We study unitary conformal field theories with a unique stress tensor and at least one higher-spin conserved current in d > 3 dimensions. We prove that every such theory contains an infinite number of higher-spin conserved currents of arbitrarily high spin, and that Ward identities generated by the conserved charges of these currents imply that the correlators of the stress tensor and the conserved currents of the theory must coincide with one of the following three possibilities: a) a theory of n free bosons (for some integer n), b) a theory of n free fermions, or c) a theory of n d−2 2 -forms. For d even, all three structures exist, but for d odd, it may be the case that the third structure (c) does not; if it does exist, it is unclear what theory, if any, realizes it. This is a generalization of the result proved in three dimensions by Maldacena and Zhiboedov [1]. This paper supersedes the previous paper by the authors [2].

We apply large N diagrammatic techniques for theories with double-trace interactions to the leading corrections to C J , the coefficient of a conserved current two-point function, and C T , the coefficient of the stress-energy tensor two-point function. We study in detail two famous conformal field theories in continuous dimensions, the scalar O(N) model and the Gross-Neveu (GN) model. For the O(N) model, where the answers for the leading

We study unitary conformal field theories with a unique stress tensor and at least one higher-spin conserved current in four dimensions. We prove that every such theory contains an infinite number of higher-spin conserved currents of arbitrarily high spin, and that Ward identities generated by the conserved charges of these currents suffice to completely fix the correlators of the stress tensor and the conserved currents to be equal to one of three free field theories: the free boson, the free fermion, and the free vector field. This is a generalization of the result proved in three dimensions by Maldacena and Zhiboedov [1].

We show that the average null energy condition implies novel lower bounds on the scaling dimensions of highly-chiral primary operators in four-dimensional conformal field theories. Denoting the spin of an operator by a pair of integers (k,k) specifying the transformations under chiral su(2) rotations, we explicitly demonstrate these new bounds for operators transforming in (k, 0) and (k, 1) representations for sufficiently large k. Based on these calculations, along with intuition from free field theory, we conjecture that in any unitary conformal field theory, primary local operators of spin (k,k) and scaling dimension ∆ satisfy ∆ ≥ max{k,k}. If |k −k| > 4, this is stronger than the unitarity bound.

Two-dimensional crystals of trapped ions are a promising system with which to implement quantum simulations of challenging problems such as spin frustration. Here, we present a design for a surface-electrode elliptical ion trap which produces a 2-D ion crystal and is amenable to microfabrication, which would enable higher simulated coupling rates, as well as interactions based on magnetic forces generated by on-chip currents. Working in an 11 K cryogenic environment, we experimentally verify to within 5% a numerical model of the structure of ion crystals in the trap. We also explore the possibility of implementing quantum simulation using magnetic forces, and calculate J-coupling rates on the order of 10^3 / s for an ion crystal height of 10 microns, using a current of 1 A