Holographic two-point functions across a PT symmetry-breaking transition

Abstract

In parity-time (PT)-symmetric non-Hermitian systems, an exceptional point, where eigenvalues and eigenvectors coalesce, typically separates a real spectrum from complex-conjugate eigenvalues. In logarithmic conformal field theories (LCFTs), such an exceptional point appears in the spectrum of scaling dimensions. Holographic models of LCFTs, such as the dipole model, are built exactly at this point. Here, we deform the holographic dipole model so that its two coalescing scaling dimensions split into either two real ones or a complex-conjugate pair. We obtain the renormalized two-point functions in closed form. They change continuously from a linear combination of two power laws, through the logarithmic form of the LCFT at the exceptional point, to log-periodic oscillations. For any pair of scalar primary operators with real mixing under dilatations, the conformal Ward identities fix this form up to two normalization constants, which the holographic dictionary determines. In PT-symmetric theories with such a dilatation operator, real and complex-conjugate scaling dimensions correspond to the PT-unbroken and PT-broken phases. In the BTZ black hole at finite temperature, the difference between the quasinormal frequencies associated with the two scaling dimensions changes from purely imaginary to purely real across the real-to-complex spectral transition. Correspondingly, the linear response changes from overdamped to underdamped, with critical damping at the exceptional point.