Weak symmetry breaking and topological order in a 3D compressible quantum liquid

We introduce a new type of 3D compressible quantum phase, in which the U(1) charge conservation symmetry is weakly broken by a rigid string-like order parameter, and no local order parameter exists. We show that this gapless phase is completely stable and described at low energy by an infinite-component Chern-Simons-Maxwell theory. We determine the emergent symmetry group, which contains U(1) 0-form planar symmetries and an unusual subgroup of the dual U(1) 1-form symmetry supported on cylindrical surfaces. Through the associated 't Hooft anomaly, we examine how the filling condition is fulfilled in the low-energy theory. We also demonstrate that the phase exhibits a kind of fractonic topological order, signified by extensively many different types of topologically nontrivial quasiparticles formed out of vortices of the weak superfluid. A microscopic model realizing the weak superfluid phase is constructed using an array of strongly coupled Luttinger liquid wires, and the connection to the field theory is established through boson-vortex duality.