Supersymmetric Hubbard operators

We develop a supersymmetric representation of the Hubbard operator which unifies the slave boson and slave fermion representations into a single U(1)× SU(1|1) gauge theory, a group with larger symmetry than the product of two U(1) gauge groups. These representations of the Hubbard operator can be used to incorporate strong Hund’s interactions in multi-electron atoms as a constraint. We show how this method can be combined with the SP (N) group to yield a locally supersymmetric large-N formulation of the t − J model. 75.30.Mb,75.20.Hr Typeset using REVTEX 1 One of the fascinating aspects of strongly correlated materials is their propensity to develop novel metallic phases in situations where local moments interact strongly with mobile electrons. Examples of such situations include metals near a metal insulator transition, metals at an anti-ferromagnetic quantum critical point and anti-ferromagnetic heavy fermion superconductors. These discoveries challenge our understanding of how spin and charge interact at the brink of magnetism. Theoretical approaches to these problems are hindered by the difficulty of capturing the profound transformation in spin correlations that develops at the boundary between antiferromagnetism and paramagnetism. Usually we model these features by representing the spin as a boson in a magnetic phase, or as a fermion in a paramagnetic phase, but by making this choice, the character of spin and charge excitations which appear in an approximate field theory is restricted and lacks the flexibility to describe the co-existence of strong magnetic correlations within a paramagnetic phase. These considerations have motivated the development of new methods to describe the spin and charge excitations of a strongly correlated material which avoid making the choice between a bosonic or fermionic spin.6–9 This paper attempts to stimulate further progress in this direction by introducing a supersymmetric representation of Hubbard operators. The method used here is an extension of the supersymmetric spin representation introduced by Coleman, Pépin and Tsvelik (CPT). Remarkably, the supersymmetry in the CPT spin representation survives the introduction of charge degrees of freedom, opening the method to a wider range of models. Hubbard operators provide a way to describe atoms in which Coulomb repulsion prevents double-occupancy of a given orbital. Suppose |a〉 ∈ {|0〉, |σ〉} describes a set of atomic states involving a charged “hole” |0〉 or a neutral spin state |σ〉 with spin component σ ∈ {1 . . .N} which for generality can have one of N possible values. The Hubbard operators are written