Higher order corrections to effective low-energy theories for strongly correlated electron systems

There is a significant recent interest in higher-order corrections to effective low-energy theories for a broad range of strongly-correlated electronic problems. We use the Hubbard model as an example to show how, to fourth order in hopping $t$, well-known perturbative approaches lead to the same effective theory, namely the $t$-$J$ model with ring exchange and various correlated hoppings. Several new issues appear in deriving higher-order low-energy effective theories. One may find amusing that the low-energy Hamiltonians obtained from different methods appear to be {\it different} in each case. However, we show that they are all connected by an additional unitary transformation that leaves the block-diagonal form invariant. We also emphasize the importance of transforming all the operators along with the Hamiltonian and demonstrate the equivalence of their transformed structure within the different approaches.