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Journal of Physics Condensed Matter

Ferromagnetism in the Hubbard model

Wolfgang von der Linden, David M. Edwards

Published Jul 1, 1991• 92 citations

Abstract

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.

DOI: 10.1088/0953-8984/3/26/014

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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.

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.

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.

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.

The extrapolation of small-cluster exact-diagonalization calculations is used to examine ferromagnetism in the one-dimensional Hubbard model with a generalized type of hopping. It is found that the long-range hopping with power decaying hopping amplitudes ${(t}_{\mathrm{ij}}\ensuremath{\sim}{q}^{|i\ensuremath{-}j|})$ stabilizes the ferromagnetic state for a wide range of electron interactions U and electron concentrations n. The critical value of the interaction strength

{U}_{c}(q)

above which the ferromagnetic state becomes stable is calculated numerically and the ground-state phase diagram of the model (in the $U\ensuremath{-}q$ plane) is presented for physically the most interesting cases.

Abstract The combination of small-cluster exact-diagonalization calculations and the quantum Monte Carlo method is used to examine ferromagnetism in the two-dimensional Hubbard model with a generalized type of hopping. It is found that the long-range hopping with exponentially decaying hopping amplitudes t ij ∼ − q Ri−Rj stabilizes the ferromagnetic state for a wide range of electron interactions U and electron concentrations n > 1. The critical value of the hopping parameter q c above which the ferromagnetic state becomes stable is calculated numerically and the ground-state phase diagram of the model is discussed for physically the most interesting cases.