Non-Hermitian polarization quantized by time-reversal symmetry

It is well known that in one-dimensional (1D) crystalline insulators, the electric polarization is a manifestation of Berry phase, which can not be quantized by time-reversal symmetry (TRS) as in Hermitian physics TRS does not induce any topological phase in one dimension. In this paper we report that even though associated with complex eigenenergies a 1D non-Hermitian insulator obeying only TRS is capable of presenting quantized bulk polarization. The underlying physical reason is unveiled: TRS guarantees the complex energies to come in pair (E, E*), and the corresponding decaying and amplifying wave functions also come in pair and have the same variation rate, hence, giving rise to a stable wannier center. The electron transport is performed by means of charge pumping process, which verifies the physical mechanism above. At last, we discuss the possible experimental implementation of the proposed model by means of twisted-π gauge flux.