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Journal of Physics A Mathematical and General

The Nagaoka state in the one-hand Hubbard model with two and more holes

Guang-Shan Tian

Published Jan 21, 1991• 14 citations
Publisher

Abstract

The infinite-U Hubbard model with two holes on a two-dimensional square lattice is explicitly studied. The author shows that the energy of the Nagaoka state and the exact ground state become degenerate in the thermodynamic limit, i.e. there exists no energy gap between the ground state and the first excited state. Finally, they discuss briefly how to generalize this result to cases in which there is a finite number of holes and the structure of the lattice is more complicated.

DOI
10.1088/0305-4470/24/2/023

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arXiv:cond-mat/9607126•Physical review. B, Condensed matter•Oct 1, 1996•0 citations
The infinite-U Hubbard model with two holes on a two-dimensional square lattice is explicitly studied. The author shows that the energy of the Nagaoka state and the exact ground state become degenerate in the thermodynamic limit, i.e. there exists no energy gap between the ground state and the first excited state. Finally, they discuss briefly how to generalize this result to cases in which there is a finite number of holes and the structure of the lattice is more complicated.
In this paper, we have rigorously shown that, in a thermodynamic limit, there always exists a one-spin-flipped state degenerate with the Nagaoka state in one-band Hubbard model on a two and three-dimensional simple cubic lattice when the number of holes [Formula: see text] (with 0≤α<1, N∧ and N e are the numbers of sites and electrons respectively) for all U (including U=∞). It is also shown that there does not exist a finite critical value U c for which, when U>U c , the Nagaoka state has the ground-state energy of the system for a two-dimensional square lattice when ρ→0.
The solution of a three-body Schr\"odinger equation (for two holes and a spin flip) shows that in the U=\ensuremath{\infty} Hubbard model the saturated ferromagnetic state with two holes is unstable (in a finite volume and for periodic boundary conditions). The ground state carries a finite momentum, and becomes degenerate with the Nagaoka state in the thermodynamic limit. This is in agreement with exact diagonalization results for small lattices where, in addition, we find that the true two-hole ground state is a singlet. As a result, our notions of ferromagnetism in the Hubbard model might need revision.
The U=\ensuremath{\infty} Hubbard model on a hypercubic lattice with exactly one hole is studied. It is seen that a degeneracy can occur in the maximal-spin sector. It is proved variationally that the Nagaoka (maximal spin) state is not a ground state, if degeneracy exists. We give an explicit construction of a variational state, an incommensurate spin-wave state, with energy lower than that of the Nagaoka state. For the case of more than one hole, a few examples are discussed where a degeneracy exists. \textcopyright{} 1996 The American Physical Society.