Hubbard Hamiltonian in the dimer representation. Large U limit

TL;DR

This paper derives an analytical form of the Hubbard Hamiltonian for a simple cubic lattice in the large U limit using dimer representation, revealing complex interactions influencing thermodynamic properties.

Contribution

It introduces a nonperturbative analytical derivation of the Hubbard Hamiltonian in the dimer representation, including a simplified form via Taylor expansion, applicable to various cluster sizes.

Findings

Derived the Hubbard Hamiltonian in the large U limit using dimer solutions.

Reproduced the exact Hubbard Hamiltonian as U approaches infinity.

Highlighted the complex interplay of magnetic interactions affecting thermodynamics.

Abstract

We formulate the Hubbard model for the simple cubic lattice in the representation of interacting dimers applying the exact solution of the dimer problem. By eliminating from the considerations unoccupied dimer energy levels in the large U limit (it is the only assumption) we analytically derive the Hubbard Hamiltonian for the dimer (analogous to the well-known t-J model), as well as, the Hubbard Hamiltonian for the crystal as a whole by means of the projection technique. Using this approach we can better visualize the complexity of the model, so deeply hidden in its original form. The resulting Hamiltonian is a mixture of many multiple ferromagnetic, antiferromagnetic and more exotic interactions competing one with another. The interplay between different competitive interactions has a decisive influence on the resulting thermodynamic properties of the model, depending on temperature, model parameters and assumed average number of electrons per lattice site. A simplified form of the derived Hamiltonian can be obtained using additionally Taylor expansion with respect to $x=\frac{t}{U}$ (t-hopping integral between nearest neighbours, U-Coulomb repulsion). As an example, we present the expansion including all terms proportional to t and to $\frac{t^2}U$ and we reproduce the exact form of the Hubbard Hamiltonian in the limit $U\to \infty $. The nonperturbative approach, presented in this paper, can, in principle, be applied to clusters of any size, as well as, to another types of model Hamiltonians.