Non-Hermitian Quasiperiodic Rings

• Non-Hermitian quasiperiodic rings are synthetic quantum systems defined by complex, nonreciprocal hopping and incommensurate onsite potentials that yield unique topological phases.
• They feature mobility rings in the complex energy plane, where extended and localized states are delineated by spectral winding numbers and PT-symmetry breaking.
• Advanced diagnostic methods such as AGR, IPR, and multifractal analysis not only track phase transitions but also pave the way for innovative quantum transport and device applications.

Non-Hermitian quasiperiodic rings constitute a class of synthetic quantum systems defined by complex and/or nonreciprocal hopping and site potentials modulated in an aperiodic (typically incommensurate) manner. The ring geometry enforces periodic boundary conditions, enabling spectral winding number diagnostics and robust topological invariants. Non-Hermiticity, implemented through onsite gain/loss or asymmetric couplings, drives symmmetry-breaking, metal-insulator transitions, and the formation of mobility edges generalized to closed contours—“mobility rings”—in the complex energy plane. The interplay of non-Hermitian effects, quasiperiodicity, dimerization, and synthetic (Abelian and non-Abelian) gauge fields yields a plethora of physical phenomena unique to ring structures, including PT-breaking, multifractality, skin effects, topological Anderson insulating phases, ultra-enhanced transport and circulating currents, and re-entrant localization transitions.

1. Model Architectures and Hamiltonians

A general non-Hermitian quasiperiodic ring is described by a second-quantized tight-binding Hamiltonian with complex-valued hopping ($t_{1}, t_{2}$), onsite quasiperiodic potential $V_{j}$, and periodic boundary conditions; prototypical forms include:

• Dimerized Non-Hermitian AAH Model: $$H = \sum_{j=1}^L \bigl[t_1 c_j^\dagger c_{j+1} + t_2 c_{j+1}^\dagger c_{j+2} + \mathrm{h.c.}\bigr] + \sum_{j=1}^{L} V_j c_j^\dagger c_j$$, with $t_1=J-\Delta$, $t_2=J+\Delta$ controlling dimerization, $V_j = V e^{i2\pi\beta j + \gamma}$ (Zhou et al., 2021).
• Flat-band/diamond models: Three-site-per-cell rings with non-Hermitian potentials, e.g., $$V_{11}(n) = V_0 e^{i\theta} \cos (2\pi\beta n + \phi + i\epsilon)$$ (Pang et al., 31 Mar 2025, Liu et al., 2023).
• Non-Abelian generalizations: Spin-1/2 chains with SU(2) gauge fields, nonreciprocal hopping, and complex onsite potential, $$H = \sum_{n=1}^{L} [J_L c_n^\dagger \sigma_0 c_{n+1} + J_R c_{n+1}^\dagger \sigma_0 c_n + V_n c_n^\dagger (\dots) c_n]$$ (Zhou, 2023, Chen et al., 16 Jul 2025).
• Other examples: Non-Hermitian Maryland model (Longhi, 2021), Fibonacci gain/loss rings (Roy et al., 9 Jan 2026), GAAH-model rings (Padhan et al., 2023, Li et al., 2024, Zheng et al., 2024).

Quasiperiodicity is introduced via irrational modulation (e.g., $\beta = (\sqrt{5} - 1)/2$) ensuring absence of translational invariance and realization of quasicrystals.

2. Mobility Edges, Rings, and the Complex Energy Plane

Hermitian quasiperiodic chains feature mobility edges as sharp real-energy transitions separating extended and localized states. In non-Hermitian rings, the concept generalizes dramatically:

• Mobility rings: Solutions of Lyapunov exponent equations (via Avila’s global theory) yield closed contours (typically circles, ellipses, or higher-genus curves) in the complex-$E$ plane:
  • Example: $|E-E_0| = R$, with $R = 1 / (\lambda e^h)$, extended states lie inside the ring, localized outside; for multi-cell/mosaic models, $|a_\kappa(E)| = 1 / (\lambda e^h)$, allowing up to $\kappa-1$ concentric rings (Li et al., 2024).
  • For non-Abelian chains, mobility rings emerge only when SU(2) gauge fields couple nontrivially to nonreciprocal hopping (i.e., $\sin\theta_l \sin\theta_r \neq 0$) (Chen et al., 16 Jul 2025).
• Mobility lines: In certain flat-band models, real segment(s) in the complex plane host extended states (e.g., $E_c \in [-2,-1] \cup [0,1]$) (Pang et al., 31 Mar 2025).
• Critical window and multifractality: Absent in most non-Hermitian models except for special flux-correlated modulations in diamond/flat-band rings (Liu et al., 2023).

In the infinite-mosaic/series limit, nested mobility rings collapse into a single topological ring (winding $w=1$), merging all point gaps (Zheng et al., 2024).

3. Topological Invariants and Spectral Winding

Topological phase transitions are encoded in spectral winding numbers that probe point-gap topology under twisted boundary conditions:

$$w(E_B) = \frac{1}{2\pi i} \int_{0}^{2\pi} \partial_\phi \ln \det[H(\phi) - E_B] d\phi$$

Key facts:

• Integer jumps of $w$ correspond to topological transitions, typically at mobility edge crossing, PT-symmetry breaking ($\mathcal{PT}$), or closure/creation of spectral loops.
• For dimerized rings (Zhou et al., 2021): $(W_+,W_-) = (0,0)$ (extended), $(-1,0)$ (mobility-edge), $(-1,-1)$ (localized).
• In mosaic and non-Abelian models, multiple basepoints can track multiple mobility rings due to higher-degree polynomials in $E$ (Li et al., 2024, Chen et al., 16 Jul 2025).
• The winding structure underpins the non-Hermitian skin effect, dictating bulk-to-boundary mode conversions.

4. Methods for Phase Diagnosis: AGR, IPR, NPR, Multifractality

Non-Hermitian rings are characterized via:

• Adjacent gap ratio (AGR): $\overline{r} \simeq 0$ (extended), $0<\overline{r}<0.4$ (mobility edge), $\overline{r}\simeq0.4$ (localized) (Zhou et al., 2021).
• Inverse participation ratio (IPR): $IPR_n = \sum_j |\psi_{n,j}|^4$; $IPR\sim1/L \rightarrow 0$ signals extended, $O(1)$ signals localized; mobility-edge phases have $max_n IPR_n>0$, $min_n IPR_n\approx0$ (Zhou et al., 2021, Chen et al., 16 Jul 2025).
• Normalized participation ratio (NPR): $NPR_n = [IPR_n \cdot L]^{-1}$ for finer diagnosis in multifractal and skin/critical regimes.
• Fractal dimension ($D_2$): Differentiates multifractal ($0<D_2<1$) from localized ($D_2\to0$) or extended ($D_2\to1$) (Liu et al., 2023).
• Entanglement entropy ($S$): Area law (localized), logarithmic scaling (extended), $S\sim \alpha\ln L$ ($0<\alpha<1$) in critical window (Zhou, 2023).

Return probability and wave-packet spread also reveal localization and transport transitions.

5. Symmetry, Gauge Fields, and Advanced Topological Phenomena

Non-Hermitian rings accommodate a range of symmetry-breaking regimes:

• PT symmetry: Non-Hermitian potentials yield PT-unbroken/real spectra at low gain/loss, PT-breaking leads to complex spectra and localization (Zhou et al., 2021, Zhou, 2023).
• Non-Abelian gauge fields: Synthetic SU(2) potentials (via commutator structure $[\sigma_x,\sigma_y]$) induce genuinely non-Abelian mobility rings and critical windows, absent in Abelian chains (Chen et al., 16 Jul 2025, Zhou, 2023).
• Re-entrant transitions: Non-Hermitian GAAH models show delocalization-localization-delocalization sequences, with winding number tracking multiple topological transitions, enabled by competition between quasiperiodic strength and onsite complex phase (Padhan et al., 2023).
• Skin effect/topological Anderson insulator: Non-reciprocal hopping ($t_{L}\neq t_{R}^{*}$) triggers skin modes, mobility rings, and distinct TAI phases with gapped extended, intermediate (mobility-edge), and localized bulks (Tang et al., 2022).

6. Transport, Currents, and Experimental Proposals

Non-Hermitian rings exhibit transport signatures and are accessible to various experimental platforms:

• Fibonacci gain/loss rings: Under balanced gain/loss topology, PT-symmetric rings show order-of-magnitude amplification of circulating currents and induced magnetic fields (up to 2 T), with parity-dependent scaling in non-PT-symmetric rings (Roy et al., 9 Jan 2026).
• Photonic waveguides: Realization of quasiperiodic non-Hermitian rings with synthetic flux, SU(2) coin steps, and measurement of localization/topological transitions via site intensity profiles.
• Ultracold atoms: Raman–induced SO coupling and site-selective dissipation allow engineering of non-Abelian potentials, mobility ring detection via time-of-flight imaging.
• Electrical circuits: Implementation of non-Hermitian diamond/flat-band models, $Z_\theta$ flux, and negative impedance converters, yielding admittance spectra analogous to ring eigenvalues (Liu et al., 2023).

7. Physical Implications and Outlook

The non-Hermitian quasiperiodic ring paradigm realizes a wealth of phenomena:

• Transition from real to complex spectra coincides with localization boundaries and topological changes.
• Mobility edges generalized to mobility rings enable spectral-winding topological protection and boundary-bulk correspondences.
• Reentrant localization transitions provide platforms for loss/gain-tunable topological switching.
• Non-Abelian rings yield double localization transitions and extended critical windows spanning multifractal, PT-breaking, and topological regimes.
• Engineered current amplification, parity effects, and switching phenomena present new opportunities for quantum device applications.

Mobility rings, spectral windings, and skin modes collectively extend the taxonomy of localization, transport, and topology far beyond the Hermitian and 1D quasiperiodic crystal paradigm, positioning non-Hermitian rings as central objects in contemporary synthetic quantum matter (Zhou et al., 2021, Pang et al., 31 Mar 2025, Li et al., 2024, Zhou, 2023, Chen et al., 16 Jul 2025, Liu et al., 2023, Longhi, 2021, Padhan et al., 2023, Roy et al., 9 Jan 2026, Tang et al., 2022).