$2$-R\'enyi CCNR Negativity of Compact Boson for multiple disjoint intervals

TL;DR

This paper computes the $2$-Rényi CCNR negativity for multiple disjoint intervals in a 2D massless compact boson, relating it to Riemann surfaces and cross-ratios, and verifies results numerically.

Contribution

It provides explicit formulas for $2$-Rényi CCNR negativity in complex geometries and extends the analysis to Dirac fermions, including numerical validation.

Findings

Derived analytical expressions for $2$-Rényi CCNR negativity.

Connected negativity to Riemann surface properties and cross-ratios.

Validated theoretical results with numerical simulations in lattice models.

Abstract

We investigate mixed-state bipartite entanglement between multiple disjoint intervals using the computable cross-norm criterion (CCNR). We consider entanglement between a single interval and the union of remaining disjoint intervals, and compute $2$-R\'enyi CCNR negativity for $2$d massless compact boson. The expression for $2$-R\'enyi CCNR negativity is given in terms of cross-ratios and Riemann period matrices of Riemann surfaces involved in the calculation. In general, the Riemann surfaces involved in the calculation of $n$-R\'enyi CCNR negativity do not possess a $Z_n$ symmetry. We also evaluate the Reflected R\'enyi entropy related to the $2$-R\'enyi CCNR negativity. This Reflected R\'enyi entropy is a universal quantity. We extend these calculations to the $2$d massless Dirac fermions as well. Finally, the analytical results are checked against the numerical evaluations in the tight-binding model and are found to be in good agreement.