Higher order effective low-energy theories

Details Higher order effective low-energy theories Higher order effective low-energy theories

2004-07-09

arxiv.org/abs/cond-mat/0407255v2

Phys. Rev. B 70, 235111 (2004)

Physics

Condensed Matter

Strongly Correlated Electrons

14 pages, no figures

Scientific paper

10.1103/PhysRevB.70.235111

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.

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