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.

Figures (20)

• Figure 1: Energy of edge (black) and bulk (red for $k_y\not=0$ and magenta for $k_y=0$) Hamiltonian $h_s$ eigenstates as a function of $k_x$. Edge and bulk eigenstates merge at the cutoff wave numbers $\pm k_1=\pm 0.0096$ nm${}^{-1}$ (for strip width $L_1=120$nm, dashed black) and $\pm k_2=\pm 0.021$ nm${}^{-1}$ (for strip width $L_2=500$ nm, solid black), marked by round black dots. The corresponding energies are $E_1=10.58$ meV and $E_2=12.59$ meV. As a comparison, results from the numerical diagonalization of tight-binding model of Sec. \ref{['tbsec']} are also plotted as black triangles for $L=120$ nm and black squares for $L=500$ nm.
• Figure 2: Cutoff wave numbers $k$ (magenta points), at which edge and bulk states merge, for several values of the strip width $L\in[100,900]$ nm, for the BHZ Hamiltonian $h_s$. The black curve corresponds to the rational fitting \ref{['kcutoffL']} of these points.
• Figure 3: Density distribution $|\psi_{\uparrow E}|^2+|\psi_{\uparrow H}|^2$ of the first (conduction) spin up ($s= 1$) edge state of the BHZ Hamiltonian $h_s$ for a strip width of $L=120$ nm as a function of $y$. The solid line corresponds to $k_x=0$ nm${}^{-1}$ ($E(0)\simeq 8.88$ meV), the dotted line to $k=0.005$ nm${}^{-1}$ and the dashed line to $k=-0.005$ nm${}^{-1}$ ($E(\pm 0.005)\simeq 9.48$ meV).
• Figure 4: Hamiltonian spectra for two strip sizes $L=100$ and $L=400$ nm as a function of the wavevector component $k_x$ for the material parameters in Table \ref{['tabla']}. Conduction states in red and valence states in blue color. The four near-gap ("edge to be") states are indicated in black color (solid and dashed). The gap closes as $L$ increases.
• Figure 5: Energy gap (logarithmic scale) of the of edge states as a function of the strip width $L$. The gap at the $\Gamma$ point $E_g^\Gamma$ (black color) shows an exponential decay, whereas the minimum gap $E_g$ (red color) exhibits oscillations with sudden drops for some critical strip widths values ($L_c\simeq 100$ and 220 nm). This time we choose the SOC parameter $\Delta_z~=~10$ meV for computational convenience.