Tripartite information at long distances
César A. Agón, Pablo Bueno, Horacio Casini
Abstract
We compute the leading term of the tripartite information at long distances for three spheres in a CFT. This falls as r^{-6\Delta} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>6</mml:mn> <mml:mi>Δ</mml:mi> </mml:mrow> </mml:msup> </mml:math> , where r <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>r</mml:mi> </mml:math> is the typical distance between the spheres, and \Delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Δ</mml:mi> </mml:math> the lowest primary field dimension. The coefficient turns out to be a combination of terms coming from the two- and three-point functions and depends on the OPE coefficient of the field. We check the result with three-dimensional free scalars in the lattice finding excellent agreement. When the lowest-dimensional field is a scalar, we find that the mutual information can be monogamous only for quite large OPE coefficients, far away from a perturbative regime. When the lowest-dimensional primary is a fermion, we argue that the scaling must always be faster than r^{-6\Delta_f} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>6</mml:mn> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> </mml:msup> </mml:math> . In particular, lattice calculations suggest a leading scaling r^(6\Delta_f+1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi>r</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> </mml:msup> <mml:mn>6</mml:mn> <mml:msub> <mml:mi>Δ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> . For free fermions in three dimensions, we show that mutual information is also non-monogamous in the long-distance regime.