Thermal Pseudo-Entropy

TL;DR

This work introduces thermal pseudo-entropy (TPE), the holomorphic extension of thermal entropy to complex inverse temperatures β+it, and shows its precise relation to pseudo-entropy via Thermofield Double dynamics. By analyzing TPE across quantum mechanics, Schwarzian theory, Random Matrix Theory, and 2D CFTs (including symmetric orbifolds), the authors uncover a deep link between the averaged real part of TPE and the spectral form factor, and reveal a universal logarithmic time dependence for continuous spectra with a gamma determined by edge-state scaling. They establish Kramers-Kronig relations between the real and imaginary parts of TPE, verify them numerically, and explore how discrete versus continuous spectra shape late-time behavior, with holographic CFTs exhibiting distinct γ = -1 signatures. The study combines exact calculations, asymptotic analyses, and numerical simulations to illuminate how TPE encodes chaotic versus integrable dynamics, offering a bridge between quantum information measures and spectral statistics with potential holographic implications.

Abstract

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.

Figures (23)

• Figure 1: Real and imaginary parts of TPE \ref{['TPE2Level']} for $\epsilon=1$ and $\beta=1/3$. Red dots indicate numerically computed imaginary part using the KK-relations.
• Figure 2: (Left) Real (blue) and imaginary (orange) parts of TPE for $\beta = 2$ and $\omega=1$ for the harmonic oscillator $N = 1$. (Right) For the Calorego-Sutherland model with $N=2$ and the same parameters. Red dots are numerical checks of the KK relations at several instances of time.
• Figure 3: Real (blue) and imaginary (orange) parts of TPE derived from \ref{['SchwPE']} for $\beta=1/3$, $S_0=100$ and $C=10$. Red dots indicate numerically computed imaginary part using the KK-relation.
• Figure 4: Log-log plot of SFF for RMT with partition function \ref{['ZRMTI1']} for $\beta = 0.1$ and $\lambda = 4$.
• Figure 5: (Left) Re-Im plot of the TPE of random matrix theory with $\beta = 0.3$ and $\lambda = 4$. The KK relation is respected. (Right) Averaged real part of TPE (shown in orange), comparing with the late time prediction \ref{['eq:RMTlate']} (in green).