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arXiv:2610.23580

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

Zuo Wang, Liang He

hep-thquant-phSubmitted Oct 16, 2026• Announced Oct 19, 2026
arXivHTMLPDF

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.

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27 pages, 3 figures

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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.
We study the non-Hermitian quantum mechanics for the discrete system. This paper gives an exact analytic single-particle solution for an NNN-site tight-binding chain with two conjugated imaginary potentials $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ at two end sites, which Hamiltonian has parity-time symmetry (PT\mathcal{P}\mathcal{T}PT symmetry). Based on the Bethe ansatz results, it is found that, in single-particle subspace, this model is comprised of two phases: an unbroken symmetry phase with a purely real energy spectrum in the region $\ensuremath{\gamma}\ensuremath{\prec}{\ensuremath{\gamma}}_{c}$ and a spontaneously broken symmetry phase with $N\ensuremath{-}2$ real and two imaginary eigenvalues in the region $\ensuremath{\gamma}\ensuremath{\succ}{\ensuremath{\gamma}}_{c}$. The behaviors of eigenfunctions and eigenvalues in the vicinity of ${\ensuremath{\gamma}}_{c}$ are investigated. It is shown that the boundary of two phases possesses the characteristics of exceptional point. We also construct the equivalent Hermitian Hamiltonian of the present model in the framework of metric-operator theory. We find out that the equivalent Hermitian Hamiltonian can be written as another bipartite lattice model with real long-range hoppings.
A parity-time (PT) transition is a spectral transition characteristic of non-Hermitian generators; it typically occurs at an exceptional point, where multiple eigenvectors coalesce. The concept of a PT transition has been extended to Markovian open quantum systems, which are described by the GKSL equation. Interestingly, the PT transition in many-body Markovian open quantum systems, the so-called \textit{Lindbladian PT (L-PT) phase transition}, is closely related to two classes of exotic nonequilibrium many-body phenomena: \textit{continuous-time crystals} and \textit{non-reciprocal phase transitions}. In this review, we describe the recent advances in the study of L-PT phase transitions. First, we define PT symmetry in three distinct contexts: non-Hermitian systems, nonlinear dynamical systems, and Markovian open quantum systems, highlighting the interconnections between these frameworks. Second, we develop mean-field theories of L-PT phase transitions for collective-spin systems and for bipartite bosonic systems with particle-number conservation. Within these classes of models, we show that L-PT symmetry can induce a breaking of continuous time-translation symmetry down to a discrete one, leading to persistent periodic dynamics. We further demonstrate that the L-PT phase transition point is typically \textit{a critical exceptional point}, where multiple collective excitation modes with zero excitation spectrum coalesce. These findings establish an explicit connection to continuous-time crystals and non-reciprocal phase transitions. Third, going beyond the mean-field theory, we analyze statistical and quantum properties, such as purity and quantum entanglement indicators of time-independent steady states for several specific models with the L-PT symmetry. Finally, we discuss future research directions for L-PT phase transitions.
We present a holographic dual of a two dimensional conformal field theory with non-hermitian but Parity-Time (PT) symmetric boundary conditions, by applying the AdS/BCFT duality and by introducing an imaginary valued scalar field localized on an end-of-the-world brane. We find that as we increase the strength of the non-hermitian PT symmetric interactions, the system experiences a spontaneous PT symmetry breaking. We also consider its Wick rotated setup as a new quantum quenched state and show that its growth of entanglement entropy can be larger than the standard results obtained from standard Cardy states.
We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time (PT\mathcal{PT}PT) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.
Conformal invariance often accompanies criticality in Hermitian systems. However, its fate in non-Hermitian settings is less clear, especially near exceptional points where the Hamiltonian becomes non-diagonalizable. Here we investigate whether a 1+1-dimensional gapless non-Hermitian system can admit a conformal description, focusing on a PT-symmetric free-fermion field theory. Working in the biorthogonal formalism, we identify the conformal structure of this theory by constructing a traceless energy-momentum tensor whose Fourier modes generate a Virasoro algebra with central charge c=−2c=-2c=−2. This yields a non-Hermitian, biorthogonal realization of a logarithmic conformal field theory, in which correlation functions exhibit logarithmic scaling and the spectrum forms Virasoro staggered modules that are characterized by universal indecomposability parameters. We further present a microscopic construction and show how the same conformal data (with finite-size corrections) can be extracted from the lattice model at exceptional-point criticality, thereby supporting the field-theory prediction.
We construct conformal interfaces having defect conformal manifolds between two-dimensional rational CFTs, at least one of which is non-unitary. In a continuous family, interfaces with real and complex spectra can be marginally deformed to each other via their defect moduli parameters. An exceptional point then arises along the trajectory in the conformal manifold, where two interface operators with complex conjugate weights coalesce into a logarithmic pair. We reconstruct the entire trajectory of this novel spectral collision, independently, from self-annulus partition functions and bulk two point correlators in the presence of the interfaces.