Multiple intermediate phases in the interpolating Aubry-André-Fibonacci model

We investigate a generalized interpolating Aubry-André-Fibonacci (IAAF) model with p-wave superconducting pairing, focusing on its localization and topological properties. Within the Aubry-André limit, we demonstrate that the system experiences transitions from a pure phase, either extended or critical, to a variety of intermediate phases and ultimately enters a localized phase with increasing potential strength. These intermediate phases include those with coexisting extended and localized states, extended and critical states, localized and critical states, and a mix of extended, critical, and localized states. Each intermediate phase exhibits at least one type of mobility edge separating different states. As the system approaches the Fibonacci limit, both the extended and localized phases diminish, and the system tends towards a critical phase. Furthermore, the model undergoes a transition from topologically nontrivial to trivial phase as potential strength increases.

Figure 1

Schematic of the quasiperiodically modulated onsite potential [Eq. (2)] for several values of $\beta =0.01$, 0.2, 5, 1000 (light to deep blue curves).

Figure 2

Phase diagram that varies with $(\mathrm{\Delta },\lambda )$ in terms of $\eta$ (a) and $\langle D\rangle$ (b). The pure phase (intermediate phase) is corresponding to the blue (red) region in panel (a). The pure phase is further distinguished into extended (deep red region) and localized phase (deep blue region) in panel (b). The critical phase is persists along $\mathrm{\Delta }=1$ with $\lambda \lesssim 0.57$, marked in red elliptic. Here, we set the parameter $\beta =0.01$ and $L=610$.

Figure 3

(a)–(c) The fractal dimension $D^{\left(n\right)}$ versus $\lambda$, where $n$ denotes $n\text{−}\mathrm{th}$ eigenstate of BdG Hamiltonian. The parameter $\mathrm{\Delta }$ is 0.5 (a), 1 (b), and 1.5 (c), respectively. Other parameters are $\beta =0.01$ and $L=610$.

Figure 4

(a)–(c) The fractal dimension $D^{\left(n\right)}$ for different $L$ at fixed $\mathrm{\Delta }=0.5$ and $\lambda =0.2$ (a), $\mathrm{\Delta }=1$ and $\lambda =0.2$ (b), and $\mathrm{\Delta }=0.5$ and $\lambda =5.8$ (c), where $n$ denotes $n\mathrm{th}$ eigenstate. (d) Finite-size extrapolation of $\langle D\rangle$ as a function $1/\log \left(2\mathrm{L}\right)$.

Figure 5

(a)–(c) The fractal dimension $D^{\left(n\right)}$ for different $L$ at fixed $\mathrm{\Delta }=0.5$ and $\lambda =1.8$ (a), $\mathrm{\Delta }=0.5$ and $\lambda =2.3$ (b), and $\mathrm{\Delta }=1.5$ and $\lambda =3.5$ (c), where $n$ denotes $n\mathrm{th}$ eigenstate. (d–f) Finite-size extrapolation of $\langle D\rangle$ as a function $1/\log \left(2\mathrm{L}\right)$ averaged over the different state zones in panels (a)–(c).

Figure 6

(a) The fractal dimension $\langle D\rangle$ averaged over all the BdG quasiparticle spectrum versus $\mathrm{\Delta }$. (b) The fractal dimension $D^{\left(n\right)}$ for different $L$ at $\mathrm{\Delta }=1$, where $n$ denotes $n\mathrm{th}$ eigenstate. The parameter $\lambda =0.2$.

Figure 7

(a), (b) Phase diagram of variable $\eta$ versus ($\mathrm{\Delta },\lambda$). The red dashed line corresponds to $\mathrm{\Delta }=0.5$. (c), (d) The fractal dimension $D^{\left(n\right)}$ versus $\lambda$, where $n$ denotes $n\mathrm{th}$ eigenstate of BdG Hamiltonian when $\mathrm{\Delta }=0.5$. The green dashed lines corresponds to $\lambda =0.2$ and 5.8, respectively. Here, the parameter $\beta =5$ (a), (c), 50 (b), (d), and $L=610$.

Figure 8

The fractal dimension $D^{\left(n\right)}$ for different $L$ at fixed $\beta =0.01$ (a), $\beta =5$ (b), and $\mathrm{\Delta }=50$ (c), where $n$ denotes $n\mathrm{th}$ eigenstate. Other parameters $\mathrm{\Delta }=0.5$ and $\lambda =0.2$ (5.8) for the upper (lower) panel of each figure.

Figure 9

(a) Phase diagram that varies with $(\beta ,\lambda )$ in terms of $Z_2$ topological invariant $M$. (b) Energy spectrum versus $\lambda$ for $L=610$. The color coding correlates to the fractal dimension $D^{\left(n\right)}$. (c) The spatial distributions of $\varphi$ and $\psi$ for the lowest excitation with $V=1.0$ (upper panel) and $V=1.5$ (lower panel) where $j$ is $j\mathrm{th}$ site. The parameter $\beta =5$ for (b), (c). Open boundary condition.

Figure 10

Finite-size extrapolation of $\langle D\rangle$ as a function $1/\log \left(2\mathrm{L}\right)$, where $\lambda =2\mathrm{\Delta }+2$ (upper panel) and $\lambda =2\mathrm{\Delta }+2.5$ (lower panel).

Figure 11

(a) The fractal dimension $D^{\left(n\right)}$ for different $L$, where $n$ denotes $n\mathrm{th}$ eigenstate. (b) Finite-size extrapolation of $\langle D\rangle$ as a function $1/\log \left(2\mathrm{L}\right)$. (c) The probability distribution of selected ($n\mathrm{th}$) eigenstate for $L=10946$, where index $j$ denotes $j\mathrm{th}$ site. The parameter $\lambda =0.2$ (5.8) for the upper (lower) panel of each figure, $\mathrm{\Delta }=0.5, \beta ={10}^4$.

Figure 12

The fractal dimension $D^{\left(n\right)}$ for different $L$, where $n$ denotes the $n\mathrm{th}$ eigenstate and $\beta =50$ (a), $\beta ={10}^3$ (c), $\beta ={10}^4$ (e). The parameters $\lambda =10$ (100) for the upper (lower) panel of (a), $\lambda =100$ (1000) for the upper (lower) panel of (c), (e). (b), (d), (f) The probability distributions of selected ($n\mathrm{th}$) eigenstate for $L=10946$, where $j$ denotes the $j\mathrm{th}$ lattice site. Other parameters are same with panels (a), (c), (e), respectively.