Bose-Fermi transformation in three-dimensional space

Abstract

A generalization of the Jordan-Wigner transformation to three (or higher) dimensions is constructed. The nonlocal mapping of spin to fermionic variables is expressed as a gauge transformation with topological charge equal to 1. The resulting fermionic theory is minimally coupled to a non-Abelian gauge field in a spontaneously broken phase containing monopoles.PACS numbers: 71.27.+a, 05.50.+q, 11.15.-q, 75.10.JmThe Jordan-Wigner (JW) transformation [1] for onedimensional spin systems has provided remarkable applications in condensed matter physics, including the twodimensional classical Ising model [2,3] and the XY spin-1/2 model [4], The counterpart in relativistic field theory, the bosonization of fermionic theories in 1+1 dimensions [5], has also opened an important field of active research.Bosonization in higher dimensions has been elusive for a long time. Relatively recent work has uncovered a Bose-Fermi transmutation in 2+1 dimensions which is experienced by the elementary excitations of the sigma model in the presence of a Chern-Simons field [6,7]. This result paved the way for the construction of the JW transformation in a lattice of two spatial dimensions, where a local fermion theory is mapped onto a system of hard-core bosons described by the Heisenberg Hamiltonian [8]. On the same basis, the bosonization scheme has been also implemented for 2+1 relativistic field theory [9].In this Letter, we propose an extension of the JW transformation to three-or more-dimensions. Here we discuss in detail the three-dimensional case. The generalization to higher dimensions is straightforward.The JW transformation relates the local spin-1/2 operators, S Z ,S ± ([S^S*] = ±S±, [5+,5-] = 2S*), to local fermionic operators, ip, ip^ ({^t/^} = 1> {ipii/>} = {^t,^t} = 0):

Algebraic approach to interacting quantum systems

“…Fradkin's approach (2d) [29] provides an alternative formulation of the original problem and has stimulated original methods to find relevant solutions in quantum Hall systems. The generalization of his approach to 3d [31] is more involved since it requires an extended Hilbert space and non-Abelian gauge transformations.…”

“…However, the deformed commutation relations of Eq. (31) indicate that the result of exchanging two anyons with different indices is the multiplication by a phase factor exp [iθ]. It is this second aspect, not related to the exclusion properties, that decides whether the particles are bosons, fermions, or anyons.…”

Generalized Jordan-Wigner Transformations

“…2, the string operators generalize [9] along the lines introduced in Ref. [14]. There is always the freedom to perform rotations in spin space to get equivalent representations to the one presented above.…”

Jordan-Wigner approach to dynamic correlations in spin ladders

“…Generalizations of the Jordan-Wigner transformation to higher dimensions have been suggested 31,32 and may be adopted for spin ladders. For spin ladders the onedimensional Jordan-Wigner transformation can be applied directly, when all spins are arranged in a onedimensional sequence.…”