Simplifying higher order perturbation theory for effective spin-1/2 models

Magnetic phenomena in Mott insulators are often understood by constructing effective spin-1/2 models. The standard way of determining these spin models is through second order perturbation theory in electron hoppings on a lattice. However, recent work on various d-orbital systems have shown that this approach is insufficient to capture the physics, when the effective spin-1/2 manifold is not well separated from higher energy levels. A recent example is provided by certain double perovskite magnets which feature competing multipolar orders. To overcome these issues there are two main options: to make long, demanding calculations to go to higher orders of perturbation theory or to resort to numerical non-perturbative methods. We propose a novel third option that allows for analytical calculations to be performed more simply. This new approach is a two-step process, where one uses usual second order perturbation theory in the hoppings to find the Hamiltonian for a higher-spin problem, then using perturbation theory to integrate out higher energy/higher spin modes to get an improved spin-1/2 model. Such an improved perturbation theory can totally reorganize which exchange couplings dominate, or can even generate entirely new types of magnetic Hamiltonians.

In this talk, I will present the basics of the method and give examples that show its effectiveness.