We study some (1+2)-dimensional field-theory models with charge-$e$ fermions. The topological solitons in the models are charge-$e$ bosons and neutral fermions. There are two kinds of the magnetic vortex in the condensed state of the charge-$e$ bosons. One carries $\frac{h}{e}$ flux and another carries $\frac{h}{2e}$ flux plus a neutral fermion. Charge-$e$ boson condensation and $\frac{h}{2e}$ fluxoids may coexist in our models. The $\frac{h}{e}$ vortex is a boson, while the $\frac{h}{2e}$ vortex is a fermion in two dimensions. Our models provide a nontrivial check of a general assertion, i.e., the $\frac{h}{2e}$ flux always has a finite energy in the superconducting state of a charge-$e$ fermion system.

• Received 26 August 1988

DOI:https://doi.org/10.1103/PhysRevB.38.12004