Single impurity-induced localization transitions in electronic systems

Abstract

Anderson localization is a fundamental phenomenon in disordered quantum systems, where transport is suppressed by wave interference from extensive randomness. Moving beyond traditional multi-impurity scenarios, we investigate impurity-induced localization phenomena in low-dimensional tight-binding systems by focusing on the properties of impurity-generated bound states. By introducing a single on-site impurity into an otherwise extended lattice, we demonstrate that the impurity can host a bound state whose spatial character undergoes a transition from extended to localized as the impurity strength surpasses a critical value. This transition pertains solely to the impurity state, while the bulk states of the host system remain extended. We characterize the localization behavior by analyzing two distinct spatial profiles of the bound states: one with symmetric decay and another with exponential decay from the impurity site. Our results highlight how a local perturbation can induce nontrivial localization behavior at the level of individual eigenstates, without implying a global localization transition of the underlying electronic system.

Figures (6)

• Figure 1: (Color online) The IPR of the bound state as a function of the impurity strength $\varepsilon_{0}$ of the 1D single-impurity model. In the presence of an infinitesimal impurity strength, the eigenstate is localized, corresponding to the size-independent IPR $\approx 1$.
• Figure 2: (Color online) Log-linear scale:
• Figure 3: (Color online) (a) The IPR (log-linear scale) of the bound state as a function of the impurity strength $\varepsilon_{0}$ in log-linear scale of the 2D single-impurity model. The nature of the bound state changes from extended to localized by tuning the impurity strength. In the inset, we enlarge the IPR region around $\varepsilon_0=1-2$ to clarify the finite-size critical regime. (b) The IPR (log-log scale) of the bound state versus system size $L$ for fixed impurity strength. The data is very well fitted by $\text{IPR}\propto L^{\alpha}$. The IPR is approximately equal to $1$ and is independent of $L$ for $\varepsilon_0=3.0$ ($\alpha=0$), varies algebraically with $L$ for $\varepsilon_0=1.5$ ($\alpha\approx0$), and decays for $\varepsilon_0=0.5$ ($\alpha<0$), corresponding to localized, critical, and extended behavior of the state, respectively.
• Figure 4: The spatial distributions of the impurity-induced bound state of the
• Figure 5: (Color online) The IPR of the impurity induced state as a function of the impurity strength $\varepsilon_{0}$ in 3D. The inset shows (on log-log scale) the IPR of the bound state as a function of linear system size $L$ at fixed impurity strength. A fit (dashed blue lines) $\text{IPR}\approx L^{\alpha}$ reveals distinct scaling exponent: for $\varepsilon_0=6.0$, $\alpha=0$ in localized regime; for $\varepsilon_0=1.5$, $\alpha\approx 0$ in the critical regime; and for $\varepsilon_0=0.5$, $\alpha<0$ in the extended regime.