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

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.

💡 Research Summary

In this work the authors investigate the emergence of non‑Hermitian topology in multi‑terminal transport through a quantum spin‑Hall (QSH) device described by the Bernevig‑Hughes‑Zhang (BHZ) model. The study begins by constructing a square‑shaped scattering region in the inverted regime (M/B > 0) that hosts a single Kramers pair of helical edge states. N semi‑infinite leads are attached uniformly around the perimeter, each lead being a copy of the BHZ lattice to ensure mode matching.

First, the authors consider the case of unpolarized leads (μ↑ = μ↓ = 0). Time‑reversal symmetry (TRS) is preserved, the transmission matrix is symmetric (Tij = Tji), and the resulting conductance matrix G is Hermitian. This reproduces the familiar reciprocal transport of a QSH bar.

Next, they introduce spin‑selective coupling by shifting the onsite energy of the spin‑up block in the leads (μ↑ ≠ 0, μ↓ = 0). This creates an imbalance between the two time‑reversed transport channels: one helical branch couples more strongly to the contacts than the other. Although the scattering region itself remains TRS‑invariant, the conductance matrix becomes non‑symmetric (Gij ≠ Gji) and thus non‑Hermitian. The structure of G directly maps onto the Hatano‑Nelson (HN) model, a prototypical non‑Hermitian chain with asymmetric nearest‑neighbour hoppings.

To diagnose the non‑Hermitian phase the authors employ three complementary indicators. (i) The adjacent‑lead conductance asymmetry ΔG = |G12 − G21|, which is experimentally accessible through simple two‑terminal measurements. (ii) The polar‑decomposition invariant wPD, obtained by subtracting the average diagonal element, performing a polar decomposition Ĝ = UP, and evaluating Tr