--- title: Non-Hermitian Quasiperiodic Rings url: https://www.emergentmind.com/topics/non-hermitian-quasiperiodic-rings type: topic --- # Non-Hermitian Quasiperiodic Rings 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, $V_j = V e^{i2\pi\beta j + \gamma}$ [2105.03302]. - **Flat-band/diamond models:** Three-site-per-cell rings with non-Hermitian potentials, e.g., $V_{11}(n) = V_0 e^{i\theta} \cos (2\pi\beta n + \phi + i\epsilon)$ [2504.08760, 2311.03166]. - **Non-Abelian generalizations:** Spin-1/2 chains with SU(2) gauge fields, nonreciprocal hopping, and complex onsite potential, $H = \sum_{n=1}^{L} [J_L c_n^\dagger \sigma_0 c_{n+1} + J_R c_{n+1}^\dagger \sigma_0 c_n + V_n c_n^\dagger (\dots) c_n]$ [2302.05710, 2507.12176]. - **Other examples:** Non-Hermitian Maryland model [2106.00230], Fibonacci gain/loss rings [2601.05743], GAAH-model rings [2306.11084, 2404.12266, 2410.04469]. 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 [2404.12266]. - For non-Abelian chains, mobility rings emerge only when SU(2) gauge fields couple nontrivially to nonreciprocal hopping (i.e., $\sin\theta_l \sin\theta_r \neq 0$) [2507.12176]. - **Mobility lines:** In certain flat-band models, real segment(s) in the complex plane host extended states (e.g., $E_c \in [-2,-1] \cup [0,1]$) [2504.08760]. - **Critical window and multifractality:** Absent in most non-Hermitian models except for special flux-correlated modulations in diamond/flat-band rings [2311.03166]. In the infinite-mosaic/series limit, nested mobility rings collapse into a single topological ring (winding $w=1$), merging all point gaps [2410.04469]. ## 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 [2105.03302]: $(W_+,W_-) = (0,0)$ (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 $E$ [2404.12266, 2507.12176]. - 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) [2105.03302]. - **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 $max_n IPR_n>0$, $min_n IPR_n\approx0$ [2105.03302, 2507.12176]. - **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