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A Model of Itinerant Ferromagnetism in Two- and Three- Dimensional Multi-orbital Systems

Yi Li, Élliott H. Lieb, Congjun Wu

Published Oct 16, 2013• 0 citations
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Abstract

We study itinerant ferromagnetism in multi-orbital 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 any generic filling factor 0 < n < 2 (0 < n < 3) in two (three) dimensions. Possible applications to p-orbital bands with ultra-cold fermions in optical lattices are discussed.

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We study itinerant ferromagnetism in multi-orbital 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 any generic filling factor 0 < n < 2 (0 < n < 3) in two (three) dimensions. Possible applications to p-orbital bands with ultra-cold fermions in optical lattices are discussed.
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.
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.
We study a class of Hubbard models in which the corresponding single-electron ground states have bulk degeneracy. We prove that the ground states of the models exhibit ferromagnetism when the electron filling factor is not more than and sufficiently close to ${\mathrm{\ensuremath{\rho}}}_{0}$=\ensuremath{\Vert}V\ensuremath{\Vert}/2\ensuremath{\Vert}\ensuremath{\Lambda}\ensuremath{\Vert} (where \ensuremath{\Vert}V\ensuremath{\Vert} is the dimension of degeneracy and \ensuremath{\Vert}\ensuremath{\Lambda}\ensuremath{\Vert} is the number of sites), and exhibit paramagnetism when the filling factor is sufficiently small. This is the first time that a three-dimensional itinerant-electron system is proved to exhibit ferromagnetism in a finite range of the electron filling factor.
We studied ferromagnetism in the one-dimensional Hubbard model with doubly degenerate atomic orbitals by means of the density-matrix renormalization-group method and obtained the ground-state phase diagrams. It was found that ferromagnetism is stable from low to high (0<n<1.75)(0<n<1.75)(0<n<1.75) electron density when the interactions are sufficiently strong. The quasi-long-range order of triplet superconductivity coexists with the ferromagnetic order for a strong Hund coupling region, where the interorbital interaction ${U}^{\ensuremath{'}}\ensuremath{-}J$ is attractive. At quarter-filling (n=1),(n=1),(n=1), the insulating ferromagnetic state appears, accompanying orbital quasi-long-range order. For low densities (n<1),(n<1),(n<1), ferromagnetism occurs owing to the ferromagnetic exchange interaction caused by virtual hoppings of electrons, the same as in the quarter-filled system. This comes from separation of the charge and spin-orbital degrees of freedom in the strong-coupling limit. This ferromagnetism is fragile against variation of band structure. For high densities (n>1),(n>1),(n>1), the phase diagram of the ferromagnetic phase is similar to that obtained in infinite dimensions. In this case, the double exchange mechanism is operative for stabilizing the ferromagnetic order and this long-range order is robust against variation of the band dispersion. A partially polarized state appears in the density region $1.68\ensuremath{\lesssim}n\ensuremath{\lesssim}1.75$ and phase separation occurs for n just below the half-filling (n=2).(n=2).(n=2).
The microscopic mechanism of itinerant ferromagnetism is a long-standing problem due to the lack of non-perturbative methods to handle strong magnetic fluctuations of itinerant electrons. We have non-pertubatively studied thermodynamic properties and magnetic phase transitions of a two-dimensional multi-orbital Hubbard model exhibiting ferromagnetic ground states. Quantum Monte-Carlo simulations are employed, which are proved in a wide density region free of the sign problem usually suffered by simulations for fermions. Both Hund's coupling and electron itinerancy are essential for establishing the ferromagnetic coherence. No local magnetic moments exist in the system as a priori, nevertheless, the spin channel remains incoherent showing the Curie-Weiss type spin magnetic susceptibility down to very low temperatures at which the charge channel is already coherent exhibiting a weakly temperature-dependent compressibility. For the SU(2) invariant systems, the spin susceptibility further grows exponentially as approaching zero temperature in two dimensions. In the paramagnetic phase close to the Curie temperature, the momentum space Fermi distributions exhibit strong resemblance to those in the fully polarized state. The long-range ferromagnetic ordering appears when the symmetry is reduced to the Ising class, and the Curie temperature is accurately determined. These simulations provide helpful guidance to searching for novel ferromagnetic materials in both strongly correlated ddd-orbital transition metal oxide layers and the ppp-orbital ultra-cold atom optical lattice systems.
Role of the double exchange for appearance of partial ferromagnetism in a semimetal is explored for a semimetallic two-band Hubbard model in one-dimension. In a strong coupling limit, a ferromagnetic exchange interaction between two neighboring triplet excitons becomes dominant compared with anti-ferromagnetic couplings and a partial spin polarization is shown to appear. Exact evaluation of magnetization is possible for the partially ferromagnetic ground state in this limit. Obtained phase diagram has a doping dependence similar to our DMRG results for finite interactions. The ferromagnetism is favored in semimetallic systems by the interplay of the intra-orbital Coulomb repulsion and the inter-orbital Hund-rule coupling.