Mechanisms of spin-gap formation due to orbital order for CaV₄O₉

The roles of orbital order in magnetic properties of two-dimensional spin-gap system for CaV₄O₉ are investigated theoretically. We assume that one of the doubly degenerated d_xz and d_yz orbitals is occupied in d¹ state of V. We consider several models with possible orbital orders. The strengths of the spin-exchange couplings for each model are determined to reproduce the temperature dependence of the uniform magnetic susceptibility in experiments by using the exact diagonalization. By the perturbation expansion, we calculate the energy dispersion of low-lying excitations and the integrated intensity of the neutron inelastic scattering. These quantities sensitively depend on the types of the orbital order. The origin of the spin gap for these models is ascribed to a generalized four-spin singlet. Although none of the models is complete to reproduce all of the reported experimental results, we point out basic mechanisms to reproduce each experimental result. & 1998 Elsevier Science B.V. All rights reserved.