Localization and entanglement characterization of edge states in HgTe quantum wells in a finite strip geometry

TL;DR

This paper uses quantum information measures like entanglement entropy and localization metrics to analyze and distinguish edge states from bulk states in HgTe quantum wells, revealing their spin and spatial properties.

Contribution

It introduces a novel approach combining localization and entanglement measures to characterize edge states in topological insulators within finite strip geometries.

Findings

Edge states show spin polarization and spatial confinement at boundaries.

Localization measures identify momentum cutoffs for edge-bulk state transition.

Entanglement entropy captures spin structure and variability of edge states.

Abstract

Quantum information measures are proposed to analyze the structure of near-gap electronic states in HgTe quantum wells in a strip geometry $(x,y)\in (-\infty,\infty)\times [0,L]$ of finite width $L$. This allows us to establish criteria for distinguishing edge from bulk states in the topological insulator phase, including the transition region and cutoff of the wave number $k_x$ where edge states degenerate with bulk states. Qualitative and quantitative information on the near-gap Hamiltonian eigenstates, obtained by tight-binding calculations, is extracted from localization measures, like the inverse participation ratio (IPR), entanglement entropies of the reduced density matrix (RDM) to the spin sector --measuring quantum correlations due to the spin-orbit coupling (SOC)-- and from correlation functions for a $y$-space partition. The analysis of IPR and entanglement entropies in terms of spin, wave number $k_x$ and position $y$, evidences a spin polarization structure and spatial confinement of near-gap wave functions at the boundaries $y=0,L$ and low $k_x$, as correspond to helical edge states. IPR localization measures provide momentum $k_x$ cutoffs from which near-gap states are no longer localized at the boundaries of the sample and become part of the bulk. Below this $k_x$-point cutoff, the entanglement entropy and the spin probabilities of the RDM also capture the spin polarization structure of edge states and exhibit a higher variability compared to the relatively low entropy of the bulk state region. For a real-space partition, the edge-state region in momentum space exhibits lower correlation modulus, but higher correlation arguments, than the bulk-state region.