Higher order effective low-energy theories

TL;DR

This paper compares three perturbative methods for deriving low-energy theories from the Hubbard model, showing their equivalence up to fourth order and discussing technical challenges and operator transformations.

Contribution

It provides a detailed comparison of perturbative approaches to derive effective low-energy theories, highlighting their equivalence and technical subtleties at higher orders.

Findings

All three methods yield the same effective t-J model up to fourth order.

Different forms of the Hamiltonian at higher orders are related by unitary transformations.

Transforming operators is crucial for demonstrating equivalence across approaches.

Abstract

Three well-known perturbative approaches to deriving low-energy effective theories, the degenerate Brillouin-Wigner perturbation theory (projection method), the canonical transformation, and the resolvent methods are compared. We use the Hubbard model as an example to show how, to fourth order in hopping t, all methods lead to the same effective theory, namely the t-J model with ring exchange and various correlated hoppings. We emphasize subtle technical difficulties that make such a derivation less trivial to carry out for orders higher than second. We also show that in higher orders, different approaches can lead to seemingly different forms for the low-energy Hamiltonian. All of these forms are equivalent since they are connected by an additional unitary transformation whose generator is given explicitly. The importance of transforming the operators is emphasized and the equivalence of their transformed structure within the different approaches is also demonstrated.