Exact ground states for the four-electron problem in a two-dimensional finite Hubbard square system

Abstract

We present exact explicit analytical results describing the exact ground state of four electrons in a two-dimensional square Hubbard cluster containing 16 sites taken with periodic boundary conditions. The presented procedure, which works for arbitrary even particle number and lattice sites, is based on explicitly given symmetry adapted base vectors constructed in r space. The Hamiltonian acting on these states generates a closed system of 85 linear equations, providing by its minimum eigenvalue, the exact ground state of the system. The presented results, described with the aim to generate further creative developments, not only show how the ground state can be exactly obtained and what kind of contributions enter in its construction, but also emphasize further characteristics of the spectrum. On this line (i) possible explications are found regarding why weak coupling expansions often provide a good approximation for the Hubbard model at intermediate couplings, or (ii) explicitly given low-lying energy states of the kinetic energy, avoiding double occupancy, suggest new roots for pairing mechanism attracting decrease in the kinetic energy, as emphasized by kinetic energy driven superconductivity theories.

Acknowledgements

This work was supported by the Hungarian Scientific Research Fund through contract OTKA-T-037212. The numerical calculations have been done at the Supercomputing Lab. of the Faculty of Natural Sciences, Univ. of Debrecen, supported by OTKA-M-041537. Z.G. kindly acknowledges test numerical results obtained from exact numerical diagonalization on the full Hilbert space provided by Sandro Sorella and Arnd Hubsch.