Symmetry Classes of Spin and Orbital Ordered States in a t_{2g} Hubbard Model on a Two-dimensional Square Lattice

This paper presents symmetry classes of the Hartree-Fock (HF) solutions of spin and orbital ordered states in a t_{2g} 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 Q_0=(0,0), Q_1=(\pi,\pi), Q_2=(\pi,0) and Q_3=(0,\pi). Each broken symmetry state is characterized by the presence of local order parameters(LOP) at each lattice site: quadrupole moment Q=(Q_2^2,Q_{12},Q_{23},Q_{31}), orbital angular momentum l=(l_1,l_2,l_3), spin density s=(s^1,s^2,s^3), spin quadrupole moment Q^{\lambda}=(Q_2^{2\lambda}, Q_{12}^{\lambda},Q_{23}^{\lambda},Q_{31}^{\lambda}) and spin orbital angular momentum l^{\lambda}=(l_1^{\lambda},l_2^{\lambda},l_3^{\lambda}) where \lambda=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^{\lambda} and l^{\lambda} can be the ground state for these parameter sets.Comment: 46 pages with 4 figure

On Local Symmetric Order Parameters of Vortex Lattice States

This paper gives a new refined definition of local symmetric order parameters (OPs)(s-wave, d-wave and p-wave order parameters) of vortex lattice states for singlet superconductivity. s-wave, d-wave and p-wave OPs at a site (m,n) are defined as A, B and E representations of the four fold rotation C_4 at the site (m,n) of nearest neighbor OPs etc. The new OPs have a well defined nature such that an OP(e.g. d-wave) at the site obtained under translation by a lattice vector (of the vortex lattice) from a site (m,n) is expressed by the corresponding OP (e.g. d-wave) at the site (m,n) times a phase factor. The winding numbers of s-wave and d-wave OPs are given.Comment: RevTeX v3.1, 5 pages with 3 figures, uses epsf.sty. to appear in Prog. Theor. Phys. Vol.101 No.3. (1999