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

TL;DR

This paper computes the replicated negativity for disjoint intervals in Ising and free compact boson conformal field theories using correlation functions and superelliptic Riemann surfaces, extending known entropy results.

Contribution

It provides an analytic method to calculate the replicated negativity for multiple intervals in specific CFTs via period matrices of superelliptic Riemann surfaces.

Findings

Derived explicit formulas for $ m{Tr}( ho_A^{T_P})^n$ in Ising and boson models.

Connected negativity calculations to Riemann surface period matrices.

Unified the negativity and entropy calculations within a geometric framework.

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.