Conformal Bounds in Three Dimensions from Entanglement Entropy

Abstract

The entanglement entropy of an arbitrary spacetime region A in a three-dimensional conformal field theory (CFT) contains a constant universal coefficient, FðAÞ. For general theories, the value of FðAÞ is minimized when A is a round disk, F0, and in that case it coincides with the Euclidean free energy on the sphere. We conjecture that, for general CFTs, the quantity FðAÞ=F0 is bounded above by the free scalar field result and below by the Maxwell field one. We provide strong evidence in favor of this claim and argue that an analogous conjecture in the four-dimensional case is equivalent to the Hofman-Maldacena bounds. In three dimensions, our conjecture gives rise to similar bounds on the quotients of various constants characterizing the CFT. In particular, it implies that the quotient of the stress-tensor two-point function coefficient and the sphere free energy satisfies CT=F0 ≤ 3=ð4π2 log 2 − 6ζ½3Þ ≃ 0.14887 for general CFTs. We verify the validity of this bound for free scalars and fermions, general OðNÞ and Gross-Neveu models, holographic theories, N ¼ 2 Wess-Zumino models and general ABJM theories.