A Superfluid at All Temperatures

We construct unitary, local, 3+1-dimensional conformal field theories with finitely many fields whose thermal state breaks a continuous global symmetry at every temperature. They are finite- N versions of the Chaudhuri--Choi--Rabinovici theories, whose planar circle of fixed points has an ordered arc. The circle is lifted at order 1/N squared, and an infinite family of fixed points on this arc reaches arbitrarily weak coupling, where the superfluid is under parametric control. We also show that weakly gauging the symmetry gives a superconductor over an exponential range of temperatures.