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Physical review. B, Condensed matter

Extension of Nagaoka’s theorem on the large- U Hubbard model

Hal Tasaki

Published Nov 1, 1989• 194 citations
Publisher

Abstract

An extension is given of Nagaoka's theorem on the existence of ferromagnetism in the large-U Hubbard model with precisely one hole. The present extension covers a large class of models with arbitrary non-negative hopping matrix elements and arbitrary spin-independent interactions.
DOI
10.1103/physrevb.40.9192

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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 prove that Nagaoka's theorem, that the ground state of the large-UUU Hubbard model with exactly one hole is ferromagnetic, holds for any balanced Hamiltonian. We argue that, in itinerant electron systems, a balanced Hamiltonian, rather than bipartite lattice, defines an unfrustrated system. The proof is valid for multiorbital models with arbitrary two-orbital interactions provided that no exchange interactions are antiferromagnetic: a class of models including the Kanomori Hamiltonian.
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 full configuration interaction quantum Monte Carlo 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, centered on individual holes. The overall total spin of the wave functions remains 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 1) number of holes.
We present an extension of Nagaoka's theorem in the SU(nnn) generalization of the infinite-UUU Hubbard model. It is shown that, when there is exactly one hole, the fully polarized states analogous to the ferromagnetic states in the SU(2) Hubbard model are ground states. For a restricted class of models satisfying the connectivity condition, these fully polarized states are the unique ground states up to the trivial degeneracy due to the SU(nnn) symmetry. We also give examples of lattices in which the connectivity condition can be verified explicitly. The examples include the triangular, Kagom\'e, and hypercubic lattices in $d\phantom{\rule{0.28em}{0ex}}(\ensuremath{\ge}\phantom{\rule{-0.16em}{0ex}}2)$ dimensions, among which the cases of d=2d=2d=2 and 3 are experimentally realizable in ultracold atomic gases loaded into optical lattices.
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.