R\'enyi entanglement asymmetry in 1+1-dimensional conformal field theories

In this paper, we consider the R\'enyi entanglement asymmetry of excited states in the 1+1 dimensional free compact boson conformal field theory (CFT) at equilibrium. We obtain a universal CFT expression written by correlation functions for the charged moments via the replica trick. We provide detailed analytic computations of the second R\'enyi entanglement asymmetry in the free compact boson CFT for excited states $\Psi=V_{\beta}+V_{-\beta}$ and $\Phi=V_{\beta}+J$ with $V_{\beta}$ and $J=i\partial\phi$ being the vertex operator and current operator respectively. We make numerical tests of the universal CFT computations using the XX spin chain model. Taking the non-Hermite fake RDMs into consideration, we propose an effective way to test them numerically, which can be applied to other excited states. The CFT predictions are in perfect agreement with the exact numerical calculations.