Non-Hermitian Topological Anderson Insulator

• Non-Hermitian Topological Anderson insulators are disorder-induced phases in non-Hermitian lattices where disorder enables nontrivial spectral winding and localized bulk states.
• They are characterized by biorthogonal topological invariants and generalized bulk-boundary correspondence that accommodate complex spectra and skin effects.
• Experimental realizations in photonic lattices and topolectrical circuits validate their potential for robust, disorder-resilient topological transport.

A non-Hermitian topological Anderson insulator is a disorder- or quasidisorder-induced topological phase of a non-Hermitian lattice system in which insulating or localized bulk states coexist with a nontrivial topological structure defined in a biorthogonal or point-gap sense. In the published literature, the term covers several closely related mechanisms: disorder-enabled topology in nonreciprocal chiral chains with zero-energy edge modes under open boundary conditions, quasiperiodicity-driven topological localization transitions in reciprocal but complex potentials, Floquet disorder-induced phases with quasienergy $0$ or $\pi/T$ boundary states, and higher-dimensional or higher-order non-Hermitian Anderson phases diagnosed by real-space invariants. Across these settings, the defining feature is not merely localization, but the emergence of topological structure specifically because disorder, quasiperiodicity, or non-Hermiticity changes the phase structure of the system (Zhang et al., 2019, Longhi, 2019).

1. Conceptual scope and relation to Hermitian Anderson topology

In Hermitian systems, a topological Anderson insulator usually denotes a phase in which disorder drives a trivial band insulator into a topological one, commonly through disorder-renormalized band parameters, mobility gaps, and conventional bulk-edge correspondence. In non-Hermitian systems, the same phrase acquires additional structure because spectra are generally complex, the distinction between line gaps and point gaps becomes essential, left and right eigenvectors must be treated biorthogonally, and the non-Hermitian skin effect can invalidate Bloch-band invariants defined under periodic boundary conditions. In that sense, a non-Hermitian topological Anderson insulator is not a single universal mechanism but a family of disorder-enabled topological phases in which localization and topology are coupled in ways unavailable to Hermitian bands (Zhang et al., 2019).

A central distinction is that the topological marker need not be a Chern number or a $\mathbb{Z}_2$ invariant. Depending on the model, it can instead be a spectral winding in the complex-energy plane, an open-bulk real-space winding number, an open-bulk Chern number, a generalized Bott index, or a biorthogonal quadrupole moment. This also means that a common misconception is too narrow: non-Hermitian topological Anderson physics is not restricted to nonreciprocal skin-effect systems. Reciprocal, $\mathcal{PT}$-symmetric quasiperiodic chains with no skin effect can also realize a non-Hermitian topological Anderson insulator through disorder-like incommensurate modulation and complex spectral winding (Longhi, 2019).

Another important refinement comes from quasiperiodic chains, where topology and localization need not coincide. In quasiperiodic Su–Schrieffer–Heeger chains, disorder-induced topological transitions can be independent of localization transitions, yielding gapped topological Anderson insulators with extended, intermediate, or localized bulk states; non-Hermitian perturbations preserve these phases while introducing real-complex spectral transitions and point-gap topology. This suggests that “Anderson” in current usage includes both random disorder and deterministic quasidisorder, provided the modulation produces the disorder-enabled topological phase structure associated with localization physics (Tang et al., 2022).

2. Canonical one-dimensional realizations

The paradigmatic disordered realization is the nonreciprocal Su–Schrieffer–Heeger chain with chiral symmetry,

$$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$

with asymmetric intercell couplings and disordered hoppings. In the regime emphasized in the original proposal, the topological phase emerges only when both moderate non-Hermiticity and disorder are present. The resulting phase is characterized by localized insulating bulk states, a disorder-averaged winding number close to unity, and two zero-energy edge modes under open boundary conditions. In the same model, non-Hermiticity can also enhance the robustness of an already topological phase by increasing the open-boundary bulk gap through a similarity transformation to a Hermitian SSH chain with strengthened intercell hopping (Zhang et al., 2019).

A distinct but equally important canonical realization is the $\mathcal{PT}$-symmetric non-Hermitian Aubry–André–Harper chain with reciprocal hopping and complex incommensurate onsite modulation,

$$\hat{H}\psi_n=J(\psi_{n+1}+\psi_{n-1})+V_n\psi_n,\qquad V_n=V_0 e^{-2\pi i\alpha n}.$$

Here the incommensurate modulation plays the role of deterministic disorder. The model undergoes an exact transition at $V_0=J$: for $V_0<J$, all states are extended and the spectrum is real; for $V_0>J$, all states are exponentially localized, the spectrum becomes complex, and the complex-energy loop carries nontrivial spectral winding. The localized phase has

$\pi/T$0

independent of energy, and because hopping is reciprocal there is no non-Hermitian skin effect, so the bulk spectrum and localization properties are boundary-condition independent. This model realizes a quasiperiodic non-Hermitian topological Anderson insulator in a setting where topology is encoded by complex-energy winding rather than edge-state counting alone (Longhi, 2019).

A broader unifying perspective is provided by disordered Hatano–Nelson chains with nonreciprocal hopping,

$\pi/T$1

In that setting, disorder can itself generate point-gap topology: a nontrivial spectral winding can arise when the logarithmic averages of right and left hoppings differ, and the sign of the Lyapunov exponent determines whether open-boundary states are skin-localized or Anderson-localized. This establishes a disorder-enabled non-Hermitian topological phase closely tied to transport and amplification (Fortin et al., 6 Sep 2025).

3. Topological characterization and generalized bulk–boundary correspondence

Because non-Hermitian spectra depend sensitively on boundary conditions, topology is often formulated directly under open boundary conditions. In the disordered SSH problem, the central invariant is a real-space open-bulk winding number. For a given disorder realization $\pi/T$2, one constructs a flattened open-boundary projector

$\pi/T$3

and defines

$\pi/T$4

This invariant is quantized in the thermodynamic limit, does not require translation symmetry, and correctly predicts zero-energy edge modes under open boundary conditions in the presence of the skin effect (Zhang et al., 2019).

In reciprocal quasiperiodic systems without skin effect, a point-gap invariant can be defined more directly through the spectral loop. In the localized phase of the $\pi/T$5-symmetric non-Hermitian Aubry–André–Harper model, the spectrum lies on an ellipse parameterized by

$\pi/T$6

and the spectral winding around a reference point $\pi/T$7 is

$\pi/T$8

For $\pi/T$9, one has $\mathbb{Z}_2$0 for points inside the ellipse, whereas for $\mathbb{Z}_2$1 the metallic spectrum is a real interval and $\mathbb{Z}_2$2. The topological content of the transition is therefore the jump in complex-energy winding at the localization threshold (Longhi, 2019).

More generally, disordered non-Hermitian chains admit a precise bulk–boundary correspondence in terms of the Lyapunov exponent. In the disordered Hatano–Nelson framework, the spectral winding around $\mathbb{Z}_2$3 obeys

$\mathbb{Z}_2$4

and the mobility-edge curve is exactly $\mathbb{Z}_2$5. This formulation unifies point-gap topology, skin localization, and Anderson localization at the level of a single disorder-averaged criterion (Fortin et al., 6 Sep 2025).

4. Localization theory, skin effect, and criticality

Localization diagnostics in non-Hermitian Anderson phases typically combine participation-ratio or inverse-participation-ratio measures with Lyapunov exponents and biorthogonal transfer-matrix methods. In the reciprocal non-Hermitian Aubry–André–Harper model, a generalized non-Hermitian Thouless relation links the inverse localization length to the complex-energy density of states,

$\mathbb{Z}_2$6

and the exact solution in the insulating phase gives the energy-independent result $\mathbb{Z}_2$7. This is a direct non-Hermitian analogue of the Hermitian Aubry–André localization law, but now in a phase with broken $\mathbb{Z}_2$8 symmetry and nontrivial spectral winding (Longhi, 2019).

In nonreciprocal disordered chains, localization is more intricate because the skin effect and Anderson localization compete. In the disordered SSH model, bulk states are immediately localized for any nonzero disorder, but under open boundary conditions the clean-system skin accumulation produces a non-monotonic localization profile: moderate disorder can sharpen zero-mode localization while large disorder destroys the skin effect and drives the open- and periodic-boundary localization measures toward one another. The same work shows that the topological transition is accompanied by an Anderson localization–delocalization transition of the zero-energy states (Zhang et al., 2019).

Direct experimental evidence for this competition has been obtained in a non-unitary split-step photonic quantum walk. There, the disorder-averaged growth-rate profile $\mathbb{Z}_2$9 peaks at finite velocity in the clean non-Hermitian regime, signalling directional flow associated with skin physics; develops a twin-peak structure at intermediate disorder, showing simultaneous skin and Anderson tendencies; and returns to a peak at $\mathcal{PT}$0 at strong disorder, indicating ordinary Anderson localization. In the same system, the biorthogonal localization length of the $\mathcal{PT}$1-quasienergy mode diverges at $\mathcal{PT}$2 and $\mathcal{PT}$3, identifying the disorder-driven topological critical points (Lin et al., 2021).

Quasiperiodic nonreciprocal chains add a further layer of structure. In a non-Hermitian SSH quasicrystal with quasiperiodically modulated intracell nonreciprocity, the Lyapunov exponent is obtained exactly as

$\mathcal{PT}$4

so there are no mobility edges, yet the system still exhibits a cascade from extended to multifractal critical and finally localized bulk states. The boundary $\mathcal{PT}$5 coincides exactly with the real–complex spectral transition, while $\mathcal{PT}$6 marks the onset of exponential localization (Zeng et al., 15 Feb 2026).

5. Extensions beyond the original one-dimensional setting

The non-Hermitian topological Anderson-insulator concept extends naturally to higher dimensions. In a two-dimensional disordered non-Hermitian Chern-insulator model with either nonreciprocal hopping or on-site gain and loss, topology is diagnosed by the disorder-averaged open-bulk Chern number and a generalized Bott index. In that setting, nonreciprocal hopping enlarges the topological region, gain and loss reduces it, and disorder-induced topological Anderson phases exist under both forms of non-Hermiticity. Localization was analyzed with right-eigenstate and biorthogonal inverse participation ratios, emphasizing that real-space open-boundary invariants are essential when the skin effect is present (Tang et al., 2020).

The higher-order generalization is the non-Hermitian higher-order topological Anderson insulator. For two-dimensional models with pseudoanti-Hermiticity or related non-Hermitian symmetry, the real-space quadrupole moment $\mathcal{PT}$7 can be used as a biorthogonal topological invariant and is quantized to $\mathcal{PT}$8 or $\mathcal{PT}$9 so long as a line gap in $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$0 remains open. Disorder can drive a trivial phase into a higher-order topological Anderson phase with corner-localized states near $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$1, and the same real-space invariant remains applicable even in models with the non-Hermitian skin effect (Liu et al., 2021).

Periodic driving adds a Floquet branch of the subject. In the periodically driven non-Hermitian SSH model, symmetric time frames restore an effective chiral structure for the Floquet operator, generalized Brillouin-zone windings retrieve bulk-boundary correspondence in the clean case, and a real-space biorthogonal winding retrieves it in the disordered case. Moderate disorder can then induce a Floquet topological Anderson insulator with boundary modes at quasienergy $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$2 or $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$3, even when the clean driven system is trivial (Wu et al., 2020).

Recent work has further broadened the class of NHTAIs. A non-Hermitian SSH quasicrystal with quasiperiodically modulated nonreciprocal intracell hopping exhibits exact topological boundaries

$$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$4

so a topological Anderson-insulator phase can appear in a clean-trivial regime and overlap with a multifractal critical bulk phase (Zeng et al., 15 Feb 2026). At the opposite extreme of disorder strength, an “anomalous” non-Hermitian topological Anderson insulator has been reported in a $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$5 lattice with ABBA-type symmetry-preserving non-Hermitian disorder, where a conventional $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$6 plateau gives way at ultra-strong disorder to a stable phase with $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$7 and an $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$8-mode zero-energy coalescence protected by a mobility gap (Ren et al., 2 Feb 2026).

6. Experimental realizations, observables, and unresolved issues

Experimental proposals and demonstrations span several synthetic platforms. Photonic lattices and waveguide arrays can realize balanced gain and loss, asymmetric couplers, non-unitary quantum walks, and quasiperiodic onsite or hopping modulations; coupled optical waveguides or resonator lattices have been proposed for the reciprocal $$H=\sum_{j}\left(m_j\,a_j^\dagger b_j + \text{h.c.}\right) + t_j^{(r)}\,a_{j+1}^\dagger b_j + t_j^{(l)}\,b_j^\dagger a_{j+1},$$9-symmetric Aubry–André–Harper realization, where the key signatures are the $\mathcal{PT}$0-breaking threshold at $\mathcal{PT}$1, abrupt changes in participation ratios, exponential localization with $\mathcal{PT}$2, and a complex spectral ellipse (Longhi, 2019). Time-multiplexed photonic quantum walks have already observed the non-monotonic NHSE–Anderson competition and disorder-induced topological transitions through biorthogonal chiral displacement and dynamical Lyapunov observables (Lin et al., 2021).

Topolectrical circuits are especially flexible because negative-impedance converters and programmable impedances can implement nonreciprocity, gain/loss, and quasiperiodic profiles in a controlled fashion. They have been proposed for one-dimensional quasiperiodic non-Hermitian SSH realizations, where edge zero modes appear as sharp impedance peaks and the admittance spectrum shows the real–complex transition, and also for strong-disorder anomalous non-Hermitian Anderson phases with ABBA-structured gain/loss supercells (Zeng et al., 15 Feb 2026, Ren et al., 2 Feb 2026). Ultracold atoms, mechanical metamaterials, and superconducting or optomechanical platforms have likewise been identified as viable settings for realizing nonreciprocal hopping, controlled loss, and disorder-resilient topological transport (Zhang et al., 2019, Fortin et al., 6 Sep 2025).

Several issues remain open across the literature. The first is disorder specificity: off-diagonal disorder that preserves chiral symmetry can stabilize non-Hermitian topological Anderson phases, whereas onsite disorder may alter edge-mode protection or invalidate the invariant being used. The second is boundary-condition dependence: in skin-effect systems, periodic-boundary spectra can be misleading, so open-boundary-compatible topology markers are mandatory. The third is finite-size scaling, especially in regimes where a band gap collapses but a mobility gap survives. Additional directions identified explicitly include interactions, nonlinearities, Floquet driving in broader non-Hermitian settings, and systematic extensions of generalized Brillouin-zone methods to truly disordered systems (Zhang et al., 2019, Tang et al., 2022).

Taken together, these developments establish the non-Hermitian topological Anderson insulator as a broad organizing concept for disorder-enabled topology in open quantum and classical lattices. Its modern form integrates point-gap topology, biorthogonal real-space invariants, generalized bulk-boundary correspondence, skin physics, and Anderson localization into a single framework. The resulting phases include reciprocal $\mathcal{PT}$3-broken quasiperiodic insulators with exact winding and localization laws, nonreciprocal disordered chiral chains with zero-energy edge states, higher-dimensional and higher-order Anderson phases, Floquet disorder-enabled topological states, and strong-disorder anomalous phases with no Hermitian analogue (Zhang et al., 2019, Longhi, 2019).