Isolated extended states and anomalous critical behavior in the generalized SSH model

Abstract

We investigate the localization properties of a generalized SSH model. Numerical and analytical results indicate the emergence of extended states protected by unbounded hopping in this model. Moreover, this protection effect is disrupted by the appearance of generalized incommensurate zeros, causing the extended phase in the system to transition into a multifractal phase. However, at the boundaries of the phase region, we still observe the existence of extended states. These extended states coincide with multifractality-enriched mobility edges, separating the multifractal phase from the localized phase. Further analysis reveals that this extended states originates from the band edge states of SSH model. In addition, these isolated extended states also influence eigenstates with nearby energies, giving rise to an anomalous extended-to-multifractal critical transition. These findings not only enrich the behavioral repertoire of eigenstates at critical points, but also offer new insights for further understanding Anderson localization and the induction of multifractal phases.

Figures (4)

• Figure 1: (a) Phase diagram of the fractal dimension with $t$ and $E$ as parameters, where $\lambda=2$ and $L=1220$, the thick black dashed line marks $t=1$. (b) Cross sections of (a) at $t=1$ for different system sizes. The inset shows the variation trend of the fractal dimension as it approaches the thermodynamic limit for the extended(blue) and localized(green) regions separated by the MEs, obtained via finite-size extrapolation, with $L=$ 288, 466, 754, 1220, 1974, 3194, 5168 and 8362. (c) The level spacings $\ln\delta^{e-o}$ (blue) and $\ln\delta^{o-e}$ (green) at $t=1$, $\lambda=2$ for a system size $L=5168$. (d1) and (d2) display the wavefunction distributions in the extended region and the localized region for $t=1$ and $\lambda=2$, respectively. Throughout, we set $J=1$ as the energy unit, the MEs are marked with thin black dashed lines.
• Figure 2: (a) Phase diagram of the fractal dimension with $t$ and $E$ as parameters, where $\lambda=2$ and $L=1220$, the thick black dashed line marks $t=3$. (b) Cross sections of (a) at $t=3$ for different system sizes. The inset shows the variation trend of the fractal dimension as it approaches the thermodynamic limit for the multifractal(blue) and localized(green) regions separated by the MMEs, obtained via finite-size extrapolation, with $L=$ 288, 466, 754, 1220, 1974, 3194, 5168 and 8362. (c) The level spacings $\ln\delta^{e-o}$ (blue) and $\ln\delta^{o-e}$ (green) at $t=3$, $\lambda=2$ for a system size $L=5168$. (d1) and (d2) display the wavefunction distributions in the multifractal region and the localized region for $t=3$ and $\lambda=2$, respectively. Throughout, we set $J=1$ as the energy unit, the MEs are marked with thin black dashed lines.
• Figure 3: (a) and (b): Detailed distribution of the fractal dimension under different boundary conditions. (c) Thermodynamic limit extrapolation of the fractal dimension for band edge states. (d) Wave functions of the band edge states.
• Figure 4: Wave functions for different $\beta$ values under the same parameters.