Non-Hermitian Quasiperiodic Lattice

• Non-Hermitian quasiperiodic lattices are tight-binding systems with incommensurate modulations, complex onsite potentials, and asymmetric hopping.
• They unify localization phenomena, multifractality, and non-Hermitian skin effects while exhibiting mobility edges and spectral winding transitions.
• Experimental platforms in photonics, cold atoms, and electrical circuits validate their role in simulating advanced quantum and classical metamaterials.

A non-Hermitian quasiperiodic lattice is a quantum or classical tight-binding system in which the hopping amplitudes, onsite potentials, or both are made non-Hermitian, and in which lattice sites are modulated by a deterministic but incommensurate (quasiperiodic) pattern. These systems unify concepts of Anderson localization, multifractality, and metal-insulator transitions from Hermitian quasiperiodic lattices with spectral topology, skin effects, and PT-symmetry phenomena unique to non-Hermitian physics. They exhibit unconventional localization transitions, re-entrant phase behavior, non-Hermitian skin effects (NHSE), mobility edges and rings in the complex-energy plane, point-gap topology, and multifractal skin criticality.

1. Fundamental Models and Hamiltonians

The canonical non-Hermitian quasiperiodic lattice generalizes the Aubry–André (AA) and Aubry–André–Harper (AAH) models by introducing non-Hermitian elements via complex onsite potentials, asymmetric (non-reciprocal) hopping, or both. The general form of a 1D non-Hermitian quasiperiodic tight-binding Hamiltonian is

$$H = -\sum_{j \neq k} \frac{J}{|j-k|^{a}} c_j^\dagger c_k + \lambda \sum_{j} V_j n_j,$$

where $J$ is the hopping amplitude, $a$ the hopping-range exponent (nearest-neighbor for $a \gg 1$, power-law for $a$ finite), $\lambda$ is the potential strength, and $V_j$ is a complex, quasiperiodic onsite potential of the form: $$V_j = \frac{\cos(2\pi\beta j+\phi)}{1 - \alpha \cos(2\pi\beta j+\phi)},$$ with irrational $\beta$ (often the golden mean), nonlinearity parameter $\alpha$, and complex phase $\phi = \theta + i h$ (the imaginary part controls non-Hermiticity) (Padhi et al., 2024).

Other models introduce:

• Asymmetric hopping (non-reciprocal Hatano–Nelson type): $J_{R} = J e^{-\alpha}$, $J_{L} = J e^{+\alpha}$, leading to non-Hermitian skin effects and spectral winding (Jiang et al., 2019, Chen et al., 30 Jan 2026).
• Complex AA/AAH potentials with phase shifts or geometric series modulation, enabling multiple mobility edges and nested spectral loops (Zheng et al., 2024, Wang et al., 2024).
• Multiband or flat-band structures (e.g., Lieb or diamond chains) exhibiting robust flat bands and novel topological localization features in the non-Hermitian regime (Jiang et al., 2024, Liu et al., 2023, Pang et al., 31 Mar 2025).
• Multi-component and spinful systems with genuine SU(2) non-Abelian gauge fields or spin-orbit coupling, generalizing the localization/topological transition to mobility rings (Chen et al., 16 Jul 2025, Zhao et al., 28 May 2025).

2. Localization Transitions and Mobility Edges

Localization in non-Hermitian quasiperiodic lattices is typically diagnosed via the inverse participation ratio (IPR), normalized participation ratio (NPR), multifractal dimension $D_2$, and direct analysis of Lyapunov exponents: $$\mathrm{IPR}_n = \sum_{j=1}^L |\psi_{j}^{n}|^4,\quad \mathrm{NPR}_n = \frac{1}{L \cdot \mathrm{IPR}_n}.$$ Extended states have $\mathrm{IPR} \sim L^{-1}$ ($\mathrm{NPR} \sim 1$), localized states have finite $\mathrm{IPR}$ ($\mathrm{NPR} \to 0$), and critical/multifractal states lie in between (Wang et al., 1 Nov 2025, Padhi et al., 2024).

A central phenomenon is the existence of non-Hermitian mobility edges or mobility rings:

• Mobility edge: a (possibly complex) energy threshold $E_c$ separating localized from extended states. In geometric-series modulated models, the mobility edge forms an ellipse in the complex energy plane (Wang et al., 2024, Zheng et al., 2024).
• Mobility ring (not a point or line, but a closed loop in the complex plane): Existence in non-Abelian systems (Chen et al., 16 Jul 2025), flat-band models (Pang et al., 31 Mar 2025), and non-Hermitian GAA models (Padhi et al., 2024). Extended states reside within a ring, localized states outside.
• Phase diagrams may contain extended (D), localized (L), intermediate (I) with contiguous energy-resolved mobility edges, or comb phases (C) where extended states appear as isolated teeth within localized backgrounds (Padhi et al., 2024).

Analytic mobility-edge criteria are model-dependent, e.g., for non-Hermitian GAA,

$$h_c = \ln\left[\frac{1 + \sqrt{1-\alpha^2}}{\alpha}\right]$$

marks the critical non-Hermitian parameter for the localization transition (Padhi et al., 2024). In long-range or dimerized models, Avila's global theory and exact duality can yield exact Lyapunov exponents and mobility-edge relations (Liu et al., 2020, Wang et al., 2024).

In 2D, such as on the Lieb lattice, the mobility edge is given by $|E|=1/\lambda$ for imaginary onsite potentials of strength $\lambda$, separating extended (purely imaginary $E$) and localized states (Jiang et al., 2024).

3. Non-Hermitian Topology: Winding Numbers and Skin Effects

Non-Hermitian spectra occupy regions in the complex energy plane, characterized by point-gap topology:

• Spectral (point-gap) winding numbers are defined for a reference point $E_0$: $$W(E_0) = \frac{1}{2\pi i} \int d\theta\, \partial_\theta \ln \det[H(\theta) - E_0],$$ where $\theta$ is a boundary or flux twist. $W \ne 0$ indicates nontrivial spectral winding, correlated with the presence of NHSE (Jiang et al., 2019, Padhan et al., 2023, Chen et al., 16 Jul 2025, Liang et al., 18 Apr 2025).
• Non-Hermitian skin effect (NHSE): Bulk eigenmodes accumulate at the boundary under OBC when spectral winding is nonzero. The transition between extended and skin-localized states is coupled to the localization transition and topological winding transitions (Chen et al., 30 Jan 2026, Zhao et al., 28 May 2025).
• Bulk–boundary–correspondence in non-Hermitian lattices is controlled by spectral winding rather than Chern numbers.

In models with multiple mobility rings/loops, the winding number can jump by $\pm 1$ as control parameters are varied, marking topological transitions synchronized with localization transitions (Chen et al., 16 Jul 2025, Zhou et al., 2021, Padhan et al., 2023). “Chiral” point-gap topology can yield pairs of skin modes at opposite edges (Liang et al., 18 Apr 2025).

4. Spectral Features, Mixed Phases, and Anomalous Intermixing

Non-Hermitian quasiperiodic systems generically display complex-valued spectra and exceptional point (EP) structures. Spectral features and intermediate phases include:

• Comb phases: Regular alternation of extended and localized states even within a single compact energy window—the spectrum displays real-energy “teeth” piercing through complex-valued localized state “loops” (Padhi et al., 2024).
• Mixed (intermediate) phases: Coexistence of extended, localized, skin, and critical states, depending on parameter mismatch between real and imaginary potential modulations or the presence of higher harmonics (Acharya et al., 2023, Wang et al., 1 Nov 2025, Pang et al., 31 Mar 2025, Liu et al., 2023).
• Fractal butterfly spectra: Sweeping an onsite offset or nonlinear parameter produces multiscale, self-similar “butterfly” patterns in the complex plane, with regions labeled by mobility edges (Wang et al., 2024, Wang et al., 1 Nov 2025).
• Spectral sensitivity and pseudospectrum: The presence of exceptional points (EPs) and non-orthogonal eigenstates leads to pronounced spectral instability near phase transitions, with $\epsilon^{1/2}$ scaling of pseudospectral clouds at criticality (Ghatak et al., 2024).

5. Interacting Models and Many-Body Non-Hermitian Localization

Beyond single-particle dynamics, interaction effects in non-Hermitian quasiperiodic lattices modify localization physics:

• Many-body localization (NHMBL) emerges via density-density interactions combined with non-Hermitian hopping and QP potentials. Phase diagrams exhibit ergodic, multifractal/intermediate, and NHMBL regimes, discerned by entropy scaling and level statistics (Chakrabarty et al., 20 Aug 2025, Tang et al., 2021).
• Interplay of long-range hopping and interactions: In power-law hopping chains, interactions destroy non-interacting mobility edges, yielding a broad intermediate regime of multifractal eigenstates (with possibly real spectra even outside the NHMBL phase) (Chakrabarty et al., 20 Aug 2025).
• Topological aspects: Many-body winding numbers, spectral reality, and skin effects can decouple such that, e.g., a nonzero spectral winding under PBC does not always produce NHSE under OBC when long-range hopping is included (Chakrabarty et al., 20 Aug 2025).

6. Experimental Realizations and Physical Implications

Physical implementations of non-Hermitian quasiperiodic lattices include:

• Photonic platforms: Waveguide arrays or coupled microrings with engineered gain/loss or complex refractive-index modulation directly realize many model Hamiltonians, permitting the observation of skin modes, spectral loops, and quantized jumps in localization (Ghatak et al., 2024, Wang et al., 2024).
• Trapped-ion and cold-atom arrays: Long-range hopping via optical control, atom losses, and lattice modulation allow for direct engineering of the non-Hermitian QP phenomena (Zhao et al., 28 May 2025, Padhi et al., 2024).
• Classical electrical circuits: Inductors, capacitors, and INICs (negative impedance converters) can simulate nonreciprocal hopping and QP site energies, making the NHSE and localization transitions accessible through admittance spectra (Liu et al., 2023, Jiang et al., 2019).
• Lindblad master equation formalism: Dissipative engineering (controlled local loss) implements non-Hermitian QP potentials in electronic and photonic lattices; quantum trajectories under Lindblad dynamics mimic non-Hermitian Hamiltonian evolution (Jiang et al., 2024).

Table: Selected Analytical Mobility-Edge Conditions in Non-Hermitian QP Models

Model Class	Mobility Edge Equation	Reference
nHGAA (short-range, complex phase)	$h_c = \ln[(1+\sqrt{1-\alpha^2})/\alpha]$	(Padhi et al., 2024)
Non-Hermitian AA (asymm. hopping)	$\Delta/J = e^{|\alpha|}$	(Jiang et al., 2019)
Geometric series, GSM potential	Ellipse in $(\Re E, \Im E)$ plane, $|\chi \pm 2\alpha| = 1+\sqrt{1-\alpha^2}$	(Wang et al., 2024)
Flat-band, Lieb lattice, 2D	$|E| = 1/\lambda$	(Jiang et al., 2024)
Long-range exp. hopping (dual, $h=0$)	$E_c = V\cosh(p) - 1$	(Liu et al., 2020)

7. Open Problems and Research Directions

Significant open questions remain:

• Multifractality and universality: The structure and scaling of multifractal exponents in comb and skin critical phases (identical $D_q$ for all eigenstates in “quasiperiodic skin criticality”) challenge standard paradigms (Chen et al., 30 Jan 2026, Padhi et al., 2024).
• Topological classification: Formal relations between non-Hermitian point-gap topology, mobility rings, and skin effect, especially in higher dimensions or for interacting systems, are active topics (Chen et al., 16 Jul 2025, Wang et al., 2024, Liang et al., 18 Apr 2025).
• Quantum dynamics and exceptional-point physics: The interplay between non-linear dynamics, exceptional-point sensitivity, and non-Hermitian-induced instabilities is under investigation (Ghatak et al., 2024).
• Experimental observation: Direct detection of uniquely non-Hermitian phenomena—such as comb phases, mobility rings, or chiral skin effects—in engineered photonic, electronic, or atomic systems is ongoing.

Non-Hermitian quasiperiodic lattices thus comprise a rich and rapidly developing field at the intersection of topological physics, localization theory, and complex spectral analysis, with direct implications for engineered quantum and classical metamaterials (Padhi et al., 2024, Pang et al., 31 Mar 2025, Chen et al., 30 Jan 2026, Wang et al., 2024, Chen et al., 16 Jul 2025, Jiang et al., 2024, Liu et al., 2023).