Subregion Spectrum Form Factor via Pseudo Entropy

TL;DR

We propose a subregion generalization of the spectral form factor using the pseudo entropy of reduced transition matrices between a thermofield double state and its time-evolved partner in a two-dimensional CFT, focusing on a single interval A. The real part of the pseudo entropy, S_A, exhibits a universal dynamical pattern akin to the spectral form factor, with subsystem-dependent Thouless and Heisenberg times given by $t_T \propto \sqrt{\beta \ell}$ and $t_H \propto \ell$, respectively. In infinite volume, Re[S_A] evolves from the thermal entanglement entropy through a dip and ramp to a plateau, eventually approaching the ground-state entanglement entropy, with late-time values matching vacuum expectations. Finite-size studies in the critical Ising model reveal integrability-induced oscillations, while holographic CFTs show self-averaging behavior, suggesting holographic and ensemble interpretations; future work includes understanding the imaginary part, non-Hermitian extensions, and the role of complex bulk geometries.

Abstract

We introduce a subsystem generalization of the spectral form factor via pseudo entropy, the von-Neumann entropy for the reduced transition matrix. We consider a transition matrix between the thermofield double state and its time-evolved state in two-dimensional conformal field theories, and study the time-dependence of the pseudo entropy for a single interval. We show that the real part of the pseudo entropy behaves similarly to the spectral form factor; it starts from the thermal entropy, initially drops to the minimum, then it starts increasing, and finally approaches the vacuum entanglement entropy. We also study the theory-dependence of its behavior by considering theories on a compact space.

Figures (3)

• Figure 1: The Euclidean path-integral for the reduced transition matrix $\mathcal{T}^{\psi|\varphi}_A$ is performed on the Euclidean cylinder with circumference $\tau_A+\tau_B$ with a cut on A.
• Figure 2: Left: The time evolution of the pseudo entropy associated with two states \ref{['in-out']} and \ref{['in-out2']} exhibits a similar behavior to the spectral form factor. Here, we consider a single interval on $R$ (let us call it $A$) as a subsystem. These log-log plots suggest that $t_T\propto\sqrt{\ell}$ and $t_H\propto\ell$, where $\ell$ is the size of subsystem $A$. Right: The inverse temperature $\beta$ dependence of the pseudo entropy. These plots suggest that $t_T\propto\sqrt{\beta\ell}$ and $t_H\propto\ell$.
• Figure 3: Numerical results for critical Ising model with various total system sizes. Here, we set $\beta=10$ and $\ell=10$ for each plot.