The U(1) nonlinear 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. Due to a destructive interference effect caused by the phase factors, a partition function near the order-disorder transition point can be dominated by vortex excitations polarized along the topological direction with the Berry phase. The proliferation of the polarized vortex excitations helps to develop an extremely anisotropic correlation of the order parameter, which has a divergent correlation length along the topological direction with the Berry phase 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, and the 1D Berry phase confers the phase factor upon each vortex loop. We derive renormalization group (RG) equations among vortex-loop fugacity, Berry phase term, and the Coulomb potential. The RG equations analyzed with approximations show that a characteristic size of the vortex loop along the topological direction becomes anomalously large near an order-disorder transition point, while the characteristic loop size within the other directions remains finite. 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 should have an intermediate quasidisordered phase between ordered and disordered phases.

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• Critical phenomena
• Exotic phases of matter
• Quantum interference effects
• Quantum phase transitions
• Topological phase transition
• Vortex interactions
• Vortices in superfluids
• XY model
• 3-dimensional systems
• Disordered systems
• Superfluids
• Type-II superconductors
• Renormalization group
• Sigma models