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Journal of Physics A Mathematical and General

A simplified proof of Nagaoka's theorem

Guang-Shan Tian

Published Jun 7, 1990• 38 citations
Publisher

Abstract

The author studies a special case of the Hubbard model in which the on-site interaction U is infinite and there is exactly one hole. Nagaoka (1966) has shown that the ground state in this case has the maximum total spin. The author gives Nagaoka's theorem another proof which is simpler, in some respects, than the original one.
DOI
10.1088/0305-4470/23/11/046

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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 consider the Hubbard model on a finite set of sites with nonpositive hopping matrix elements and infinitely strong on-site repulsion. Nagaoka's theorem states that in this model the relative ground state in the sector with one unoccupied site is maximally ferromagnetic. We show that this phenomenon is a consequence of a combinatorial coincidence valid in the one-hole regime only. In the case of more than one hole there is no reason to expect maximally ferromagnetic ground states. We prove this claim for the case of two holes for models defined on a class of graphs which contains all tori that are not too small.
We consider the Hubbard model on a finite set of sites with nonpositive hopping matrix elements and infinitely strong on-site repulsion. Nagaoka's theorem states that in this model the relative ground state in the sector with one unoccupied site is maximally ferromagnetic. We show that this phenomenon is a consequence of a combinatorial coincidence valid in the one-hole regime only. In the case of more than one hole there is no reason to expect maximally ferromagnetic ground states. We prove this claim for the case of two holes for models defined on a class of graphs which contains all tori that are not too small.