Abstract
We consider the symmetry resolution of relative entropies in the 1+1 dimensional free massless compact boson conformal field theory (CFT) which presents an internal U(1) symmetry. We calculate various symmetry resolved Rényi relative entropies between one interval reduced density matrices of CFT primary states using the replica method. By taking the replica limit, the symmetry resolved relative entropy can be obtained. We also take the XX spin chain model as a concrete lattice realization of this CFT to perform numerical computation. The CFT predictions are tested against exact numerical calculations finding perfect agreement.
Highlights
We consider the symmetry resolution of relative entropies in the 1+1 dimensional free massless compact boson conformal field theory (CFT) which presents an internal U (1) symmetry
We will mainly focus on the symmetry resolution of relative entropies in CFT
We define all needed concepts concerning symmetry resolved relative entropy and summarise the known results of the symmetry resolved entanglement entropy which will be useful
Summary
Let’s briefly review the replica trick to compute the relative entropies of two reduced density matrices of excited states in 1+1 dimensional CFT. In contrast to the ground state case, the corresponding path-integral representation of the density matrix ρ = |Ψ Ψ| presents two additional insertions of Ψ(−i∞) and Ψ†(i∞). In this way, we end up with a n-sheeted Riemann surface Rn and tr(ρnΨ) is given by a 2n-point function on Rn [45]. The other type of primary field in this theory is the derivative operator i∂φ with conformal dimension (h, h) = (1, 0). The Renyi relative entropies between the ground state and the vertex operator are given by α2 sin πx.
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