Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs

TL;DR

This work analyzes Rényi entropy (RE) for locally excited states in 2D CFTs under thermal and boundary effects. Using the replica trick, it shows that at low temperature the RE contributions separate into a thermal piece and a local-excitation piece, effectively a sum rule, and extends this framework to BCFTs where the boundary alters time evolution but not the maximal RE value. In 2D rational CFTs with a boundary, the RE exhibits plateaus where the increase equals the logarithm of the primary operator’s quantum dimension, $\Delta S^{(n)}_A=\log d_O$, explained via a quasi-particle picture with the boundary acting as an infinite barrier. The results are illustrated in free-boson and Ising models and generalized to arbitrary rational CFTs, highlighting the role of fusion and boundary conformal blocks in determining RE while showing boundary conditions do not affect the maximal RE.

Abstract

We study Rényi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs). Firstly we consider locally excited states obtained by acting primary operators on a thermal state in low temperature limit. The Rényi entropy is summation of contribution from thermal effect and local excitation. Secondly, we mainly study the Rényi entropy of locally excited states in 2D CFT with a boundary. We show that the evolution of Rényi entropy does not depend on the choice of boundary conditions and boundary will change the time evolution of Rényi entropy. Moreover, in 2D rational CFTs with a boundary, we show that the Rényi entropy always coincides with the log of quantum dimension of the primary operator during some periods of the evolution. We make use of a quasi-particle picture to understand this phenomenon. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards the boundary.

Figures (4)

• Figure 1: This figure is to show our setup in two dimensional complex plane $\omega= x+i t$. The system will be triggered at $x=-L$ and there are left- and right-moving quasi-particle at $t=0$. This setup is same as the one in 2D CFTs without boundary He:2014mwa.
• Figure 2: This figure is to show our setup in two dimensional left half plane $\omega=x+i t$ with a boundary $x=0$. The system will be triggered at $x=-L$ and there are left- and right-moving quasi-particle at $t=0$.
• Figure 3: Fusion transformation in Ising model. Where $i$ and $\bar{i}$ presented in the figure stands for the position $z_i$ and $\bar{z}_i$ respectively. $F_{00}[\sigma]$ denotes the fusion transformation and $0$ corresponds to the identity operator.
• Figure 4: To show the time evolution of the entangled quasi-particles pairs. The boundary $x=0$ works as reflector. These red points stand for quasi-particles and the blue wave lines correspond to the entanglement between them.