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Phase transitions in a three dimensional $U(1) \times U(1)$ lattice London superconductor: Metallic superfluid and charge-4e superconducting states

We consider a three-dimensional lattice $U(1) \times U(1)$ and $[U(1)]^N$ superconductors in the London limit, with individually conserved condensates. The $U(1) \times U(1)$ problem, generically, has two types of intercomponent interactions of different characters. First, the condensates are interacting via a minimal coupling to the same fluctuating gauge field. A second type of coupling is the direct dissipationless drag represented by a local intercomponent current-current coupling term in the free energy functional. In this work, we present a study of the phase diagram of a $U(1) \times U(1)$ superconductor which includes both of these interactions. We study phase transitions and two types of competing paired phases which occur in this general model: (i) a metallic superfluid phase (where there is order only in the gauge invariant phase difference of the order parameters), (ii) a composite superconducting phase where there is order in the phase sum of the order parameters which has many properties of a single-component superconductor but with a doubled value of electric charge. We investigate the phase diagram with particular focus on what we call ``preemptive phase transitions.'' These are phase transitions unique to multicomponent condensates with competing topological objects. A sudden proliferation of one kind of topological defects may come about due to a fluctuating background of topological defects in other sectors of the theory. For $U(1) \times U(1)$ theory with unequal bare stiffnesses where components are coupled by a non-compact gauge field only, we study how this scenario leads to a merger of two $U(1)$ transitions into a single $U(1) \times U(1)$ discontinuous phase transition. We also report a general form of vortex-vortex bare interaction potential and possible phase transitions in an $N$-component London superconductor with individually conserved condensates.