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A variety of analytical approaches have been developed for the study ofquantum spin systems in two dimensions, the notable ones being spin-waves,slave boson/fermion parton constructions, and for lattices with one-to-onelocal correspondence of faces and vertices, the 2D Jordan-Wigner (JW)fermionization. Field-theoretically, JW fermionization is implemented throughChern-Simons (CS) flux attachment. For a correct fermionization of latticequantum spin-$1/2$ magnets, it is necessary that the fermions obey mutualbosonic (anyonic) statistics under exchange - this is not possible to implementon arbitrary 2D lattices if fermionic matter couples only to the lattice gaugefields. Enlarging the gauge degrees of freedom to include the dual latticeallows the construction of consistent mutual Chern-Simons field theories. Herewe propose a mutual CS theory where the microscopic (spin) degrees of freedomare represented as lattice fermionic matter additionally coupled to specificcombinations of dual lattice gauge fields that depend on the local geometry. Weillustrate the use of this method for understanding the properties of ahoneycomb Kitaev model subjected to a strong Zeeman field in the $z$-direction.Our CS gauge theory framework provides an understanding why the topologicalphase is degraded at lower (higher) critical fields for the ferro- (antiferro-)magnetic Kitaev interaction. Additionally, we observe an effectivelyone-dimensional character of the low-excitations at higher fields in the$z$-direction which we also confirm by spin-wave calculations.

Jordan-Wigner fermionization of quantum spin systems on arbitrary 2D lattices: A mutual Chern-Simons approach