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International Journal of Modern Physics B

ON THE INSTABILITY OF THE FERROMAGNETIC STATE IN THE U=∞ LIMIT OF THE HUBBARD MODEL

Benoît Douçot, R. Rammal

Published Dec 1, 1989• 10 citations
Publisher

Abstract

The U=∞ limit of the Hubbard model is studied using a variational approach in which the spins are in a static coherent state. We find that the Nagaoka Ferromagnetic state is no longer the ground state as soon as two holes are present, and this for various geometries. We suggest that this instability also remains for a finite density of holes.

DOI
10.1142/s0217979289001123

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Annalen der Physik•Jan 1, 1995•13 citations
The U=∞ limit of the Hubbard model is studied using a variational approach in which the spins are in a static coherent state. We find that the Nagaoka Ferromagnetic state is no longer the ground state as soon as two holes are present, and this for various geometries. We suggest that this instability also remains for a finite density of holes.
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.
Abstract The instability of the fully polarized ferromagnetic ground state (Nagaoka state) of the Hubbard model on the square lattice is investigated. We use single spin flip variational wave functions including majority spin correlation effects and calculate spin flip energies in the thermodynamic limit. With very local wave functions and with moderate numbers of variational parameters we reproduce the best known estimate for the critical hole density δcr = 0.29 and we obtain an estimate of Ucr = 63 t for the critical coupling which is considerably better than the best estimate of Ucr = 42 t previously known. The simplicity of our wave functions makes the physical origin of the various aspects of the instability particularly transparent.
A variational approach has been used to study the stability of the Nagaoka state in the Hubbard model with infinite interaction. In this approach, real Fermi operators are replaced by auxiliary ones; an effective Hamiltonian is constructed in terms of these auxiliary operators. A two-chain (ladder) system is considered as an example. For this model, we show that the Nagaoka state is unstable with respect to the creation of two-phase states: ferromagnetic and singlet spin states. The instability occurs at a finite density of holes.
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.
Abstract We discuss two single spin flip variational wave functions describing spin wave excitations which were proposed earlier by Shastry, Krishnamurthy and Anderson (SKA) and by Basile and Elser (BE), respectively, in order to investigate the instability of the fully polarized ferromagnetic state (Nagaoka state) in the infinite U Hubbard model. We calculate the energy of these variational states for the square lattice and for multiple chains. At the zone boundary in the vicinity of the point (0, π) the spin wave energy is reduced substantially by the binding of the spin up hole to the flipped down spin. For the square lattice this leads to a critical hole density of δcr = 0.407 for the SKA spin wave and of δcr = 0.322 for the BE spin wave which implies remarkable improvements in comparison to the corresponding scattering states investigated previously.