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

TL;DR

This paper extends the theory of electric polarization to one-dimensional non-Hermitian systems, revealing how polarization can be quantized under certain symmetries and energy gap types, leading to new topological phases.

Contribution

It introduces a generalized polarization framework for non-Hermitian chains and demonstrates polarization quantization depending on gap types and symmetries.

Findings

Biorthogonal Wilson loop is unitary in the thermodynamic limit.

Polarization quantization depends on the type of energy gap (real or imaginary).

Numerical models support the theoretical predictions.

Abstract

We generalize the modern theory of electric polarization to the case of one-dimensional non-Hermitian systems with 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, the associated Wilson-loop operator is biorthogonally unitary in the thermodynamic limit, thereby leading to real-valued electronic positions that allow for a clean definition of polarization. Non-Hermitian polarization can be quantized in the presence of certain symmetries, as for Hermitian insulators. Different from the latter case, though, 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 counter-intuitive example is the 1D non-Hermitian chain with time-reversal symmetry only, where non-Hermitian polarization is quantized in presence of an imaginary line-gap. We propose two specific models to provide numerical evidence supporting our findings.