---
title: Non-Hermitian Quasiperiodic Lattice
url: https://www.emergentmind.com/topics/non-hermitian-quasiperiodic-lattice
type: topic
---

# Non-Hermitian Quasiperiodic Lattice

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) [2412.04344].

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 [1901.09399, 2601.23015].
- **Complex AA/AAH potentials** with phase shifts or geometric series modulation, enabling multiple mobility edges and nested spectral loops [2410.04469, 2404.11020].
- **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 [2402.14370, 2311.03166, 2504.08760].
- **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 [2507.12176, 2505.22319].

## 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 [2511.00350, 2412.04344].

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 [2404.11020, 2410.04469].
- **Mobility ring** (not a point or line, but a closed loop in the complex plane): Existence in non-Abelian systems [2507.12176], flat-band models [2504.08760], and non-Hermitian GAA models [2412.04344]. 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 [2412.04344].

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 [2412.04344]. In long-range or dimerized models, Avila's global theory and exact duality can yield exact Lyapunov exponents and mobility-edge relations [2012.10029, 2404.11020].

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 [2402.14370].

## 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 [1901.09399, 2306.11084, 2507.12176, 2504.13770].
- **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 [2601.23015, 2505.22319].
- **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 [2507.12176, 2105.03302, 2306.11084]. “Chiral” point-gap topology can yield pairs of skin modes at opposite edges [2504.13770].

## 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” [2412.04344].
- **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 [2306.09036, 2511.00350, 2504.08760, 2311.03166].
- **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 [2404.11020, 2511.00350].
- **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 [2410.09185].

## 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 [2508.14724, 2101.05505].
- **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) [2508.14724].
- **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 [2508.14724].

## 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 [2410.09185, 2404.11020].
- **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 [2505.22319, 2412.04344].
- **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 [2311.03166, 1901.09399].
- **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 [2402.14370].

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

| Model Class                            | Mobility Edge Equation                                            | Reference         |
|----------------------------------------|-------------------------------------------------------------------|-------------------|
| nHGAA (short-range, complex phase)     | $h_c = \ln[(1+\sqrt{1-\alpha^2})/\alpha]$                        | [2412.04344]      |
| Non-Hermitian AA (asymm. hopping)      | $\Delta/J = e^{|\alpha|}$                                        | [1901.09399]      |
| Geometric series, GSM potential        | Ellipse in $(\Re E, \Im E)$ plane, $|\chi \pm 2\alpha| = 1+\sqrt{1-\alpha^2}$ | [2404.11020]      |
| Flat-band, Lieb lattice, 2D            | $|E| = 1/\lambda$                                                | [2402.14370]      |
| Long-range exp. hopping (dual, $h=0$)  | $E_c = V\cosh(p) - 1$                                            | [2012.10029]      |

## 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 [2601.23015, 2412.04344].
- **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 [2507.12176, 2404.11020, 2504.13770].
- **Quantum dynamics and exceptional-point physics**: The interplay between non-linear dynamics, exceptional-point sensitivity, and non-Hermitian-induced instabilities is under investigation [2410.09185].
- **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 [2412.04344, 2504.08760, 2601.23015, 2404.11020, 2507.12176, 2402.14370, 2311.03166].

Source: https://www.emergentmind.com/topics/non-hermitian-quasiperiodic-lattice