Topological effect on order-disorder transitions in U(1) sigma models

U(1) non-linear sigma model (NLSM) with a one-dimensional (1D) Berry phase is studied by a renormalization group theory. Order-disorder transition in U(1) NLSMs in $D \!\ (\ge 2)$-dimensional space ($d+1$-dimensional spacetime; $d\ge 1$) is instigated by the proliferation of vortex excitations, where the 1D Berry phase term confers finite phase factors upon those vortex excitations that have finite projection in a subspace complementary to a topological direction with the 1D Berry phase. A destructive interference effect caused by the phase factors may help to develop an intermediate quasi-disorder phase between ordered and disorder phases, which has a divergent order-parameter correlation length along the topological direction, and a finite correlation length along the other directions. In order to explore such a possibility in $D=3$, we develop a perturbative renormalization group theory of a 3D model of vortex loops, in which loop segments interact via a $1/r$ Coulomb interaction. We derive renormalization group (RG) equations among vortex-loop fugacity, Berry phase term, and the Coulomb potential. Approximate analyses of the RG equations show that near an order-disorder transition point, vortex loops are anomalously elongated along the topological direction. Utilizing a duality mapping to a lattice model of a type-II superconductor under a magnetic field, we also argue that a global phase diagram of the 3D U(1) sigma model with 1D Berry phase must have the quasi-disorder phase between ordered and disorder phases.