Unitary transformations of the group~$U(N)$ and diagonalization of multilevel Hamiltonians

V. V. Val'kov

Teoretičeskaâ i matematičeskaâ fizika, Tome 76 (1988) no. 1, pp. 143-152

Résumé

Simple unitary transformations of the group $U(N)$ that make it possible to diagonalize the Hamiltonian of an $N$-level quantum system are proposed. The use of Hubbard operators to construct the generators of the group ensures the possibility of obtaining in explicit form the laws of transformation of the Hubbard operators under unitary rotations. Recursion relations connecting the parameters of the Hamiltonian in the new basis to its parameters in the original basis are found. The proposed form of the general method of unitary transformations is demonstrated by the example of strongly anisotropic ferromagnets with a large number of levels of magnetoactive ions. The developed approach has made it possible to trace in a unified manner the transition of easy-plane ($S=2$) and cubic ($S=4$) ferromagnets to the quadrupole phase with increasing anisotropy constants and to determine the values of these constants.