Information Theoretic Signatures of Localization and Mobility Edges in Quasiperiodic Systems

Abstract

We investigate localization transitions and mobility edge phenomena in one-dimensional quasiperiodic lattice models using an information theoretic framework based on the Tsallis entropy of single particle eigenstates.We employ the Tsallis entropy as a continuous, normalized functional of wavefunction amplitudes, where the entropic index $q$ provides a tunable sensitivity to different regions of the probability distribution, enhancing the contribution of localized peaks ($q>1$) or extended components ($q<1$). Building on this framework, we introduce an entropy-gradient susceptibility defined from the energy dependence of the Tsallis entropy, which probes variations in eigenstate structure across the spectrum. We show that this quantity clearly distinguishes global localization transitions from mobility edge physics. In the Aubry Andre model, where all eigenstates undergo a uniform transition, the entropy varies smoothly, resulting in a broad crossover in the susceptibility. In contrast, in systems hosting mobility edges, including a quasiperiodically modulated Su Schrieffer Heeger chain and the generalized Aubry Andre model, the coexistence of localized and extended states produces sharp spectral variations, leading to a pronounced and system size independent peak. The qualitative behavior persists over a broad range of the entropic parameter $q$, with systematic variations reflecting its role as a tunable probe of spectral structure. Our results establish an information theoretic approach that leverages the continuous $q$ dependence of Tsallis entropy to construct a derivative based measure of spectral heterogeneity, providing a complementary and physically transparent diagnostic of mobility edge phenomena beyond conventional state resolved measures.

Figures (14)

• Figure 1: Normalized Tsallis entropy $\tilde{S_q}$ as a function of quasiperiodic potential strength $\lambda$ for the Aubry–André (AA) model at different entropic orders $q$, with system size $L=600$. The entropy decreases monotonically with increasing $\lambda$, reflecting the simultaneous localization of all eigenstates at the critical point $\lambda_c= 2t$
• Figure 2: Correlation between normalized Tsallis entropy $\tilde{S}_q$ and inverse participation ratio (IPR) for a representative parameter set in a SSH Hamiltonian for different q values. States with large IPR correspond to low entropy, confirming consistency between entropic and conventional localization measures.
• Figure 3: The scaling behaviour of (a) $\tilde{S_q}$ at $q=1.5$ and (b) IPR as a function of $\lambda$ for different $L$ for AA model, where in the case of $\tilde{S_q}$ scaling the $\lambda_c$ is denoted as the point where $\tilde{S_q}$ starts decreasing from maximum value $1$ and for $IPR$ scaling $\lambda_c$ is denoted as the point where the $IPR$ datas for different $L$ first collapsed.
• Figure 4: Entropy-gradient susceptibility $\chi_q(\lambda)$ of the GAA model as a function of the quasiperiodic potential strength $\lambda$ for $q=1.5,2,2.5,3$. For all entropic orders, the susceptibility develops a pronounced peak at intermediate coupling, signaling the coexistence of localized and extended eigenstates associated with a mobility edge. While the peak height and width depend systematically on $q$, its position remains nearly unchanged, demonstrating that the mobility-edge signature is robust for $q \ge 1$, with systematic deviations for $q < 1$ due to enhanced rare-state weighting.
• Figure 5: Entropy-gradient susceptibility $\chi_q(\lambda)$ of the GAA model as a function of the quasiperiodic potential strength $\lambda$ for $q=1.5$ for $L=600, 800, 1000$. The peak height and width do not depend on $L$; its position also remains unchanged, demonstrating that the mobility-edge signature is robust against system size.