Abstract
Contrary to the common wisdom, local bosonizations of fermionic systems exist in higher dimensions. Interestingly, resulting bosonic variables must satisfy local constraints of a gauge type. They effectively replace long distance exchange interactions. In this work we study in detail the properties of such a system which was proposed a long time ago. In particular, dependence of the constraints on lattice geometry and fermion multiplicity is further elaborated and is now classified for all two dimensional, rectangular lattices with arbitrary sizes. For few small systems the constraints are solved analytically and the complete spectra of reduced spin hamiltonias are shown to agree with the original fermionic ones. The equivalence is extended to fermions in an external Wegner $Z_2$ field. It is also illustrated by an explicit calculation for a particular configuration of Wegner variables. Finally, a possible connection with the recently proposed web of dualities is discussed.
Highlights
Relation between fermionic and spin degrees of freedom is an old subject [1,2], but it still attracts a fair amount of interest
IV, we show that constraints can be interpreted as the condition that certain Z2 gauge field hidden in the bosonic theory is trivial
One needs additional constraints for above spins to render the exact correspondence. This can be traced to the fact that original fermionic operators S and Sobey additional relations, not present in spatial dimension one. These will have to be imposed as constraints on physical states in the spin system
Summary
Relation between fermionic and spin degrees of freedom is an old subject [1,2], but it still attracts a fair amount of interest. There has been a lot of progress in these directions recently It has been shown [5,6] that fermions in space dimension d can be exactly mapped to a local generalized gauge theory on the dual lattice, with Z2 gauge variables associated to (d − 1)dimensional objects ( an Ising model for d 1⁄4 1, standard gauge theory with modified Gauss’ law for d 1⁄4 2 and the so-called higher gauge theories for d ≥ 3). The standard way to derive this equivalence is via the Jordan-Wigner transformation [1] Direct generalization of this method to higher dimensions leads to nonlocal spin-spin interactions. V and discuss a very attractive potential relation with the rapidly developing family of dualities in (2 þ 1) dimensions
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