Strong-coupling critical behavior in three-dimensional lattice Abelian gauge models with charged $N$-component scalar fields and $\mathrm{SO}\left(N\right)$ symmetry

We consider a three-dimensional lattice Abelian Higgs gauge model for a charged $N$-component scalar field $\varphi$, which is invariant under $\text{SO}\left(N\right)$ global transformations for generic values of the parameters. We focus on the strong-coupling regime, in which the kinetic Hamiltonian term for the gauge field is a small perturbation, which is irrelevant for the critical behavior. The Hamiltonian depends on a parameter $v$, which determines the global symmetry of the model and the symmetry of the low-temperature phases. We present renormalization-group predictions, based on a Landau-Ginzburg-Wilson effective description that relies on the identification of the appropriate order parameter and on the symmetry-breaking patterns that occur at the strong-coupling phase transitions. For $v=0$, the global symmetry group of the model is $\text{SU}\left(N\right)$; the corresponding model may undergo continuous transitions only for $N=2$. For $v\ne 0$, i.e., in the $\text{SO}\left(N\right)$ symmetric case, continuous transitions (in the Heisenberg universality class) are possible also for $N=3$ and 4. We perform Monte Carlo simulations for $N=2,3,4,6$, to verify the renormalization-group predictions. Finite-size scaling analyses of the numerical data are in full agreement.