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Chinese Physics Letters

Low-Density Ferromagnetism in the Hubbard Model

Jin An, Chang-De Gong

Published Dec 13, 2000• 2 citations
Publisher

Abstract

We study analytically the stability of the saturated ferromagnetic ground state (the Nagaoka state) of the Hubbard model in the low electron density limit with the mean-field approach. We find that the critical U c is just 4π t , where t is the hopping integral between the nearest neighbours, i.e., when U is larger than 4π t , the paramagnetic state is unstable with respect to the ferromagnetic state, and vice versa. PACS: 75. 30. Kz, 71. 10. Fd

DOI
10.1088/0256-307x/18/1/338

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We study analytically the stability of the saturated ferromagnetic ground state (the Nagaoka state) of the Hubbard model in the low electron density limit with the mean-field approach. We find that the critical U c is just 4π t , where t is the hopping integral between the nearest neighbours, i.e., when U is larger than 4π t , the paramagnetic state is unstable with respect to the ferromagnetic state, and vice versa. PACS: 75. 30. Kz, 71. 10. Fd
We investigate the instability of the saturated ferromagnetic ground state (Nagaoka state) in the Hubbard model on various lattices in dimensions d=2d=2d=2 and d=3d=3d=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.
A single-band Hubbard model with nearest and next-nearest neighbor hopping is studied for d=1, 2, 3, using both analytical and numerical techniques. In one dimension, saturated ferromagnetism is found above a critical value of U for a band structure with two minima and for small and intermediate densities. This is an extension of a scenario recently proposed by M\"uller-Hartmann. For three dimensions and nonpathological band structures, it is proven that such a scenario does not work. \textcopyright{} 1996 The American Physical Society.
Accepted for publication in J. Phys. A A Hubbard model with a Nd-fold degenerate single-particle ground state has ferromagnetic ground states if the number of electrons is less or equal to Nd. It is shown rigorously that the local stability of ferromagnetism in such a model implies global stability: The model has only ferromagnetic ground states, if there are no single spin-flip ground states. If the number of electrons is equal to Nd, it is well known that the ferromagnetic ground state is unique if and only if the single-particle density matrix is irreducible. We present a simplified proof for this result. PACS-numbers: 75.10.Lp, 71.10.Fd 1
Ferromagnetism in the one-dimensional Hubbard model with the next-nearest-neighbor hopping is explored by using the exact-diagonalization method in a small cluster and the equation-of-motion method in the thermodynamic limit with electron density n = 3/2. With these two complementary methods, it is found that an intermediate value of the next-nearest-neighbor hopping amplitude t1 tends to stabilize the fully polarized ferromagnetic state under the condition that the on-site coulomb interaction U is sufficiently large in our model. The ground-state phase diagram of the model is presented in the t1—U plane.
The stability of the ferromagnetic state with complete spin alignment against a single spin reversal is studied in the square-lattice Hubbard model with nearest-neighbour hopping. A variational ansatz is used, which is exact in one dimension and by comparison with accurate results for small clusters yields almost exact results in the two-dimensional case. The ferromagnetic region in the (W/U, delta ) phase diagram is mapped out, where W is the bandwidth, U the on-site electron interaction and delta is the number of holes per atom. This region is considerably smaller than in previous variational calculations and it is rigorously concluded that the state of complete spin alignment is unstable when delta >0.29, for all U, and when W/U>0.19, for all delta . The nature of the instability, and of the low-lying excitations in the ferromagnetic state for varying delta and W/V, is discussed.
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 Nh{\mathit{N}}_{\mathit{h}}Nh​ 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 Nh{\mathit{N}}_{\mathit{h}}Nh​\ensuremath{\gg}${\mathit{N}}_{\mathrm{\ensuremath{\Lambda}}}^{1/2\mathit{d}}$.