Bounds on corner entanglement in quantum critical states

The entanglement entropy in many gapless quantum systems receives a contribution from the corners in the entangling surface in 2+1d, which is characterized by a universal function a(theta) depending on the opening angle theta, and contains pertinent low energy information. For conformal field theories (CFTs), the leading expansion coefficient in the smooth limit theta -> pi yields the stress tensor two-point function coefficient C-T. Little is known about a(theta) beyond that limit. Here, we show that the next term in the smooth limit expansion contains information beyond the two- and three-point correlators of the stress tensor. We conjecture that it encodes four-point data, making it much richer. Further, we establish strong constraints on this and higher-order smooth-limit coefficients. We also show that a(theta) is lower-bounded by a nontrivial function multiplied by the central charge C-T, e.g., a(pi/2) >= (pi(2) ln 2)C-T/6. This bound for 90-degree corners is nearly saturated by all known results, including recent numerics for the interacting Wilson-Fisher quantum critical points (QCPs). A bound is also given for the Renyi entropies. We illustrate our findings using O(N) QCPs, free boson and Dirac fermion CFTs, strongly coupled holographic ones, and other models. Exact results are also given for Lifshitz quantum critical points, and for conical singularities in 3+1d.