Cumulant Expansion of Localized-Electron Model for Antiferromagnetic Insulators

A many-body perturbation theory is developed for the calculation of the antiferromagnetic ground state and lower excited states of one-electron atoms in the insulating state. It is assumed that the interatomic distances are moderately small and that the interactions are well described by exchange coupling, but that distortion of atoms in the lattice is still negligible. The method is essentially the Heitler-London approach: A set of Slater determinants is constructed from nonorthogonal atomic orbitals with all possible spin arrangements. The unperturbed ground state $|0〉$ has an alternating spin configuration, while excited states are generated one by one from $|0〉$ by interchanges of spins and by projecting out the lower excited states. The resulting orthonormal states introduce the concept of quasielectron (or quasihole) states and define creation and destruction operators of those quasiparticles. Although the quasiparticles are correlated with each other through the overlap integrals in the projected wave functions, the energy matrix elements can be decomposed into linked clusters of localized electrons and holes. It is found that the clusters thus generated are cumulants. The expression for the energy matrix is then transformed into the form of an effective Hamiltonian consisting of cumulants and of creation and destruction operators of quasifermions, and the concept of the exchange interaction is defined in terms of those operators. The new formulation not only facilitates the use of the familiar many-body perturbation theory, but also eliminates difficulties in handling spin variables in the Heisenberg theory of magnetism. If only the two-body Coulomb and exchange diagrams are retained and the higher-order diagrams are neglected, however, the effective Hamiltonian takes a form reminiscent of the anisotropic exchange Hamiltonian, $H_{\mathrm{spin}}=-{\Sigma }_{k,h}{J}_{\mathrm{kh}}[{S_k}^z{S_h}^z+\lambda ({S_k}^x{S_h}^x+{S_k}^y{S_h}^y\left)\right]$, which is used in the generalized Heisenberg theory and which includes the Ising model in its limit $\lambda \to 0$. Finally, it is shown that excitations accompanying electron transfers are readily included in the present method as additional diagrams by an extension of the prescriptions for the generation of excited spin states. This yields, in its first approximation, the intra-atomic interaction and the electron-transfer interaction, leading to the Hamiltonian assumed by Hubbard and Anderson.