Theory of antiferro-quadrupolar ordering in TmTe

A theory of the antiferro-quadrupolar (AFQ) ordering in TmTe is presented o n the basis of a classical description of multipoles. The phase diagram of the AFQ state and its relation to the type of crystal field and the form of multipolar interactions are studied in detail. It is shown that the Gamma( 3)-AFQ order parameter in a crystal field with the (001)-easy-axis type is uniquely consistent with the magnetic-held anisotropy observed in TmTe. A c orrelation is present between the spontaneous and field-induced antiferroma gnetic orders through the magnetic interaction in the AFQ state; it leads t o a natural interpretation of the phase diagram of TmTe. A large contributi on from octupoles on the AFQ transition temperature is predicted in the hig h-field region due to the cooperative relation of octupoles and the magneti c field.