Pseudo Rényi Entanglement Entropies For an Excited State and Its Time Evolution in a 2D CFT

TL;DR

This work extends the study of entanglement measures in a 2D CFT of free massless scalars by computing the second and third pseudo Rényi entanglement entropies (PREE) for a locally excited state $|\psi\rangle$ and its time-evolved state $|\phi\rangle = e^{-iHt}|\psi\rangle$. Using the transition-matrix framework, the authors derive explicit forms for $\Delta S_A^{(2,3)}$ across finite and semi-infinite entangling regions for several primary operators $\mathcal{O}_i$, highlighting that PREE are generally complex for $t\neq 0$ and reduce to real Rényi entropies at $t=0$. They present operator-dependent expressions involving cross-ratios and UV cutoffs $a,a'$, revealing distinct dynamical behavior, including zeros for certain operators and nontrivial time evolution and late-time saturation for others. The results emphasize qualitative differences between PREE and REE in local quench scenarios, offering insights into entanglement swapping, quasi-particle pictures, and potential extensions to higher dimensions, descendants, deformations, and finite-temperature settings.

Abstract

In this paper, we investigate the second and third pseudo Rényi entanglement entropies (PREE) for a locally excited state $| ψ\rangle $ and its time evolution $| φ\rangle = e^{- i H t} | ψ\rangle$ in a two-dimensional conformal field theory whose field content is a free massless scalar field. We consider excited states which are constructed by applying primary operators at time $t=0$, on the vacuum state. We study the time evolution of the PREE for an entangling region in the shape of finite and semi-infinite intervals at zero temperature. It is observed that the PREE is always a complex number for $t \neq 0$ and is a pure real number at $t=0$. Moreover, we discuss on its dependence on the location $x_m$ of the center of the entangling region.

Figures (19)

• Figure 1: Replica trick for the calculation of the n-th REE of a locally excited state $| \psi \rangle$ in eq. \ref{['psi']}. The entangling region is a finite interval $A \in [0, \infty]$. The difference with the replica trick for the n-th REE of the vacuum state, is that there are two operators $\mathcal{O}_i(w_{2k})$ and $\mathcal{O}_i^\dagger(w_{2k-1})$ on the k-th sheet.
• Figure 2: $\Delta S_A^{(2)}$ as a function of $t$ for $l= 1$, $L=2$. The values of $\Delta S_A^{(2)}$ in the time intervals $0 < t < l$, $l < t < l+L$ and $t > l+L$ are shown in blue, orange and purple, respectively. At late times it saturates to zero.
• Figure 3: $\Delta S_A^{(2)}$ as a function of $t$ for the operator $\mathcal{O}_2$, $l= 2$, $L=4$ and First Row) $a= a^\prime =0$Second Row) $a= a^\prime =1$Third Row) $a=1, a^\prime =3$Fourth Row) $a=2, a^\prime =1$. The values of $\Delta S_A^{(2)}$ in the time intervals $0 < t < l$, $l < t < l+L$ and $t > l+L$ are shown in blue, orange and purple, respectively. At late times, it saturates to a constant real value given by eq. \ref{['DeltaS2-late-time']} which is shown by the dashed black line.
• Figure 4: The second REE of the state $| \phi \rangle$, i.e. $\Delta S_A^{(2)}$, as a function of $t$ for $l= 2$, $L=4$ and Left) $a^\prime =0$Middle) $a^\prime =1$Right) $a^\prime =3$. The values of $\Delta S_A^{(2)}$ in the time intervals $0 < t < l$, $l < t < l+L$ and $t > l+L$ are shown in blue, orange and purple, respectively.
• Figure 5: $\Delta S_A^{(2)}$ as a function of $x_m$ for the operator $\mathcal{O}_2$, $t=0$ and Left) $l=0$, $L=10$, $a=1$, $a^\prime=2$Right) $l=4$, $L=20$, $a=3$, $a^\prime=1$. This Figure is the same as Figure 9 in ref. Nakata:2020luh. It should be pointed out that $\Delta S_A^{(2)}$ is real at $t=0$. Moreover, the plots are symmetric around the insertion point of the operator $\mathcal{O}_2$ at $t=0$.