Imaginary HTEE: Probing Criticality and Coherence

Abstract

We study the imaginary part of the holographic timelike entanglement entropy (HTEE) in a charged asymptotically AdS$_5$ black-hole background whose dual field theory exhibits a critical point in the presence of a chemical potential $μ$ and temperature $T$. Identifying the imaginary part of the HTEE with the imaginary part of the pseudoentropy, we derive an analytical expression for this quantity and investigate its behavior on the stable and unstable thermodynamic branches. We find that the imaginary HTEE vanishes in the uncharged limit and is induced entirely by a nonzero chemical potential, signaling the emergence of quantum coherence between the initial and final states. As the dimensionless ratio $μ/T$ approaches its critical value, we obtain an inverse-square-root divergence in the derivative of the imaginary HTEE and extract the critical exponent $θ_I = 1/2$ analytically. We further show that the same exponent governs the temperature response at fixed chemical potential. Our results establish the imaginary part of the HTEE as a non-local quantum-information probe of criticality in strongly coupled systems, and they imply that the initial and final states become maximally coherent at the critical point.