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

TL;DR

This paper provides an exact analytical solution for the ground state of four electrons in a 16-site two-dimensional Hubbard model, revealing insights into spectral characteristics and pairing mechanisms.

Contribution

It introduces a symmetry-adapted basis and a systematic method to exactly determine the ground state for finite Hubbard clusters with arbitrary even particle numbers.

Findings

Exact ground state obtained for four electrons in a 16-site Hubbard cluster.

Insights into why weak coupling expansions are effective at intermediate couplings.

Identification of low-lying energy states relevant to pairing mechanisms.

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 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.