Generalized Aubry-André-Harper model with power-law quasiperiodic potentials

TL;DR

We address localization in a generalized Aubry–André–Harper model with a power-law quasiperiodic potential and nonreciprocal hopping. The approach diagonalizes the single-particle Hamiltonian via BdG-like formalism and analyzes IPR, NPR, fractal dimensions, and DOS across Hermitian and non-Hermitian regimes for $p=1$–$4$. In the Hermitian case, $p\geq3$ yields a mixed phase with mobility edges and universal high-IPR state positions given by $x_n = nβ - ⌊nβ⌋$. In the non-Hermitian case, PT transitions coincide with the extended-to-localized boundary for $p=1,2$, but align with the mixed-to-localized boundary for $p\geq3$. These results reveal how power-law quasiperiodic potentials and nonreciprocal hopping shape localization phenomena and phase structure in quasiperiodic systems, with implications for non-Hermitian transport control.

Abstract

We investigate a generalized Aubry-André-Harper (AAH) model with non-reciprocal hopping and power-law quasiperiodic potentials $V(i) = V\left[ \cos(2πβi) \right]^p$. Our study reveals that the interplay between nonreciprocity, quasiperiodicity, and the power-law exponent $p$ gives rise to a variety of phase transitions and localization phenomena. In the Hermitian case, the system undergoes a direct transition from extended to localized phases for $p=1, 2$, while for \(p \geq 3\), an intermediate mixed phase emerges, characterized by the coexistence of extended and localized states and the presence of mobility edges. Importantly, we find that high inverse participation ratio (IPR) states appear at specific energy levels, whose positions are accurately described by the universal relation \(x_n = nβ- \lfloor nβ\rfloor\), with a mirror-symmetric spatial distribution. In the non-Hermitian regime, the energy spectrum becomes complex and the \(\mathcal{PT}\) transition coincides with the extended-to-localized phase boundary for \(p = 1, 2\), whereas for \(p \geq 3\), \(\mathcal{PT}\)-symmetry breaking occurs at the mixed-to-localized phase transition. This work reveals how power-law quasiperiodic potentials and non-reciprocal hopping govern phase transitions, providing new insight into localization phenomena of quasiperiodic systems.

Figures (7)

• Figure 1: Fractal dimensions of the Hermitian AAH model with $L = 4000$. Panels (a)-(d) show the fractal dimensions $D_2$ of eigenstates for exponents $p = 1, 2, 3,$ and $4$, respectively. The white dashed lines in (c) and (d) indicate the locations of eigenstates exhibiting large IPR values as explained in the main text.
• Figure 2: Energy differences in the $p = 3$ Hermitian AAH model with $L = 4000$. Panels (a)-(c) display the energy difference $\Delta E_{2m-1}=E_{2m}-E_{2m-1}$ (red dots) and $\Delta E_{2m}=E_{2m+1}-E_{2m}$ (blue dots) as functions of the eigenstate index $m/L$ for $V = 0.5$, $1.33$, and $3$ respectively. In panel (b), the two black dashed lines mark the positions of the mobility edges at $m/L = 0.14575$ and $0.8545$.
• Figure 3: The $\text{IPR}^{(m)}$ of Hermitian AAH model as a function of the eigenstate index $m/L$ for different values of $p$ and $V$ at $L = 4000$. (a) $p = 1$, $V = 1$; (b) $p = 2$, $V = 1$; (c) $p = 3$, $V = 1$; (d) $p = 4$, $V = 2$. The distinct vertical lines indicate the positions of eigenstates with high IPR values.
• Figure 4: Phase diagram of the non-Hermitian AAH model. (a) Phase diagram obtained from the averaged inverse participation ratio $\overline{\text{IPR}}$ for $L = 987$ and $p = 1$; the white dashed line denotes $g = 0.3$. (b) $\overline{\text{IPR}}$ , $\overline{\text{NPR}}$, and $\zeta$ as functions of $V$ at $g = 0.3, p=1$; the left vertical axis corresponds to $\overline{\text{IPR}}$ and $\overline{\text{NPR}}$, while the right vertical axis corresponds to $\zeta$, The black dashed line marks the phase transition from extended to localized states at $V_c = 2.7$. (c) Phase diagram based on $\overline{\text{IPR}}$ for $L = 987$, $p = 3$; the white dashed line indicates $g = 0.3$. (d) $\overline{\text{IPR}}$ , $\overline{\text{NPR}}$, and $\zeta$ as functions of $V$ at $g = 0.3, p=3$; the gray-shaded region $1.8 < V < 3.55$ denotes the mixed phase where extended and localized states coexist.
• Figure 5: Spectral properties of the non-Hermitian AAH model at $L = 987$ and $g = 0.3$. Panels (a)-(c) correspond to $p = 1$; (a) imaginary parts of the eigenvalues $\operatorname{Im}(E_m)$ as a function of $V$; (b) density of states (DOS) versus $V$; (c) maximum imaginary part of the eigenvalues $\max(\operatorname{Im}(\text{E}_\text{m}))$ with respect to $V$. In panels (a)-(c), the black dashed line indicates the $\mathcal{PT}$ transition at $V = 2.7$. Panels (d)-(f) show the same quantities as in (a)(b)(c) at $p=3$, where the red and black dashed lines in (d) and (f) mark the extended-to-mixed state transition and the $\mathcal{PT}$ transition at $V = 1.8$ and $V = 3.55$, respectively.