Physical review. B, Condensed matter

Stability of the Nagaoka state in the one-band Hubbard model

Guang-Shan Tian

Published Sep 1, 1991• 26 citations

Abstract

We discuss the stability of the saturated ferromagnetic state of the one-band Hubbard model in the thermodynamic limit. We prove rigorously that the Nagaoka state is stable on a d-dimensional (d=2,3) simple-cubic lattice if the number of holes ${\mathit{N}}_{\mathit{h}}$ is less than ${\mathit{N}}_{\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{\alpha}}}$, 0\ensuremath{\le}\ensuremath{\alpha}1/2d. Finally, we explain briefly why the Nagaoka state is probably unstable when ${\mathit{N}}_{\mathit{h}}$\ensuremath{\gg}${\mathit{N}}_{\mathrm{\ensuremath{\Lambda}}}^{1/2\mathit{d}}$.

DOI: 10.1103/physrevb.44.4444

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Physical review. B, Condensed matter

Stability of the Nagaoka state in the one-band Hubbard model

Guang-Shan Tian

Published Sep 1, 1991• 26 citations

Publisher

Abstract

{\mathit{N}}_{\mathit{h}}

is less than

{\mathit{N}}_{\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{\alpha}}}

, 0\ensuremath{\le}\ensuremath{\alpha}1/2d. Finally, we explain briefly why the Nagaoka state is probably unstable when

{\mathit{N}}_{\mathit{h}}

\ensuremath{\gg}

{\mathit{N}}_{\mathrm{\ensuremath{\Lambda}}}^{1/2\mathit{d}}

DOI: 10.1103/physrevb.44.4444

Related papers

VARIATIONAL STUDY ON INSTABILITY OF NAGAOKA STATE IN A THERMODYNAMIC LIMIT

Shun-Qing Shen, Zhaoming Qiu

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.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.

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We consider the question of ferromagnetism in the one-band Hubbard model on the square lattice with infinite on-site repulsion using exact diagonalization techniques. Our results show a nonmonotonic behavior as a function of the number of holes ${N}_{h}$ away from half filling. We find that the spin per electron is generally finite but less than its maximum value and, for a fixed hole number, increases with increasing system size. We interpret this as indicating that the inifinite-U Hubbard model is ferromagnetic in the thermodynamic limit for a finite fraction of holes, but that there are nonferromagnetic states which are very close in energy.We consider the question of ferromagnetism in the one-band Hubbard model on the square lattice with infinite on-site repulsion using exact diagonalization techniques. Our results show a nonmonotonic behavior as a function of the number of holes ${N}_{h}$ away from half filling. We find that the spin per electron is generally finite but less than its maximum value and, for a fixed hole number, increases with increasing system size. We interpret this as indicating that the inifinite-U Hubbard model is ferromagnetic in the thermodynamic limit for a finite fraction of holes, but that there are nonferromagnetic states which are very close in energy.

Physical review. B, Condensed matterSep 1, 198985 citations

{\mathit{N}}_{\mathit{h}}

is less than

{\mathit{N}}_{\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{\alpha}}}

, 0\ensuremath{\le}\ensuremath{\alpha}1/2d. Finally, we explain briefly why the Nagaoka state is probably unstable when

{\mathit{N}}_{\mathit{h}}

\ensuremath{\gg}

{\mathit{N}}_{\mathrm{\ensuremath{\Lambda}}}^{1/2\mathit{d}}

We investigate the instability of the saturated ferromagnetic ground state (Nagaoka state) in the Hubbard model on various lattices in dimensions

d=2

and

d=3

. A variational resolvent approach is developed for the Nagaoka instability both for $U=\ensuremath{\infty}$ and for $U<\ensuremath{\infty}$ which can easily be evaluated in the thermodynamic limit on all common lattices. Our results significantly improve former variational bounds for a possible Nagaoka regime in the ground-state phase diagram of the Hubbard model. We show that a pronounced particle-hole asymmetry in the density of states and a diverging density of states at the lower band edge are the most important features in order to stabilize Nagaoka ferromagnetism, particularly in the low-density limit.

{N}_{h}

away from half filling. We find that the spin per electron is generally finite but less than its maximum value and, for a fixed hole number, increases with increasing system size. We interpret this as indicating that the inifinite-U Hubbard model is ferromagnetic in the thermodynamic limit for a finite fraction of holes, but that there are nonferromagnetic states which are very close in energy.