Symmetry indicators for inversion-symmetric non-Hermitian topological band structures

We characterize non-Hermitian band structures by symmetry indicator topological invariants. Enabled by crystalline inversion symmetry, these indicators allow us to short-cut the calculation of conventional non-Hermitian topological invariants. In particular, we express the three-dimensional winding number of point-gapped non-Hermitian systems, which is defined as an integral over the whole Brillouin zone, in terms of symmetry eigenvalues at high-symmetry momenta. Furthermore, for time-reversal symmetric non-Hermitian topological insulators, we find that symmetry indicators characterize the associated Chern-Simons form, whose evaluation usually requires a computationally expensive choice of smooth gauge. In each case, we discuss the non-Hermitian surface states associated with nontrivial symmetry indicators.

Figure 1

Inversion symmetry indicator for $w_{3\mathrm{D}}$. (a),(c) Bulk (gray) and surface (blue) spectra in the complex plane for a non-Hermitian topological insulator with $w_{3\mathrm{D}}=0$ (a) and $w_{3\mathrm{D}}=1$ (c) [43]. (b),(d) Band structure of the corresponding Hermitian doubles along high-symmetry lines of the Brillouin zone. All bands are twofold-degenerate, and each pair has equal inversion eigenvalues at the inversion-symmetric momenta (the inversion eigenvalues of the occupied bands are shown in red). By Eq. (13), the presence (absence) of a double band inversion in (d) [in (b)] indicates the presence (absence) of anomalous non-Hermitian surface states in (c) [in (a)].