To help you understand Eq. (11.169), consider a two-site lattice with the Hamiltonian in Eq. (11.163). Assume that
min∣V1±V2∣≫t,
where t=t12=t21. For t=0, the Hamiltonian is diagonal in the Sz basis, and the four energy levels are widely separated. For the case of small but non-zero t, use the methods described in Appendix I to find a local quantum circuit (unitary transformation) which will perturbatively eliminate t to lowest order. With this in hand, transform the Sz operators
Li=USizU†
and explicitly obtain an expression for the terms given schematically in Eq. (11.169). Appendix I focuses on low-energy states, but the methods readily generalize to the present problem, where we are considering all four energy eigenstates of this small lattice.