Pseudo Entropy in $U(1)$ gauge theory

TL;DR

This work investigates pseudo entropy in a free $U(1)$ gauge theory in $d=4$, building excited states from field-strength components at different Euclidean times and measuring the difference $\Delta S_A^{(n)}$ from the ground-state Rényi entropy. Using the replica trick, the authors first develop the conformal-scalar framework in $d=4$ and then apply it to Maxwell theory, computing two- and four-point functions on replica surfaces to obtain $\Delta S_A^{(2)}$ for various excitations. They find that boundary effects dominate $\Delta S_A^{(2)}$, with a strong dependence on the ratio of Euclidean times near the boundary and vanishing contributions far from it; real-time continuation drives the quantity toward $\log 2$ as left- and right-moving modes become maximally entangled. Subtleties arise in certain excitations (e.g., $F_{ry}$) where divergences or complex values appear near the boundary, reflecting non-Hermitian aspects of the reduced transition matrix. The results suggest a general boundary-anchored structure for pseudo-entropy in gauge theories and motivate extensions to gravity, fermions, and higher-spin fields, as well as connections to operator properties and real-time quenches.

Abstract

We study the properties of pseudo entropy, a new generalization of entanglement entropy, in free Maxwell field theory in $d = 4$ dimension. We prepare excited states by the different components of the field strengths located at different Euclidean times acting on the vacuum. We compute the difference between the pseudo Rényi entropy and the Rényi entropy of the ground state and observe that the difference changes significantly near the boundary of the subsystems and vanishes far away from the boundary. Near the boundary of the subsystems, the difference between pseudo Rényi entropy and Rényi entropy of the ground state depends on the ratio of the two Euclidean times where the operators are kept. To begin with, we develop the method to evaluate pseudo entropy of conformal scalar field in $d=4$ dimension. We prepare two states by two operators with fixed conformal weight acting on the vacuum and observe that the difference between pseudo Rényi entropy and ground state Rényi entropy changes only near the boundary of the subsystems. We also show that a suitable analytical continuation of pseudo Rényi entropy leads to the evaluation of real-time evolution of Rényi entropy during quenches.

Figures (10)

• Figure 1: The first plot shows the variation of $\Delta S^{(2)}_A$ with respect to subsytem size with fixed UV cutoffs. Blue line: $\ell=20$, orange line: $\ell=10$, green line : $\ell=4$. The second plot shows the variation of $\Delta S^{(2)}_A$ with respect to UV cutoffs at a fixed subsytem size of $\ell=20$. Blue line: $a = 4$, $a' = 6$, orange line: $a = 2$, $a' = 8$, green line: $a = 0.1$, $a' = 9.9$.
• Figure 2: $\Delta S^{(2)}_A$ as a function of the center of the operators.
• Figure 3: Real-time evolution of $\Delta S^{(2)}_A$. Blue line : $x=2$, ; orange line $x=4$; green line $x=6$. We keep $\epsilon=0.1$ in all cases.
• Figure 4: $\Delta S^{(2)}_A$ as a function of the center of the operators $F_{r\theta}$.
• Figure 5: $\Delta S_A^{(2)}$ as a function for same components $F_{r\theta}$ of the field strength . Blue line : $x=2$, ; orange line $x=4$; green line $x=6$. We keep $\epsilon=0.01$ in all cases.