Non-Hermitian topology of quantum spin-Hall systems to detect edge-state polarization

TL;DR

The paper investigates whether non-Hermitian topology can emerge in time-reversal-symmetric quantum spin-Hall systems modeled by the BHZ Hamiltonian. It shows that non-Hermitian transport arises only when there is directional imbalance in multi-terminal scattering, which can occur via spin-selective lead coupling or an out-of-plane Zeeman field, yielding a Hatano–Nelson–like conductance matrix with $T_{ij}\neq T_{ji}$ and a non-Hermitian skin effect. The authors employ the summed probability density of right eigenvectors and the polar-decomposition invariant $w_{\mathrm{PD}}$ as complementary diagnostics to quantify edge-state polarization and non-Hermitian topology. They further demonstrate that spin-mixing disorder can suppress the skin effect, driving a crossover to a trivial phase, while in-plane fields preserve reciprocity and out-of-plane fields induce nonreciprocity depending on edge-state polarization. Collectively, these results provide a transport-based method to probe edge-state spin polarization and contact selectivity in QSH devices and establish non-Hermitian skin effect as a diagnostic tool in this setting, with implications for spintronic applications.

Abstract

We study the non-Hermitian topology of multi-terminal transport in a quantum spin-Hall device described by the Bernevig-Hughes-Zhang model. We show that breaking time-reversal symmetry alone does not imply non-reciprocal transport or a non-Hermitian conductance matrix. Instead, non-Hermitian topology arises only when transport becomes directionally imbalanced. We identify two distinct mechanisms that generate such a response: spin-selective coupling at the contacts and an out-of-plane Zeeman field that unbalances the counter-propagating helical edge modes. We show, for unpolarized leads, that the spin polarization-dependent response to Zeeman fields, provides a transport-based probe of the intrinsic spin polarization of the helical edge states. Moreover, we demonstrate that non-Hermitian skin effect is more sensitive than conductance elements to detect the spin polarization of the edge states. Our results clarify the conditions required for non-Hermitian topology in quantum spin-Hall transport and establish non-Hermitian skin effect as a diagnostic tool for spin-selective coupling and edge-state polarization.

Figures (5)

• Figure 1: Setup and emergence of non-Hermiticity with spin-polarized leads. (a) Square-shaped QSH device (blue) described by the BHZ model, connected to eight terminals (red) distributed along its perimeter. The direction of helical edge states is shown using black arrows. (b) Multi-terminal conductance matrix $G_{ij}$ for unpolarized leads; $G$ is Hermitian, consistent with time-reversal symmetry. (c,d) Conductance matrices for spin-polarized leads with ($\mu_{\uparrow}=+5$, $\mu_{\downarrow}= 0$) and ($\mu_{\uparrow}=0$, $\mu_{\downarrow}=5$), respectively. The asymmetry between $G_{ij}$ and $G_{ji}$ indicates the onset of spin-selective, directional transport and renders $G$ non-Hermitian even though the underlying Hamiltonian is Hermitian. (e,f) Corresponding summed probability density (SPD) of the right-eigenvectors of $G$, $\mathrm{SPD}=\sum_n|v_n|^2$, plotted on a logarithmic scale along the ordered lead index. The exponential accumulation of SPD toward opposite boundaries in the two cases shows the spin-dependent non-Hermitian skin effect.
• Figure 2: Disorder-averaged transport signatures of spin-selective coupling. (a) Disorder-averaged adjacent-lead conductance asymmetry $\langle \Delta G \rangle = |\langle G_{12}\rangle-\langle G_{21}\rangle|$ (left axis) and polar-decomposition invariant $\langle w_{\rm PD}\rangle$ (right axis) plotted on a logarithmic scale, shown versus the lead-polarization parameter $\mu_{\uparrow}$ for a device with $N=32$ leads. (b) Disorder-averaged non-Hermitian skin-effect strength $\langle \Delta \mathrm{SPD}\rangle$, defined from the coarse-grained boundary imbalance of the summed probability density (SPD) (see Eq. \ref{['eq:dSPDm']} with $m=3$), plotted on a logarithmic scale. (c) Noise-normalized step sensitivity $\mathcal{S}$ of $\langle \Delta G\rangle$ and $\langle \Delta \mathrm{SPD}\rangle$, quantifying the statistical significance of their response to incremental changes in $\mu_{\uparrow}$. All quantities are averaged over $n_l=1000$ disorder realizations; error bars denote the standard error of the mean; $\delta/|M| = 0.1$. Sensitivities are evaluated between adjacent polarization points $\mu_{\uparrow}$ and plotted on a logarithmic scale.
• Figure 3: Zeeman-field response for probing the spin polarization of QSH edge states. (a) Adjacent-lead conductances $G_{12}$ and $G_{21}$ (in units of $e^2/h$) as a function of an in-plane Zeeman field $B_x$ for a 32-terminal device with unpolarized leads ($\mu_{\uparrow}=\mu_{\downarrow}=0$). Although $B_x$ breaks time-reversal symmetry and suppresses the quantized edge conductance, reciprocity ($G_{12}=G_{21}$) is preserved. (b) Corresponding SPD profile of the conductance matrix, which remains uniform, confirming that the conductance matrix stays topologically trivial. (c) Adjacent-lead conductances $G_{12}$ and $G_{21}$ (in units of $e^2/h$) for unpolarized leads under an out-of-plane field $B_z$. Increasing $B_z$ produces pronounced non-reciprocity. (d) Corresponding SPD profile, showing the emergence of a non-Hermitian topological phase with strong boundary localization once one helical branch is pushed away from the Fermi level (see Appendix \ref{['app:ribbon']}).
• Figure 4: Impact of spin–mixing disorder on non-Hermitian transport. (a,b) Disorder-averaged adjacent-lead conductances ($G_{12}$, $G_{21}$ in units of $e^2/h$) for a 32-terminal device as a function of the disorder strength $\delta/|M|$ for unpolarized ($\mu_{\uparrow}=\mu_{\downarrow}=0$) and polarized ($\mu_{\uparrow}= 4,$$\mu_{\downarrow}=0$) leads, respectively. The curves are obtained from the disorder-averaged conductance matrix $\langle {G} \rangle$. (c,d) Corresponding summed probability density (SPD) profiles for the same parameters, indicating the presence or absence of the non-Hermitian skin effect. (e,f) Disorder-averaged polar-decomposition invariant $\langle w_{\mathrm{PD}} \rangle$ plotted against the disorder strength $\delta/|M|$. Panels (a,c,e) correspond to unpolarized leads; panels (b,d,f) correspond to polarized leads.
• Figure 5: Ribbon band structure for the discretized BHZ model on a square lattice (infinite along $y$ and 100 unit cells along $x$). A color scale depicting the probability density of states $\Psi$ integrated over the first 50 unit cells is used such that states localized on the left and right ends of the ribbon are shown in blue and red, respectively. Bulk states are shown in a faded green color. (a) Zeeman field $B_z$ set to zero: helical edge states in blue and red traverse the bulk gap, with one pair on each edge. (b) $B_z=0.5$: one branch of the helical pair is pushed away from the Fermi level, leaving the opposite branch at the Fermi level ($E = 0$).