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

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.