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arXiv:0909.2082 · Physical Review B

Nagaoka ferromagnetism in large-spin fermionic and bosonic systems

Seiji Miyashita, Masao Ogata, Hans A. De Raedt

Published Nov 24, 2009• 6 citations
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Abstract

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.

DOI
10.1103/physrevb.80.174422

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We study the magnetic properties of itinerant quantum magnetic particles, described by a generalized Hubbard model with large spin (S>1/2)(S>1/2)(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=1/2S=1/2S=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>1/2S>1/2S>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+12S+12S+1 types of particles and we study a SU(2S+1)\text{SU}(2S+1)SU(2S+1) invariant model, the degeneracy of the ground state is given by the multiplets of the fully symmetric Young tableau of SU(2S+1)\text{SU}(2S+1)SU(2S+1) if the state with maximum total spin belongs to the ground state.
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 investigate Nagaoka ferromagnetism in the two-dimensional Hubbard model with one hole using the spin-adapted [SU(2)\text{SU}(2)SU(2) conserving] full configuration interaction quantum Monte Carlo method. This methodology gives us access to the ground-state energies of all possible spin states SSS of finite Hubbard lattices, here obtained for lattices up to 26 sites for various interaction strengths (UUU). The critical interaction strength, Uc{U}_{c}Uc​, at which the Nagaoka transition occurs is determined for each lattice and is found to be proportional to the lattice size for the larger lattices. Below Uc{U}_{c}Uc​, the overall ground states are found to favour the minimal total spin (S=12S=\frac{1}{2}S=21​), and no intermediate spin state is found to be the overall ground state on lattices larger than 16 sites. However, at Uc{U}_{c}Uc​, the energies of all the spin states are found to be nearly degenerate, implying that large fluctuations in total spin can be expected in the vicinity of the Nagaoka transition.
Whether spin-independent Coulomb interaction can be the origin of a realistic ferromagnetism in an itinerant electron system has been an open problem for a long time. Here we study a class of Hubbard models on decorated lattices, which have a special property that the corresponding single-electron Schrödinger equation has NdN_{\rm d}Nd​-fold degenerate ground states. The degeneracy NdN_{\rm d}Nd​ is proportional to the total number of sites $\absΛ$. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to $ρ_0=N_{\rm d}/(2\absΛ)$, and paramagnetism when the filling factor is sufficiently small. An important feature of the present work is that it provides examples of three dimensional itinerant electron systems which are proved to exhibit ferromagnetism in a finite range of the electron filling factor.
Following Nagaoka, the ground state of the large-U nearly half-filled Hubbard model is ``ferromagnetic.'' However, because of the anisotropic exchange energy JA{\mathit{J}}_{\mathit{A}}JA​, the physical Nagaoka ground state is nondegenerate and comprises a totally disordered spin state, i.e., a linear combination of all possible spin arrangements. For this ground state the susceptibility is Pauli-like but of a magnitude associated with a narrow band of width \ensuremath{\sim}JA{\mathit{J}}_{\mathit{A}}JA​ while the holons, i.e., the charge excitations, correspond to spinless fermions with a band of width 2zt. Associated with the spinon sector are very poorly defined spin waves, i.e., bosons, with a quadratic dispersion relation. These spin waves do not become soft at any finite temperature, rather, they soften as they approach the metal-insulator transition, i.e., as the number of holes n0{\mathit{n}}_{0}n0​ becomes small. Examined is the problem of constructing a mean-field slave-boson approach for the present problem.
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 NNN-component fermions is presented, and Nagaoka's theorem is also generalized to SU(N)\text{SU}(N)SU(N)-symmetric fermion systems on nonseparable graphs.
We study itinerant ferromagnetism in multiorbital Hubbard models in certain two-dimensional square and three-dimensional cubic lattices. In the strong coupling limit where doubly occupied orbitals are not allowed, we prove that the fully spin-polarized states are the unique ground states, apart from the trivial spin degeneracies, for a large region of filling factors. Possible applications to ppp-orbital bands with ultracold fermions in optical lattices, and electronic 3d3d3d-orbital bands in transition-metal oxides, are discussed.