Phase transitions and composite order in $\mathrm{U}(1)^N$ lattice London models

The phase diagrams and the nature of the phase transitions in multicomponent gauge theories with an Abelian gauge field are important topics with various physical applications. While an early renormalization-group-based study indicated that the direct transition from a fully ordered to a fully disordered state is continuous for $N = 1$ and $N > 183$, recently it was demonstrated that the transition is discontinuous for $N = 2$. We quantitatively study the dependence on $N$ of the degree of discontinuity of this transition. Our results suggest that the transition is discontinuous at least up to $N = 7$. Furthermore, we demonstrate that, at increased coupling strength, the phase transitions of the neutral and charged sectors of the model split, which for $N > 2$ yields a new phase with composite order. The transition from the composite-order phase to the fully disordered phase is then also discontinuous, at least for $N = 3$ and $N = 4$. Via a duality argument, this indicates that van der Waals-type interaction between directed loops may be responsible for the discontinuous phase transitions in these models.