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arXiv:1210.4774 · Physical Review A

Nagaoka states in the SU( n ) Hubbard model

Hosho Katsura, Akinori Tanaka

Published Jan 17, 2013• 38 citations

Abstract

We present an extension of Nagaoka's theorem in the SU($n$) generalization of the infinite-$U$ 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($n$) 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 $d=2$ and 3 are experimentally realizable in ultracold atomic gases loaded into optical lattices.

DOI: 10.1103/physreva.87.013617

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We present an extension of Nagaoka's theorem in the SU(

n

) generalization of the infinite-

U

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(

n

) 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=2

and 3 are experimentally realizable in ultracold atomic gases loaded into optical lattices.

We study the magnetic properties of itinerant quantum magnetic particles, described by a generalized Hubbard model with large spin

(S>1/2)

, which may be realized in optical lattices of laser-cooled atom systems. In fermion systems (half-integer spins), an extended form of Nagaoka ferromagnetism may be realized. However, as novel aspects of the large-spin cases, we found that the condition on the lattice connectivity is more stringent than in the case of

S=1/2

particles and that the system shows a peculiar degenerate structure of the ground state in which the ferromagnetic state is included. In contrast, it turns out that the ground state of itinerant bosonic systems (integer spins) has a degenerate structure similar to that of fermion system with

S>1/2

regardless of the shape, connectivity, or filling of the lattice, and that the state with the maximum total spin is always one of the ground states. Because the system consists of

2S+1

types of particles and we study a

\text{SU}(2S+1)

invariant model, the degeneracy of the ground state is given by the multiplets of the fully symmetric Young tableau of

\text{SU}(2S+1)

if the state with maximum total spin belongs to the ground state.

The U=\ensuremath{\infty} Hubbard model on a hypercubic lattice with exactly one hole is studied. It is seen that a degeneracy can occur in the maximal-spin sector. It is proved variationally that the Nagaoka (maximal spin) state is not a ground state, if degeneracy exists. We give an explicit construction of a variational state, an incommensurate spin-wave state, with energy lower than that of the Nagaoka state. For the case of more than one hole, a few examples are discussed where a degeneracy exists. \textcopyright{} 1996 The American Physical Society.

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 apply a result from graph theory to prove exact results about itinerant ferromagnetism. Nagaoka's theorem is extended to all nonseparable graphs except single polygons with more than four vertices by applying the solution to the generalized 15-puzzle problem, which studies whether the hole's motion can connect all possible tile configurations. This proves that the ground state of a $U\ensuremath{\rightarrow}\ensuremath{\infty}$ Hubbard model with one hole away from the half filling on a two-dimensional honeycomb lattice or a three-dimensional diamond lattice is fully spin polarized. Furthermore, the condition of connectivity for

N

-component fermions is presented, and Nagaoka's theorem is also generalized to

\text{SU}(N)

-symmetric fermion systems on nonseparable graphs.