitian Hamiltonians ($H \neq H^\dagger$), which naturally arise in open, dissipative, or amplifying environments. Unlike Hermitian systems, non-Hermitian models possess complex energy spectra, exhibit distinct right and left eigenvectors, and support phenomena such as exceptional points, non-Hermitian skin effects, and novel topological invariants. The mathematical and physical structures underlying non-Hermitian topology substantially generalize conventional topological band theory, necessitating new classifications, gap types, and bulk–boundary correspondences.

1. Mathematical Framework of Non-Hermitian Systems

A general non-Hermitian single-particle Hamiltonian $H$ satisfies $H \neq H^\dagger$. The spectrum $\{\lambda_i\}$ is generically complex-valued, and there exist two biorthogonal eigenbases:

• Right eigenvectors: $H|\psi_i^R\rangle = \lambda_i |\psi_i^R\rangle$
• Left eigenvectors: $\langle\psi_i^L|H = \lambda_i \langle\psi_i^L|$

Biorthogonal normalization imposes $\langle\psi_i^L|\psi_j^R\rangle = \delta_{ij}$ and completeness $$\sum_i |\psi_i^R\rangle\langle\psi_i^L| = \mathbb{I}$$ (Ding et al., 2022). The physical relevance of non-Hermitian operators covers a wide array of situations: open quantum systems coupled to baths, systems with gain/loss, effective Hamiltonians for quasiparticles with finite lifetimes, and engineered nonreciprocal couplings in photonic, acoustic, or electrical networks (Alvarez et al., 2018).

2. Topological Invariants, Complex Gaps, and Band Geometry

Point-Gap and Line-Gap Topologies

The non-Hermitian spectrum does not admit a canonical ordering; thus, two gap concepts arise:

• Point gap: the complex energy manifold does not contain a reference energy $E_\text{ref}\in\mathbb{C}$, i.e., $H(k) - E_\text{ref}$ invertible for all $k$. This structure underpins winding-number invariants.
• Line gap: the spectrum avoids a line (e.g., $\mathrm{Re}\,E=0$ or $\mathrm{Im}\,E=0$) in $\mathbb{C}$, allowing an analogue of Chern numbers and polarization (Kawabata et al., 2018).

Band Topology and Invariants

For a non-Hermitian Bloch Hamiltonian $H(k)$:

• Biorthogonal Berry connection and phase:

$$\mathcal{A}_i(k) = i\langle\psi^L_i(k)|\nabla_k|\psi^R_i(k)\rangle, \qquad \Phi_i = \oint_C \mathcal{A}_i(k)\,dk$$

The resulting Berry phase $\Phi_i$ can be complex, with its real part governing geometric phases and the imaginary part capturing net amplification or attenuation (Ding et al., 2022).

• Point-gap winding number (in 1D):

$$W(E_\text{ref}) = \frac{1}{2\pi i} \int_{BZ} dk\,\partial_k \ln \det[H(k) - E_\text{ref}]$$

This counts how many times $\det[H(k)-E_\text{ref}]$ encircles the origin as $k$ traverses the Brillouin zone. $W\neq0$ characterizes nontrivial point-gap topology (Ding et al., 2022).

• Non-Hermitian Chern number (isolated band $i$, line gap):

$$F_i(k) = \partial_{k_x}\mathcal{A}_{i,y}(k) - \partial_{k_y}\mathcal{A}_{i,x}(k)$$

$C_i = \frac{1}{2\pi} \iint_{BZ} d^2k\, F_i(k)$

Quantization is maintained under a line gap and appropriate symmetry (e.g., class A) (Ding et al., 2022).

Higher-Order and Multiband Structures

Non-Hermitian systems with multiple bands possess complex band braiding and Möbius- or Penrose-like composite loop topologies, characterized by winding, vorticity, and generalized (nonadiabatic) Zak phases. These composite topologies are quantized when associated composite loops in the complex-energy space encircle odd numbers of exceptional points, yielding new insulating and metallic phases with no Hermitian analogue (Nehra et al., 2022).

3. Exceptional Points and Their Topological Properties

Exceptional points (EPs) are singularities in parameter or momentum space where two or more eigenvalues and corresponding eigenvectors coalesce, rendering $H(k_{\rm{EP}})$ defective (Bergholtz et al., 2019). For a simple order-2 EP at $(k_\text{EP},\lambda_\text{EP})$, the eigenvalue manifold locally obeys a Puiseux expansion: $$\lambda(k) = \lambda_\text{EP} + a_1 (k-k_\text{EP})^{1/2} + \cdots$$ Encircling an EP in parameter space permutes the Riemann sheets of the complex spectrum.

• Topological charge (vorticity) of EPs: Encircling an order-$p$ EP, the eigenvalues and eigenvectors permute such that the phase of $\arg\det[H(k)-\lambda_\text{EP}]$ acquires a winding $W=m/p$ for $m\in\mathbb{Z}$. For a simple EP, $v = \pm 1/2$ (Ding et al., 2022, Bergholtz et al., 2019).
• EPs are generically stable in two dimensions (codimension-2) and define the boundaries or nodes of Fermi arc or Seifert surfaces, which are open, non-Hermitian analogues of Fermi surfaces, terminating at EPs or exceptional lines (Bergholtz et al., 2019).

4. Non-Hermitian Skin Effect, Non-Bloch Band Theory, and Bulk–Boundary Correspondence

Skin Effect and Breakdown of Bulk–Boundary Correspondence

The non-Hermitian skin effect (NHSE) is a bulk accumulation of extensive eigenstates at system boundaries under open boundary conditions, driven by nonzero point-gap winding in models with nonreciprocal hopping (Ding et al., 2022, Alvarez et al., 2018). Under the NHSE, conventional Bloch band theory fails, and periodic and open boundary spectra are no longer related by analytic continuation (Kunst et al., 2018).

Generalized Brillouin Zone and Non-Bloch Topology

Restoration of bulk–boundary correspondence requires replacing $e^{ik}\to\beta\in\mathbb{C}$ satisfying $|\beta|=r$ (the generalized Brillouin zone, GBZ), where all roots of $\det[H(\beta)-E]=0$ accumulate on $|\beta|=r$ (Alvarez et al., 2018):

$$W_\text{non-Bloch} = \frac{1}{2\pi i} \int_{GBZ} d\beta\, \partial_\beta \ln\det[H(\beta)-E_\text{ref}]$$

This invariant predicts the number and localization of skin modes in the open system exactly (Kunst et al., 2018). In higher dimensions, similar constructions with multidimensional complex tori partially recover edge or corner mode counts (Ding et al., 2022).

Transfer Matrix and Riemann Surface Approach

Real-space transfer matrix techniques reveal that the condition for the skin effect is $\det T(\epsilon)\neq1$ for the transfer matrix $T(\epsilon)$. Vanishing determinant signals real-space exceptional points whose order scales with system size, unifying skin, boundary, and EP physics (Kunst et al., 2018). The OBC bulk bands trace out cuts on a two-sheeted Riemann surface over $\epsilon$, enabling a new invariant: the winding of edge-state loops around branch points matches the count of protected boundary modes.

5. Symmetries, Classification, and Topological Phase Diagrams

Symmetry Extension and AZ† Classes

Non-Hermitian generalizations of symmetries (TRS, PHS, chiral symmetry) lead to a refinement of the Altland–Zirnbauer classification to 38 symmetry classes. This “ramification” occurs due to distinctions among transposition, complex conjugation, and Hermitian conjugation on $H$ (Kawabata et al., 2018, Bergholtz et al., 2019). The presence or absence of real and/or point gaps, and specific symmetry constraints, determines the topological structure:

Gap Type	Invariant	Symmetry Constraints
Point gap	Integer winding; $W \in\mathbb{Z}$	None (class A), or others
Line gap	Biorthogonal Chern; $C\in\mathbb{Z}$	Chiral, TR, PHS, AZ†

• For a line gap, the topological classification reproduces the Hermitian periodic table.
• For a point gap, non-Hermitian-specific invariants classify spectral loops and composite band structures (Kawabata et al., 2018).

Model Examples and Phase Diagrams

Explicit models include:

• Nonreciprocal SSH chain: Exhibits NHSE and phase boundaries set by $|t_1\pm \gamma/2|=t_2$ (Ding et al., 2022).
• Non-Hermitian Chern insulators: Complexified mass or hopping leads to nonzero Chern number and chiral edge amplification or attenuation (Ding et al., 2022, Banerjee et al., 2022).
• PT-symmetric models: Retain real spectra up to PT-breaking EPs (Ding et al., 2022).

Phase diagrams are determined by gain/loss and coupling, with point/line-gap closures marking topological transitions.

6. Experimental Realizations, Physical Manifestations, and Applications

Non-Hermitian topology has been experimentally accessed in a range of platforms:

• Photonic systems: Engineered gain/loss, non-reciprocal ring resonators, time-multiplexed synthetic lattices, and photonic crystals showing skin modes, exceptional points, Fermi arcs, and enhanced topological sensing (Parto et al., 2023, Xu et al., 23 Dec 2025).
• Cold atomic and quantum gas systems: Measurement of spectral winding in dissipative Rydberg gases as a function of interaction and dissipation (Zhang et al., 30 Sep 2025).
• Electronic and topolectrical circuits: Skin modes detected as impedance peaks; programmable boundary conditions probe defects and exceptional-point physics.
• Classical mechanical and acoustic metamaterials: Realization of skin effect and edge/corner-localized modes via nonreciprocal couplings (Banerjee et al., 2022, Bergholtz et al., 2019).

Physical consequences include exponential sensitivity in non-Hermitian topological sensors, room-temperature realization of skin effect, and amplified response at exceptional points. These properties have implications for nonreciprocal transmission, robust lasing, and topological quantum information transport (Parto et al., 2023, Xu et al., 23 Dec 2025).

7. Outlook: Extensions, Open Problems, and New Directions

Current and emerging research challenges include:

• Generalizing to interacting many-body systems, nonlinear dynamics, and Liouvillian frameworks (Banerjee et al., 2022).
• Complete GBZ theory beyond 1D: Non-Bloch band theory in higher dimensions remains an area of active research (Ding et al., 2022).
• New topological phases: Homotopy theory reveals braid, frame, and separation-gap topologies, as well as fragile and non-Abelian invariants in multiband and $\mathcal{PT}$-symmetric systems (Yang et al., 2023).
• Higher-order topology: Non-Hermitian analogues of second- and higher-order topological insulators and superconductors, with corner and hinge skin modes multiplexed by symmetry (Ji et al., 2023, Li et al., 2021).
• Exotic geometry: Exceptional topology on nonorientable parameter spaces (e.g., Klein bottle), with nontrivial constraints on EP monopoles and Fermi arc orientation (Xu et al., 23 Dec 2025).
• Fundamental links to Hermitian topology: Via effective non-Hermitian boundary Hamiltonians and self-energy approaches, connecting $d$-dimensional NH topology to $(d+1)$-dimensional Hermitian invariants (Hamanaka et al., 2024, Wanjura et al., 23 Sep 2025).

Experimental advances continuously broaden non-Hermitian topology’s scope, with implications for quantum technologies, robust device engineering, and the mathematical foundation of topological phases. The interplay between symmetry, complex band structures, and boundary phenomena remains central to the ongoing expansion of this field.