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Physical review. B, Condensed matter

Auxiliary-space variational approach for theU=∞ Hubbard model: The stability of a ferromagnetic state

V. Ya. Krivnov, А. А. Овчинников

Published Jun 1, 1992• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.45.12996

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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.
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.
We present different numerical calculations based on variational quantum Monte Carlo simulations supporting a ferromagnetic ground state for finite and small hole densities in the two-dimensional infinite-U Hubbard model. Moreover, by studying the energies of different total spin sectors, these calculations strongly suggest that the paramagnetic phase is unstable against a phase with a partial polarization for large hole densities delta approximately 0.40 with evidence for a second-order transition to the paramagnetic large doping phase.
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.