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.

Exact ground state for the four-electron problem in a 2D finite honeycomb lattice

“…The technique we apply, which has never been used in the study of 2D materials with hexagonal repeat units, has been described in details in Ref. 24 , where it has been successfully utilized in deriving the four-electron ground state for finite 2D Hubbard model on a square lattice. The method works for singlet ground states |Ψ g provided by an arbitrary even number of electrons, N, whose Hilbert space is H. The procedure itself is based on the identification of a small subspace S in which the ground state is placed giving rise to the exact, explicit and handable expression of the multielectronic ground state wave function.…”

“…For example, in case of the 2D square Hubbard system analyzed in Ref. 24 , it was shown that for N = 4 electrons and N Λ = 4 × 4 = 16 sites, for which the Hilbert space dimensionality is Dim(H) = 14400, the subspace S containing |Ψ g has only the dimension Dim(S) = 85.…”

“…Hence, in the process of deducing |Ψ g for the square system in Ref. 24 , working in S instead of H, one has a 170 times of dimensionality reduction (i.e. two orders of magnitude).…”

“…Besides experimental observations 6,12 , theoretical investigations for the four-particle cases are also present, mostly by numerical descriptions using exact diagonalization 10 , or effective theories 11 . However, connected to, and originating from the search for techniques leading to non-approximated results for non-integrable systems in one 4,5,20,21 , two 22 , and three 23 dimensions, also exact results are present for the four particle problem in the 2D square Hubbard case 24 , or Hubbard ladders 25 .…”

“…Starting from these requirements, in the present paper we present exact four-particle ground states for 2D honeycomb samples with periodic boundary conditions taken in both directions. The method 24 is based on deducing a small subspace S containing the |Ψ g ground state wave function in exact terms, followed by the non-approximated calculation of different ground state characteristics. The technique itself starts from a wave vector |1 containing the most interacting particle configuration translated to each site of the lattice and added.…”