Canonical perturbation expansion of the Hubbard model

We have incorporated the projection operators to the canonical transformation to derive an analytical infinite perturbation-expansion series. This canonical perturbation expansion (CPE) is valid if the unperturbed Hamiltonian $H_0$ and the perturbation $H_1$ can be expressed as $H_0=\Sigma j^{}{P}_{j}H{P}_j$ and $H_1=\Sigma {j\ne k}^{}{P}_{j}H{P}_k$, where $P_j$ is the projection operator corresponding to a group of closely spaced effective one-electron orbital energies $E_{j\mu }$ with $\mu =1, 2, \dots , d_j$, and if $|E_{j\mu }-E_{j\nu }|\ll |E_{j\omega }-E_{k\delta }|$ with $j\ne k$. We have shown that the CPE is equivalent to the time-dependent perturbation theory. An extremely simple effective Hamiltonian $\tilde{H}$ is obtained when the CPE is applied to the $s$-band Hubbard model at the atomic limit. An explicit form of $\tilde{H}$ to the eighth order is given, and the magnetic interaction in $\tilde{H}$ is of the form of Heisenberg exchange ${\overrightarrow{\mathrm{S}}}_i·{\overrightarrow{\mathrm{S}}}_j$, including far neighbors. We then use this form to compute the antiferromagnetic groundstate energy to the seventh order. Our result is compared with other works.