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Some results on the mutual information of disjoint regions in higher dimensions

John Cardy
(
- Oxford U., Theor. Phys. and
- Oxford U.
)

Apr 30, 2013

21 pages

Published in:

J.Phys.A 46 (2013) 285402

Published: Jun 25, 2013

e-Print:

1304.7985 [hep-th]

DOI:

10.1088/1751-8113/46/28/285402

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Abstract: (IOP)

We consider the mutual Rényi information of disjoint compact spatial regions A and B in the ground state of a d + 1-dimensional conformal field theory (CFT), in the limit when the separation r between A and B is much greater than their sizes RA, B. We show that in general , where α is the smallest sum of the scaling dimensions of operators whose product has the quantum numbers of the vacuum, and the constants depend only on the shape of the regions and universal data of the CFT. For a free massless scalar field, where α = d − 1, we show that is proportional to the capacitance of a thin conducting slab in the shape of A in d + 1-dimensional electrostatics, and give explicit formulae for this when A is the interior of a sphere Sd − 1 or an ellipsoid. For spherical regions in d = 2 and 3 we obtain explicit results for C(n) for all n and hence for the leading term in the mutual information by taking n → 1. We also compute a universal logarithmic correction to the area law for the Rényi entropies of a single spherical region for a scalar field theory with a small mass.

Note:

17 pages, 1 figure. v2: corrected assertion that x_epsilon>2x_sigma in general; new section added on massive theory. v.3 (final): typos corrected, references added

03.67.Mn
11.25.Hf
05.70.Jk
field theory: scalar
field theory: conformal
operator: dimension
scaling: dimension
higher-dimensional
entropy

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