Fermionized dual vortex theory for magnetized kagom\'{e} spin liquid

Inspired by the recent quantum oscillation measurement on the kagom\'{e} lattice antiferromagnet in finite magnetic fields, we raise the question about the physical contents of the emergent fermions and the gauge fields if the U(1) spin liquid is relevant for the finite-field kagom\'{e} lattice antiferromagnet. Clearly, the magnetic field is non-perturbative in this regime, and the finite-field state has no direct relation with the U(1) Dirac spin liquid proposal at zero field. We here consider the fermionized dual vortex liquid state as one possible candidate theory to understand the magnetized kagom\'{e} spin liquid. Within the dual vortex theory, the $S^z$ magnetization is the emergent U(1) gauge flux, and the fermionized dual vortex is the emergent fermion. The magnetic field polarizes the spin component that modulates the U(1) gauge flux for the fermionized vortices and generates the quantum oscillation. Within the mean-field theory, we discuss the gauge field correlation, the vortex-antivortex continuum and the vortex thermal Hall effect.