Symmetry Classes of Spin and Orbital Ordered States in a t2g Hubbard Model on a Two-dimensional Squar Lattice

This paper presents symmetry classes of the Hartree-Fock (HF) solutions of spin and orbital ordered states in a t2g Hubbard model on a two-dimensional square lattice. Using a group theoretical bifurcation theory of the Hartree-Fock equation, we obtained many types of broken symmetry solutions which bifurcate from the normal state through one step transition in cases of commensurate ordering vectors Q0 = (0, 0), Q1 = (π, π), Q2 = (π, 0) and Q3 = (0, π). Each broken symmetry state is characterized by the presence of local order parameters(LOP) at each lattice site: quadrupole moment Q = (Q2, Q12, Q23, Q31), orbital angular momentum l = (l1, l2, l3), spin density s = (s , s, s), spin quadrupole moment Q = (Q 2 , Q λ 12, Q λ 23, Q λ 31) and spin orbital angular momentum l λ = (l 1 , l λ 2 , l λ 3 ) where λ = 1, 2, 3. We performed numerical calculations for some parameter sets. Then we have found that many types of non-collinear magnetic orbital ordered states having LOP:Q and l can be the ground state for these parameter sets.