Replicated Entanglement Negativity for Disjoint Intervals in the Ising and Free Compact Boson Conformal Field Theories

Abstract

We calculate the $N$ interval replicated negativity for the Ising and free compact boson conformal field theory using correlation functions of branch point twist fields. For some subset $P\subset N$, this is a calculation of $\rm{Tr}(\rho_A^{T_P})^n$ where $n$ is the replica index. This can be reformulated as a calculation of partition functions over superelliptic Riemann surfaces, and for the models in question, this partition function can be expressed in terms of the period matrix of this surface. We detail how to construct the period matrices for these surfaces, giving an analytic expression for $\rm{Tr}(\rho_A^{T_P})^n$. The results are expressed such that when $P = \emptyset$, which corresponds to calculating the R\'enyi entropy, the formulas aligns with known results for the $N$ interval R\'enyi entropy.