Aspects of N-partite information in conformal field theories

TL;DR

This paper derives new theoretical results for the N-partite information in conformal field theories, confirms predictions with lattice simulations, and explores positivity and holographic relations across various models and dimensions.

Contribution

It provides explicit formulas for N-partite information in CFTs, verifies long-distance scaling, and demonstrates positivity and holographic equality in specific settings.

Findings

N-partite information scales as r^{-2NΔ} at large distances.

Explicit formula for I_4 in terms of lower-point functions.

Numerical evidence supports positivity of I_N in free scalar theories.

Abstract

We present several new results for the $N$-partite information, $I_N$, of spatial regions in the ground state of $d$-dimensional conformal field theories. First, we show that $I_N$ can be written in terms of a single $N$-point function of twist operators. Using this, we argue that in the limit in which all mutual separations are much greater than the regions sizes, the $N$-partite information scales as $I_N \sim r^{-2N\Delta}$, where $r$ is the typical distance between pairs of regions and $\Delta$ is the lowest primary scaling dimension. In the case of spherical entangling surfaces, we obtain a completely explicit formula for the $I_4$ in terms of 2-, 3- and 4-point functions of the lowest-dimensional primary. Then, we consider a three-dimensional scalar field in the lattice. We verify the predicted long-distance scaling and provide strong evidence that $I_N$ is always positive for general regions and arbitrary $N$ for that theory. For the $I_4$, we find excellent numerical agreement between our general formula and the lattice result for disk regions. We also perform lattice calculations of the mutual information for more general regions and general separations both for a free scalar and a free fermion, and conjecture that, normalized by the corresponding disk entanglement entropy coefficients, the scalar result is always greater than the fermion one. Finally, we verify explicitly the equality between the $N$-partite information of bulk and boundary fields in holographic theories for spherical entangling surfaces in general dimensions.