A Hubbard-like model with SU(4) symmetry for electrons with twofold orbital degeneracy is studied extensively. Exact solution in one dimension is derived by means of Bethe ansatz, where the sites are supposed to be occupied by at most two electrons. The features of ground state and excited states for repulsive coupling are shown. For finite
N number of electrons, the configurations of quantum numbers are given explicitly and the spectra of excitations are obtained by solving Bethe-ansatz equation numerically. For infinite
N, the ground state and various kinds of low-lying excitations are obtained on the basis of thermodynamics limit.
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) 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), the insulating ferromagnetic state appears, accompanying orbital quasi-long-range order. For low densities
(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), 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).We study the elementary excitations of the ferromagnetic spin-orbital model which represents the low-energy excitations for strongly correlated electrons described by the Hubbard Hamiltonian with a twofold degeneracy. At most two and four spin-and-orbital bound states have been found in one and two dimensions, respectively. For a completely degenerate system, the spin-and-orbital bound states are the lowest-lying elementary excitations in most of the Brillouin zone and for Ising-like interactions they can have much lower energy than the other excitations of the model and might thus be responsible for the instability of the ground state. This might lead to quantum disordered states in both spin and orbital space, as indicated by exact solutions obtained for small clusters.
The enhancement of the spin-space symmetry from the usual
SU(2) to
SU(N) with
N>2 is promising for finding nontrivial quantum spin liquids, but the realization of
SU(N) spin systems in real materials is still challenging. Although there is a proposal in cold atomic systems, in magnetic materials with a spin-orbital degree of freedom it is difficult to achieve the
SU(N) symmetry by fine tuning. Here we propose a new mechanism by which the
SU(4) symmetry emerges in the strong spin-orbit coupling limit. In
d1 transition metal compounds with edge-sharing anion octahedra, the spin-orbit coupling gives rise to strongly bond-dependent and apparently
SU(4)-breaking hopping between the
Jeff=3/2 quartets. However, in the honeycomb structure, a gauge transformation maps the system to an
SU(4)-symmetric Hubbard model, which means that the system has a hidden symmetry in spite of its large spin-orbit coupling. In the strong repulsion limit at quarter filling, as expected in
α-ZrCl
3, the low-energy effective model is the
SU(4) Heisenberg model on the honeycomb lattice, which cannot have a trivial gapped ground state and is expected to host a gapless spin-orbital liquid. In such quantum spin-orbital liquids, both the spin and orbital degrees of freedom become fractionalized and correlated together at low temperature due to the strong frustrated interactions between them. Similarly to spinons in pure quantum spin liquids, quantum spin-orbital liquids can host not only spinon excitations, but also fermionic "orbitalon" excitations at low temperature, which we have named here in distinction from orbitons in the symmetry-broken Jahn-Teller phases.
The present paper studies the symmetries of the Hubbard model of electrons with generally n-fold orbital degeneracy. It is shown that SUd(2n) and SUc(2n) symmetries hold, respectively, for the model with completely repulsive or attractive on-site interaction and that with partly attractive interactions. An extended Lieb–Mattis transformation is given to map these two symmetries into each other. The subsymmetry SUd(e)(n)⊗SUd(o)(n) is found to be shared by the two models with arbitrary chemical potential μ. By assuming at most two electrons on each site it is found that SUd(2n)P and SUc(2n)P both exist in each kind of the two models and consequently lead to a larger symmetry SUd(2n)P×SUc(2n)P. Another underlying symmetry (SUc(e)(2)P×⋯×SUc(e)(2)P)⊗(SUc(o)(2)P×⋯×SUc(o)(2)P) is also revealed for the unified U model under the excluding. The symmetry is valid for the partially attractive model with chemical potential μ=−U.