Generalized Dyson-Maleev representation of doped and undoped Heisenberg antiferromagnets

Abstract

The doped and undoped Heisenberg antiferromagnet is described within a rigorous and unified theoretical framework based on ideas of Dyson. The unified description is achieved by means of a particular representation of the relevant operators, referred to as generalized Dyson-Maleev (DM) representation, where the kinematic constraints are fully taken into account by means of a projection operator. The latter, whose explicit analytic form is derived, commutes with the Hamiltonian, and its main effect is that the representation vanishes on the whole unphysical subspace. On the physical subspace, our representation is similar to that proposed by Schmitt-Rink et al., and coincides with the latter in the linear spin-wave approximation. In the absence of holes, the generalized DM representation properly reduces to the ordinary DM representation of spin-1/2 operators. One of the main results of our approach is that the projection operator has no effect on the eigenvalues, but does affect the eigenstates and, hence, expectation values and correlation functions. As in the original Dyson theory, there is thus a clear separation of kinematic and dynamic effects. Finally, the case of a single hole is treated in some detail. Guided by the formal similarity with the Fröhlich polaron model, the Hamiltonian is subject to a unitary transformation, which is the lattice version of the Jost transformation, well-known in polaron theory. The result of the transformation is that the Hamiltonian assumes diagonal form in the hole operators.