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Journal of Experimental and Theoretical Physics

Absence of saturated ferromagnetism in the two-dimensional Hubbard model with two holes for U=∞

Yu. V. Mikhaǐlova

Published Mar 1, 1998• 1 citation
Publisher

Abstract

For a square Hubbard lattice with infinite repulsion energy U the following exact result has been obtained: the ferromagnetic state with maximum spin is not the ground state of the system if the number of holes is equal to two.
DOI
10.1134/1.558501

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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 Nh{N}_{h}Nh​ 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.
Ferromagnetic domains in the large-UUU Hubbard model with a few holes: an FCIQMC study
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.
We investigate the magnetization of the ground state of the one-dimensional flat-band Hubbard model as a function of the electron filling, using exact diagonalization techniques. Our calculation shows that the systems with odd n holes added to half-filling bands for a sufficiently large U (the on-site Coulomb repulsion energy) have a saturated ferromagnetic ground state, which is nondegenerate except for the trivial spin degeneracy. Dependences of the critical value of U on the size and on λ (a parameter of the system) are also investigated. For all values of λ, it is found that the ground state is a saturated ferromagnetic state if U is sufficiently large. Moreover, we find that the critical U for the saturated ferromagnetic ground state is almost independent on the system size, at least, for the system near quarter-filling. These facts indicate that even if the system size go to infinity, the ferromagnetic ground state can be exist for a finite U .
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.