Electric conductivity in non-Hermitian holography

Abstract

We study the phase structure and charge transport at finite temperature and chemical potential in the non-Hermitian \mathcal{PT}𝒫𝒯-symmetric holographic model of [SciPost Phys. 9, 032 (2020)]. The non-Hermitian \mathcal{PT}𝒫𝒯-symmetric deformation is realized by promoting the parameter of a global U(1) symmetry to a complex number. Depending on the strength of the deformation, we find three phases: stable \mathcal{PT}𝒫𝒯-symmetric phase, unstable \mathcal{PT}𝒫𝒯-symmetric phase, and an unstable \mathcal{PT}𝒫𝒯-symmetry broken phase. In the three phases, the square of the condensate and also the spectral weight of the AC conductivity at zero frequency are, respectively, positive, negative, and complex. We check that the Ferrell-Glover-Tinkham sum rule for the AC conductivity holds in all the three phases. We also investigate a complexified U(1) rotor model with \mathcal{PT}𝒫𝒯-symmetric deformation, derive its phase structure and condensation pattern, and find a zero frequency spectral weight analogous to the holographic model.