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

A variety of analytical approaches have been developed for the study of quantum spin systems in two dimensions, the notable ones being spin-waves, slave boson/fermion parton constructions, and for lattices with one-to-one local correspondence of faces and vertices, the 2D Jordan-Wigner (JW) fermionization. Field-theoretically, JW fermionization is implemented through Chern-Simons (CS) ﬂux attachment. For a correct fermionization of lattice quantum spin-1 / 2 magnets, it is necessary that the fermions obey mutual bosonic (anyonic) statistics under exchange - this is not possible to implement on arbitrary 2D lattices if fermionic matter couples only to the lattice gauge ﬁelds. Enlarging the gauge degrees of freedom to include the dual lattice allows the construction of consistent mutual Chern-Simons ﬁeld theories. Here we propose a mutual CS theory where the microscopic (spin) degrees of freedom are represented as lattice fermionic matter additionally coupled to speciﬁc combinations of dual lattice gauge ﬁelds that depend on the local geometry. We illustrate the use of this method for understanding the properties of a honeycomb Kitaev model subjected to a strong Zeeman ﬁeld in the z -direction.