Electric polarization and its quantization in one-dimensional non-Hermitian chains

We generalize the modern theory of electric polarization to the case of one-dimensional (1D) non-Hermitian systems with a line-gapped spectrum. In these systems, the electronic position operator is non-Hermitian even when projected into the subspace of states below the energy gap. However, in the framework of biorthogonal quantum mechanics, the associated Wilson-loop operator is unitary in the thermodynamic limit, thereby leading to real-valued electronic positions that allow for a clear definition of polarization. Non-Hermitian polarization can be quantized in the presence of certain symmetries, as for Hermitian insulators. Differently from the latter case, however, in this regime polarization quantization depends also on the type of energy gap, which can be either real or imaginary, leading to a richer variety of topological phases. The most counterintuitive example is the 1D non-Hermitian chain with time-reversal symmetry only, where non-Hermitian polarization is quantized in the presence of an imaginary-line gap. We propose two specific models to provide numerical evidence supporting our findings.