Thermal Pseudo-Entropy

In this work, we develop a generalisation of the thermal entropy to complex inverse temperatures, which we call the thermal pseudo-entropy. We show that this quantity represents the pseudo-entropy of the transition matrix between Thermofield Double states at different times. We have studied its properties in various quantum mechanical setups, Schwarzian theory, Random Matrix Theories, and 2D CFTs, including symmetric orbifolds. Our findings indicate a close relationship between the averaged thermal pseudo-entropy and the spectral form factor, which is instrumental in distinguishing chaotic and integrable models. Moreover, we have observed a logarithmic scaling of this quantity in models with a continuous spectrum, with a universal coefficient that is sensitive to the scaling of the density of states near the edge of the spectrum. Lastly, we found the connection between the real and imaginary parts of the thermal pseudo-entropy through the Kramers-Kronig relations.