Non-Hermitian AAH Lattice Phenomena

• The non-Hermitian AAH lattice is a one-dimensional quasiperiodic system with complex, nonreciprocal couplings that induce unique localization and topological phenomena.
• Key insights include the emergence of skin effects, mobility edges, and re-entrant phases as well as decoupled spectral and topological transitions driven by non-Hermiticity.
• Analytical methods like self-duality and Lyapunov exponents, combined with experimental realizations in photonic and cold atom systems, validate its rich phase structure and dynamic observables.

A non-Hermitian Aubry-André-Harper (AAH) lattice is a one-dimensional quasiperiodic tight-binding lattice where either the onsite potential, the hopping amplitudes, or both, have non-Hermitian (complex, nonreciprocal, or spatially varying gain/loss) structure. This class of models extends the canonical Hermitian AAH (almost-Mathieu) system—famous for its sharply tuned delocalization-localization and topological phase transitions—into a regime where the interplay of quasiperiodicity, non-Hermitian symmetry breaking, spectral topology, and unconventional bulk-edge correspondence gives rise to a wealth of analytically tractable and experimentally relevant phenomena, including skin effects, mobility edges, re-entrant phases, and nontrivial winding invariants.

1. Core Non-Hermitian AAH Hamiltonians

A generic non-Hermitian AAH Hamiltonian for spinless fermions (or bosons) on a 1D lattice of length $L$ may be written as:

$$H=\sum_{n=1}^{L-1}\bigl[t_{R}^{(n)} c_{n+1}^\dagger c_n + t_{L}^{(n)} c_n^\dagger c_{n+1}\bigr] + \sum_{n=1}^L V_n c_n^\dagger c_n$$

where

• $t_{R}^{(n)}$ and $t_{L}^{(n)}$ are the (possibly nonreciprocal/complex) right/left hopping amplitudes,
• $V_n$ is a complex, generally quasiperiodic onsite potential, e.g., $V_n = V_0 e^{-2\pi i\alpha n}$ or $i\lambda \cos(2\pi\alpha n+\phi)$,
• $\alpha$ is irrational (e.g., the inverse golden ratio) for true quasiperiodicity.

Variants include additional dimerization, power-law hopping terms, spatially patterned imaginary gauge fields, spin-dependent hoppings, or time-dependent driving. Non-Hermiticity is encoded in $t_R \neq t_L^*$ or $\Im V_n \neq 0$ (nonreciprocal transport, gain/loss, complex modulation).

2. Spectral and Localization Phase Transitions

The transition from extended to localized states in non-Hermitian AAH lattices sharply departs from the Hermitian archetype:

• PT-Symmetric Non-Hermitian AAH: For an on-site potential $V_n = V_0 e^{-2\pi i\alpha n}$, the model is PT-symmetric. For $V_0 < J$ (hopping), all eigenvalues are real and extended; for $V_0 > J$, all eigenvalues are complex (elliptical spectral locus) and all states are exponentially localized. At $V_0=J$ both a PT-breaking transition and a topological winding number jump occur; the localization length is $\xi = [\ln(V_0/J)]^{-1}$ and independent of the energy $E$ (Longhi, 2019).
• Nonreciprocal Hopping (Hatano-Nelson-type): With $t_R \neq t_L$ and a real quasiperiodic potential, the localization threshold is $V_c=J_L$. Below $V_c$, extended states exhibit a non-Hermitian skin effect (all eigenstates pile up at one boundary under OBC); above $V_c$, eigenstates are Anderson localized (Longhi, 2021, Li et al., 2023). The transition is discontinuous in dynamical observables such as wavepacket spreading exponent $\delta$ and ballistic velocity $v$ (Longhi, 2021).
• Non-Hermitian AAH with Off-Diagonal Modulation: Including an incommensurate modulation of the hopping ($t_n$), a combined phase criterion emerges: $r=\sqrt{V^2+\lambda^2}$; for $r<t$, all states are extended, for $r>t$, all states localize. Critically, the real-complex spectral boundary and the mobility edge no longer coincide for incommensurate $t_n$ (PT symmetry is broken everywhere except in certain commensurate cases) (Chen et al., 2022).
• Generalized Nonreciprocal AAH with Interactions: When both nonreciprocal hopping and complex potentials are present, the localization (mobility edge), complexification, and topological transitions can become partially decoupled. For nonreciprocal hopping, all these boundaries coincide, while for complex onsite potentials, the real-to-complex spectral transition can precede the localization and topological transitions (Tang et al., 2021).
• Mobility Edges and Mixed Phases: For power-law or dimerized cases, analytical self-duality relations and Lyapunov exponent criteria yield mobility edges—energy boundaries between extended and localized (or multifractal) states. The presence of mobility edges is often indicated by simultaneous coexistence of real and complex eigenvalues, and characterized by spectral winding numbers (Liu et al., 2020, Zhou et al., 2021, Wang et al., 17 Nov 2025, Peng et al., 2023). Dimerization and power-law potentials further introduce reentrant phase transitions and cascades of intermediate (mixed) phases (Han et al., 2021, Wang et al., 17 Nov 2025).

Table: Summary of Non-Hermitian AAH Phase Boundaries (Selected Cases)

Model Variant	Critical Point/Condition	Extended–Localized/Complex Transition
PT-symmetric complex potential (Longhi, 2019)	$V_0/J = 1$	Coincident PT and localization transition
Nonreciprocal hopping (Longhi, 2021, Li et al., 2023)	$V_c = J_L$	Coincident mobility edge, skin effect, complexification
Off-diagonal modulation (Chen et al., 2022)	$V^2 + \lambda^2 = t^2$	Real-complex and mobility edge decoupled
Power-law potential, $p\ge3$ (Wang et al., 17 Nov 2025)	Two critical $V$: $E\to M$, $M\to L$	Mobility edge phase between extended/localized
Dimerized, staggered potential (Zhou et al., 2021, Han et al., 2021)	$\gamma_{c1,2}(\Delta)$; see text	Reentrant/mobility edges, winding jumps

3. Topological Invariants and Bulk–Edge Correspondence

Non-Hermitian AAH lattices support several types of spectral/topological invariants:

• Winding Numbers: The number of times the complex spectrum encircles a chosen base energy (point-gap topology) as a parameter (e.g., boundary twist or flux) is varied. Winding numbers may signal topological transitions even when Anderson localization is absent, and are integral in distinguishing topologically distinct extended, mixed, and localized phases (Longhi, 2019, Liu et al., 2020, Zhou et al., 2021, Chen et al., 2022, Han et al., 2021).
• Non-Hermitian Zak Phase (Bulk Polarization): For models with non-Hermitian PH symmetry (anti-symmetry), a quantized Zak phase (0 or $\pi$) can be defined via the biorthogonal Berry connection and a Wilson loop over the Brillouin zone, classifying phases with and without edge modes (Zhu et al., 2022).
• Edge/Domain-Wall Modes: Unique to certain non-Hermitian AAH lattices (notably with imaginary potential of period divisible by four), spectrally isolated edge states with purely imaginary eigenenergies can exist and be robust even when the real-line gap closes. These are stabilized by non-Hermitian PH symmetry and quantified by a $\mathbb{Z}_2$ polarization (Zhu et al., 2022).
• Skin Effect: In nonreciprocal and related models, all bulk eigenstates may become localized at an edge under OBC, a breakdown of conventional bulk–boundary correspondence. The skin effect can coexist or compete with Anderson localization, leading to distinct "area-law" entanglement phases (Li et al., 2023).

4. Intermediate, Re-entrant, and Mixed Phases

Non-Hermitian AAH lattices realize a variety of phase structures not possible in the Hermitian limit:

• Re-entrant Delocalization–Localization: As the non-Hermiticity parameter (e.g., imaginary phase $h$) is varied, the system may traverse extended $\to$ mixed $\to$ localized $\to$ mixed $\to$ extended phases, with distinct winding numbers for each phase. Both real-complex spectral transitions and nontrivial topology can re-enter at large non-Hermiticity due to analytic structure of the quasiperiodic modulation (Padhan et al., 2023).
• Mobility Edges and Mixed Phases: Mixed phases with coexisting extended and localized states arise in dimerized or power-law AAH chains, marked by stepwise topological winding jumps and appearance of non-Hermitian mobility edges, analytically tracked by self-duality and Lyapunov exponent calculations (Liu et al., 2020, Wang et al., 17 Nov 2025, Zhou et al., 2021, Han et al., 2021).
• Spectral–Topological Decoupling: In the presence of both nonreciprocity (asymmetric hopping) and a complex potential, Anderson localization and topological (spectral winding) transitions can become decoupled, especially under different boundary conditions—bulk localization need not coincide with a topological winding number change (Cai, 2020).

5. Dynamical, Entanglement, and Physical Observables

• Transport and Dynamics: In non-Hermitian AAH lattices, localization or skin transitions lead to abrupt changes in quantum diffusion exponent, light-cone velocity, and relaxation dynamics (e.g., algebraic vs exponential damping in open systems with gain/loss). Surprising phenomena such as disorder-enhanced ballistic transport or quantized jumps in propagation emerge in certain regimes (Longhi, 2021, Ghatak et al., 2024, He et al., 2021).
• Entanglement Transitions: The coexistence and competition between non–Hermitian skin effect and Anderson localization leads to two distinct area-law entanglement scaling phases and a volume-law entanglement critical line, aligning with the spectral-topological transition (Li et al., 2023). In the many-body context, real-complex transitions, topological winding changes, and many-body localization cross at (or near) the same critical point for nonreciprocal cases but separate in complex-potential cases (Tang et al., 2021, Li et al., 2023).
• Experimental Observation: Non-Hermitian AAH lattices have been realized in discrete-time photonic quantum walks, topo-electrical circuits, optical mesh lattices, and cold atom momentum lattices. Observables include direct spectral imaging, winding number extraction, voltage/impedance profiles, and dynamical preparation of edge/localized modes (Lin et al., 11 Aug 2025, Liu et al., 2020, Cai, 2020, He et al., 2021, Miao et al., 11 Jan 2026).

6. Model Generalizations and Future Directions

• Dimerization and Hopping Engineering: Introducing dimerized hopping or staggered potential splits single transition points into multiple transitions, engineering cascades of reentrant/mixed phases and enabling mobility-edge design (Zhou et al., 2021, Han et al., 2021).
• Imaginary Gauge Fields and Edge-State Steering: Spatially fluctuating imaginary gauge fields permit mapping to Hermitian models with modified wavefunction envelopes, enabling static and dynamical steering of in-gap edge states (Miao et al., 11 Jan 2026).
• Spinful and Driven Systems: With spin–orbit coupling and periodic magnetic textures, models exhibit spectral fractality ("butterfly maps") coupled to spin-polarization polarization, enriching the topological and localization phenomena (Padhi et al., 8 Jan 2026). Time-periodic (Floquet) driving can unbreak non-Hermitian symmetry and stabilize real quasienergy spectra and Floquet topological phases (Blose, 2019).
• Nonlinearity and Open Dynamics: Gain saturation and Lindbladian open-system descriptions introduce new damping transitions, nonlinear restoration of delocalized transport, and hybridization of skin/localized modes in meta-material and laser implementations (Ghatak et al., 2024, He et al., 2021, Zhu et al., 2022).

7. Summary and Outlook

Non-Hermitian Aubry-André-Harper lattices, by generalizing quasiperiodic order into the non-Hermitian regime, provide a fertile platform for analytically tractable and experimentally accessible studies of localization, topology, spectral singularities, and dynamical phenomena. Distinctive features include reentrant and mixed topological phases, nontrivial winding formalism, robust mobility edges with analytically trackable boundaries, interplay of skin and Anderson localization, and precise bulk-edge correspondences (or their breakdown). The synergy between analytical methods—self-duality, Lyapunov exponents, winding calculation, biorthogonal perturbation—and experimentally feasible platforms such as photonic quantum walks, topolectrical circuits, and cold atom lattices ensures continuing impact on the physics of localization, topology, and open quantum matter (Longhi, 2019, Cai, 2020, Liu et al., 2020, Zhou et al., 2021, Zhu et al., 2022, Padhan et al., 2023, Wang et al., 17 Nov 2025, Miao et al., 11 Jan 2026, Padhi et al., 8 Jan 2026, Ghatak et al., 2024, Lin et al., 11 Aug 2025, Li et al., 2023, Han et al., 2021).