Ambiguity in the statistics of single-component winding vortex in a two-band superconductor

There are four topologically different paths to interchange two single-component winding vortices in a two-band superconductor having a weak inter-band interaction, where each band has its own superconducting component and its own phase, θν, other than the ordinary quantum phase, Θ (ν is the index of the band and Θ is defined at the region of θ1=θ2.). The acquired phase for the interchange of two single-winding vortices depends on the path, and three values, Θsoliton, −Θsoliton and 0, are observed. Θsoliton is the phase-slip accompanying the inter-band phase difference soliton (i-soliton). The interchange requires the creation and annihilation of the i-soliton and gives rise to an ambiguity in the statistics.