Localization Tensor Revisited: Geometric-Probabilistic Foundations and a Structure-Factor Criterion under Periodic Boundaries

Abstract

We revisit the localization tensor (LT) from geometric and probabilistic perspectives and construct extensions that are naturally compatible with periodic boundary conditions (PBC), without redefining the position operator. In open boundary conditions, we show that the LT can be written exactly as the covariance of a bivariate probability distribution built from density-density correlations. This leads to two conceptually distinct extensions to PBC: (i) a geometric one based on the Riemannian center (Frechet mean) on the circle, and (ii) a metric-free one based on the mutual information I, which treats the configuration space purely as a probability space. We then relate the LT to the static structure factor by identifying the diagonal part, Cpp, as a "localization function" C(p), whose small-momentum behavior determines the LT in the thermodynamic limit. This clarifies why the LT is sensitive to transitions out of the extended phase but by itself cannot distinguish Anderson-type localization from dimerization: both share the same low-momentum asymptotics. We show that the finite-momentum behavior of C(p), together with an inverse participation ratio (IPR)-based upper bound valid in localized phases, provides a sharp criterion that discriminates localization from dimerization. These results are illustrated on the Su-Schrieffer-Heeger and Aubry-Andre models, with and without interactions, and suggest that structure factor-based probes offer robust and experimentally accessible diagnostics of localized and dimerized phases under PBC.

Figures (5)

• Figure 1: The various definitions of the localization tensor $\lambda$ for different values of the dimerization parameter $\delta$ in the Su-Schrieffer-Heeger model (\ref{['eqHd']}) are shown. (a) We set $N = 200$ and $M = N/2$ (half filling), using different line styles to distinguish between the definitions $\lambda_r$, $\lambda_{\mathrm{LF}}$, and $\lambda_{\mathrm{RC}}$. (b) Different line styles represent the results for $\lambda_{\mathrm{MI}}$ with varying $N$. (c) The results for $\lambda_{\mathrm{LF}}$ with different $N$.
• Figure 2: The different definitions of localization tensor $\lambda$ for various values of the localization parameter $\delta$ in Aubry-Andrë model (\ref{['eqHAA']}). We take $\beta=(\sqrt{5}-1)/2$, $V=0$ (non-interacting) and $J=1$. (a) Here $N=610$, filling $M=N/2$, different types of lines are used to distinguish different definitions of $\lambda_{r}$, $\lambda_{\rm LF}$ and $\lambda_{\rm RC}$. (b) The results of $\lambda_{\rm MI}$ under different $N$.
• Figure 3: The behavior of several different localization tensors in the ground state of the AA model described by equation (\ref{['eqHAA']}). Our calculations are based on exact diagonalization and are performed at half filling. We set $J = 1$, $V = 0.25$, $\beta = (\sqrt{5}-1)/2$ and vary $\delta$ to drive the localization phase transition. Panel (a-c) are the three different type of localization tensor discussed in the main text, and Panle (d) shows the scaling behavior to confirm the convergence of the numerical calculations. We choose $\delta = 3.0$ to represent the localized phase while $\delta = 1.0$ to the extended phase.
• Figure 4: The value of localization function $\tilde{C}(p)$ varying with $\delta$ under different models, and we use different types of lines to distinguish $\tilde{C}(p)$ of different $\delta$. (a) The SSH model (\ref{['eqHd']}) with the size $N=610$ and $M=N/2$. Inset: The asymptotic behavior of curves approaching $0^+$. (b) The AA model (\ref{['eqHAA']}) with the size $N=610$ and $M=N/2$. We take $\beta=(\sqrt{5}-1)/2$, $V=0$ and $J=1$. Inset: The asymptotic behavior of $\tilde{C}(p)$ when $p \to 0^+$.
• Figure 5: The localization function of AA model (\ref{['eqHAA']}) under different $\delta$ with the size $N=610$ and $M=20$, and we use different types of lines to distinguish different $\delta$, the curve of $\tilde{C}(p)$ are with graphical markers, while the unmarked lines give the upper bounds(\ref{['eqCpBound']}) of $\tilde{C}(p)$. Here $\beta=(\sqrt{5}-1)/2$, $V=0$ and $J=1$.