Non-Hermitian Topological Phases

Updated 13 July 2025

Non-Hermitian topological phases are quantum or classical states defined by non-conjugate Hamiltonians, featuring complex spectra and exceptional points.
They are analyzed using generalized gap conditions and topological invariants that predict phenomena like the skin effect and looped edge modes.
Experimental platforms such as photonic lattices and ultracold atoms validate these phases, paving the way for robust directional transport devices.

Non-Hermitian topological phases are quantum or classical states of matter in open, driven, or dissipative systems whose governing Hamiltonians are not equal to their Hermitian conjugate. These phases display phenomena, such as nontrivial spectral winding, exceptional points, anomalous edge and boundary modes, and the non-Hermitian skin effect, which have no direct Hermitian analog. The paper of non-Hermitian topology has yielded systematic classification frameworks, new types of topological invariants, robust transport signatures, and broadly applicable experimental proposals, forming a distinct and rapidly developing branch of topological band theory.

1. Theoretical Foundations and General Framework

Central to non-Hermitian topology is the lack of conventional Hermiticity: Hamiltonians $H$ satisfy $H \neq H^\dagger$ . This leads to spectra that are generally complex-valued and eigenstates that may lack orthogonality or even coalesce at exceptional points (EPs)—singularities where both eigenvalues and eigenvectors merge.

General classification principles distinguish non-Hermitian topology from its Hermitian counterpart in two fundamental ways:

Dynamical, Not Ground State, Classification: Non-Hermitian topology is inherently "dynamical." Unlike Hermitian phases defined via ground state wavefunctions, the topology is encoded in the spectrum and eigenstates of $H$ (or the full nonunitary time-evolution operator), capturing both amplified and decaying modes (1802.07964).
Generalized Gap Condition: The notion of a "band gap" is replaced by a more general invertibility condition. For a reference ("base") energy $E_B$ , a point-gap requires that $\det[H(\mathbf{k}) - E_B] \neq 0$ for all momenta $\mathbf{k}$ ; a line-gap is defined with respect to an entire line in the complex plane (e.g., the real or imaginary axis) (1812.09133). This gap condition supports the definition of topological equivalence: continuous deformations of $H$ are allowed only if the gap remains open everywhere.

A powerful result is that any invertible non-Hermitian $H(\mathbf{k})$ can be "unitarized" via polar decomposition, $H(\mathbf{k}) = U(\mathbf{k}) P(\mathbf{k})$ (with $U$ unitary and $P$ positive-definite Hermitian), enabling mapping into known K-theory and Clifford algebra classification schemes (1802.07964).

2. Topological Invariants, Symmetry Classes, and Periodic Tables

The non-Hermitian extension of the Altland–Zirnbauer (AZ) symmetry classes leads to an expanded periodic table of topological phases (1812.09133). Notably:

Symmetry Generalization: Unlike Hermitian systems, charge conjugation and chiral symmetry must be defined via transposition, not complex conjugation, due to the lack of Hermiticity; additional "AZ $^\dagger$ " classes and internal symmetries such as pseudo-Hermiticity and sublattice symmetry also emerge, enlarging the periodic table to 38 symmetry classes (1812.09133).
Topological Invariants:
- Winding Number: In 1D, the spectral winding number is defined as
$w = \int_{-\pi}^{\pi} \frac{dk}{2\pi i} \partial_k \log \det[H(k)]$

This invariant counts how many times the spectrum winds around the base energy in the complex plane, generalizing the concept of the Chern number or Berry phase (1802.07964). - $\mathbb{Z}_2$ Index: In zero- or one-dimensional systems with (antiunitary) symmetry, a $\mathbb{Z}_2$ index is given by the sign of $\det H$ ; such indices can characterize models as diverse as quantum channels and dissipative discrete time crystals (1802.07964). - Biorthogonal and Real-Space Invariants: Under open boundary conditions, winding numbers must be redefined on the generalized Brillouin zone (GBZ). For systems with disorder, real-space invariants—such as $\nu = \frac{1}{2\pi i} \mathrm{Tr}\{ C Q[Q, X]\}$ , where $C$ is a chiral operator and $Q$ is a projector—offer robust characterization (1908.01172).
Dimensional and Gap-Type Dependence: The presence or absence of non-Hermitian topology depends not just on symmetry class but also on gap type (point or line). K-theory formulas for complex and real classifying spaces govern the periodic table, with Bott periodicity and gap-type distinctions leading to a rich pattern of possible phases across spatial dimensions (1802.07964, 1812.09133).
Crystalline and Nonsymmorphic Symmetries: When crystalline symmetries (especially nonsymmorphic ones) are included, entirely new topological classes can emerge—most notably, $\mathbb{Z}_2$ and $\mathbb{Z}_4$ phases with anomalous, complex-valued boundary spectra, unachievable in either purely Hermitian or non-Hermitian systems with only internal symmetry (2504.20743).

3. Paradigmatic Models and Unique Non-Hermitian Phenomena

The non-Hermitian Hatano–Nelson model, non-Hermitian Su–Schrieffer–Heeger (SSH) chain, non-Hermitian Chern insulator, and their higher-dimensional or synthetic variants serve as canonical examples and demonstrate key effects:

Model	Key Non-Hermitian Features	Notable Phenomena
Hatano–Nelson	Asymmetric hopping; no Hermitian analog	Spectral winding, skin effect, delocalization
Non-Hermitian SSH	Chiral and sublattice symmetry, asymmetric terms	Fractional winding, edge mode asymmetry
Non-Hermitian Chern-like	Complex potential or hopping in 2D	Real-energy edge modes, new invariants
Nonsymmorphic extensions	Fractional translation + NH physics	$\mathbb{Z}_2$ , $\mathbb{Z}_4$ invariants, looped edge spectra

Exceptional Points (EPs): Non-Hermitian systems universally harbor parameter regimes where eigenvalues and eigenvectors simultaneously coalesce, yielding non-diagonalizable Hamiltonians and spectrally singular behavior (2212.06478).
Non-Hermitian Skin Effect (NHSE): Under open boundaries, the spectral winding induces macroscopic accumulation of eigenstates at one edge—a phenomenon absent in Hermitian systems (1812.02186, 2212.06478). The “skin effect” can only be properly captured by solving for eigenstates on the GBZ, not the conventional Brillouin zone.
Anomalous (Looped) Boundary States: In certain symmetry classes (especially with nonsymmorphic symmetry), edge or surface states can trace out noncontractible loops in the complex energy plane, and can be protected by $\mathbb{Z}_2$ or $\mathbb{Z}_4$ invariants unique to the interplay of non-Hermiticity and crystalline symmetry (2504.20743).

4. Bulk–Boundary Correspondence, Disorder, and Criticality

Breakdown and Restoration of Bulk–Boundary Correspondence: In generic non-Hermitian systems, the conventional correspondence between bulk topological number and protected edge states fails due to the NHSE. However, using the GBZ and redefining winding numbers along nonunit modulus contours restores a predictive correspondence (1812.02186, 2212.06478).
Disorder and Non-Hermitian Topological Anderson Insulators: Disorder can not only destroy but also induce non-Hermitian topological phases. For example, non-reciprocity combined with disorder may yield a non-Hermitian topological Anderson insulator with a robust, disorder-averaged winding number and protected zero modes in an otherwise insulating bulk (1908.01172).
Criticality and Topological Phase Transitions: Non-Hermitian topological phase transitions can exhibit both critical and non-critical behavior. In certain cases, edge state correlation lengths diverge at the transition (with exponents $\nu = 1$ corresponding to the Hermitian universality class), while in others, transitions between fractional and integer winding numbers (such as $1/2 \to 1$ ) may be non-critical, with edge localization unaffected by the transition (2208.14400).

5. Many-Body, Floquet, and Machine Learning Approaches

Many-Body Non-Hermitian Topology: The notion of intrinsic non-Hermitian topology extends beyond free-fermion models. Robust spectral winding survives strong interactions, with many-body invariants defined as the winding of the determinant of the full Hamiltonian under flux threading. For steady-state properties, distinct invariants characterize both the band structure and the quantum state, leading to decoupled phase transitions (2202.02548, 2311.03043).
Periodically Driven (Floquet) Non-Hermitian Phases: In driven systems where chiral symmetry and translation invariance are broken, a systematic method using similarity transformations can restore effective chiral symmetry, enabling the definition of predictive winding numbers. This enables the synthesis and control of edge modes at both $0$ and $\pi/T$ quasienergy, as well as the creation of Floquet topological Anderson insulators under disorder (2003.08055).
Machine Learning and Unsupervised Discovery: Supervised and unsupervised neural network methods have proven capable of accurately classifying non-Hermitian topological phases, both in clean and disordered systems, using winding number data or even raw projective matrices. Advanced unsupervised algorithms now can classify phases without relying on explicit invariants, instead learning topological distinctions from path-connectedness in parameter space under symmetry constraints and automatically constructing a periodic table under open-boundary conditions (2009.06476, 2010.14516, 2412.20882).

6. Field-Theoretical and Transport Perspectives

A “spatial” topological field theory for non-Hermitian systems leads to distinctive physical predictions:

In one dimension, the winding number manifests as unidirectional transport (an "anomaly" in gauge invariance) and underlies the skin effect.
In three dimensions, a chiral magnetic skin effect appears, with the particle flow along the direction of a real-space synthetic magnetic field.
In two dimensions, spatial textures can induce anomalous boundary-localized currents and density pileups, exemplifying a new class of boundary anomalies tied to non-Hermitian topology (2011.11449).

Transport phenomena, such as the Hall conductance in non-Hermitian Chern insulators, depart from conventional quantization, with dissipation-induced corrections and possible nonuniversal response (2212.06478).

7. Applications, Experimental Platforms, and Outlook

Experimental advancements allow for the realization and measurement of non-Hermitian topological phenomena:

Platforms: Photonic lattices (with controlled gain/loss), ultracold atoms (with engineered dissipation), electronic circuits, mechanical and acoustic metamaterials, and frequency-modulated lasers have each demonstrated instances of non-Hermitian skin effect, exceptional points, and robust boundary modes (1901.08060, 1905.09460, 2212.06478).
Technological Implications: Non-Hermitian topological insulators support robust, controllable edge modes even in non-equilibrium and noisy environments, promising for topological lasers, directional transport devices, and optical or acoustic sensors.
Frontiers: Future challenges include full understanding of many-body effects, the development of machine learning approaches to discover hidden non-Hermitian topology, the classification under intricate crystalline symmetries, and the engineering of periodically driven and higher-order topological phases unique to non-Hermitian systems.

In summary, non-Hermitian topological phases constitute a broad and cohesive field with novel theoretical structures, a comprehensive and symmetry-rich classification, anomalous boundary physics, and concrete potential for experimental realization and device design. They serve as a robust testbed for exploring how fundamental symmetries, topology, and non-Hermitian dynamics intersect in open quantum and classical systems.