We consider three-dimensional (3D) lattice Abelian Higgs (AH) models, with compact U(1) gauge variables coupled to a doubly charged $N$-component complex scalar field (CLAH). We focus on their phase transitions between the disordered-confined (DC) and ordered-deconfined (OD) phases. When they are continuous, they are expected to belong to the 3D AH universality class associated with the stable charged fixed point (CFP) of the renormalization-group flow of the 3D AH field theory (AHFT), or scalar electrodynamics, describing $N$-component complex scalar fields minimally coupled to a U(1) gauge field. This CFP exists only for a sufficiently large number of components, i.e., $N\ge N_d^{*}$, where the integer $N_d^{*}$ depends on the spatial dimension $d$ (for example $N_4^{*}=183$). To estimate $N_3^{*}$, we look for the minimum number $N_{\text{cL}}$ of scalar components of 3D doubly charged CLAH models developing continuous transitions along their DC-OD transition line. For this purpose, we present finite-size scaling analyses of Monte Carlo simulations for $N\in [4,10]$, up to lattice sizes $L\approx 100$. The results provide evidence of continuous DC-OD transitions for $N=10$, and weak first-order transitions for $N\le 7$. They are not conclusive for $N=8$, 9. Therefore, we estimate $N_{\text{cL}}=9(1)$, which also provides an estimate of $N_3^{*}$ under the reasonable hypothesis that the DC-OD transitions are within the attraction domain of the CFP of the 3D AHFT for any $N\ge N_3^{*}$.

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• Critical phenomena
• Gauge theories
• Lattice field theory
• Phase transitions
• Renormalization group
• Finite-size scaling
• Lattice gauge theory
• Monte Carlo methods