Quantum Dimension as Entanglement Entropy in 2D CFTs

TL;DR

This work studies entanglement entropy in excited states of two-dimensional CFTs, focusing on states produced by primary operators acting on the vacuum. Using the replica method and conformal/fusion transformations, it demonstrates that after time evolution under causality, the late-time increase in both Renyi and von Neumann entropies equals $ΔS_A^{(n)}=\log d_a$ for rational CFTs, where $d_a$ is the quantum dimension of the inserted primary. The result generalizes to products of primaries as $ΔS_A^{(n)}=\sum_a n_a \log d_a$, highlighting a tight link between dynamical entanglement growth and fusion algebra. These findings connect a dynamical entanglement signal to topological data in 2D rational CFTs and point toward potential holographic interpretations and further extensions to more general theories.

Abstract

We study entanglement entropy of excited states in two dimensional conformal field theories (CFTs). Especially we consider excited states obtained by acting primary operators on a vacuum. We show that under its time evolution, entanglement entropy increases by a finite constant when the causality condition is satisfied. Moreover, in rational CFTs, we prove that this increased amount of (both Renyi and von-Neumann) entanglement entropy always coincides with the log of quantum dimension of the primary operator.

Figures (3)

• Figure 1: The $n$-sheeted space $\Sigma_n$. The red interval describes the subsystem A.
• Figure 2: The time evolution of $\Delta S^{(2)}_A$ for ${\mathcal{O}}_2$. We set $l = 1, L = 1$.
• Figure 3: The fusion transformations to obtain $\Delta S^{(n)}_A$.