Analytical block-diagonalization of matrices using unit spherical tensors

Abstract

A new method for block-diagonalizing large Hamiltonian matrices, in closed form, is described. The method is based on (i) a general unitary transformation due to Slichter, and (ii) Fano's unit spherical operatorsU Q K (Ii,Ii). The method is illustrated with a simple three spin 1/2 dipolar coupled spin system, characterized by off-block-diagonal unit spherical tensorsU 0 2 (3/2,1/2,α) andU 0 2 (3/2,1 /2′,α). In addition, it is pointed out that any Hamiltonian matrix can be re-labelled in terms of fictitious spin labels, enabling a wide variety of unit spherical tensors to be used in block-diagonalization. For example, a single spin 5/2 matrix can be re-labelled using three spin labels 1/2, 1/2′, and 1/2″, respectively. Thus the tensor algebra required to block-diagonalize a 6 x 6 matrix is determined solely by the properties of the Pauli spin matrices. Finally, it is shown that re-labelling within the unit spherical tensor framework provides a unifying framework for standard basis operators, fictitious spin 1/2 and 1 operators, and others. The fictitious spin 1 / 2 unit spherical operators discussed in this paper differ from those of Vega and Pines.