Symmetry classes of mean field solutions in the double degenerate e(g) Hubbard model of manganese oxides

This paper describes symmetry properties of mean field (unrestricted Hartre e-Fock) solutions of spin and orbital ordered phases in a double degenerate e(g) Hubbard model including antiferomagnetic interaction with t(2g) elect rons in manganese oxides. We consider spin (axial along z axis) and orbital ordered states with four types of ordering vectors Q(F) = Q(o) = (0,0,0), Q(A) = Q(1) = (0,0,pi), Q(C) = Q(2) = (pi,pi ,0) and Q(G) = Q(3) = (pi,pi,p i) without charge order, spin canting and Jahn-Teller distortion. We obtain ed invariance groups G(ij)(i,j = F, A, C, G) for 4 (for spin orders)x4 (for orbital orders) = 16 states. The canonical form of the mean field Hamilton ian for G(ij) state is obtained from its symmetry condition for the invaria nce group G(i,j). We found that the mean field Hamiltonian includes Hamilto nian elements besides eg spin operators Ses and orbital pseudo-spin operato rs T. We performed numerical calculations of 16 ordered states for e(g) ele ctron numbers per site n(e) = 1.0 similar to 0.1 at zero temperature and fo und sequential change of the spin ordered structures such as A --> F --> A --> C --> G with hole doping for a parameter set. Spin canted states derive d from non-canted G(ij) (i,j = F, A, C, G) states are discussed.