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 time evolution of Rényi entropy is affected by the boundary, but does not depend on the boundary condition. Moreover, we show that the maximal value of Rényi entropy always coincides with the log of quantum dimension of the primary operator. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards it.
Highlights
We would like to review the thermal effect and boundary effect respectively
We study Renyi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs)
In 2D rational CFTs with a boundary, we show that the maximal value of the Renyi entropy always coincides with the log of quantum dimension of the primary operator during some periods of the evolution
Summary
We would like to study the local excitation of thermal state. We consider a system with temperature T = 1/β and assume the excitation is local at x = −L by primary operator O shown in figure 1. Which can be taken as the reduced density matrix related to the vacuum and first excited state respectively, where we normalize the vacuum energy to be zero, B is the complementary part of subsystem A. The second terms of (2.6) is still given by in ǫ → 0 ne−βE1 1−n ψ′ (z−∞)ψ′ (z+∞) C1. The sum of the second and third term is the same as the thermal correlation [8] for the short interval limit. (2.6) is the summation over the thermal correction of the vacuum state and local excitation in the vacuum state in low temperature limit. With a different method [8], we can reproduce the Renyi entropy [18] for short interval in the low temperature with taking limit ǫ → 0 for general 2D rational CFT
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