Programs for generating Clebsch-Gordan coefficients of SU(3) in SU(2) and SO(3) bases
Abstract
Computer codes are developed to calculate Clebsch-Gordan coefficients of SU(3) in both SU(2)- and SO(3)-coupled bases. The efficiency of this code derives from the use of vector coherent state theory to evaluate the required coefficients directly without recursion relations. The approach extends to other compact semi-simple Lie groups. The codes are given in subroutine form so that users can incorporate the codes into other programs. Nature of physical problem: The group SU(3) and its Lie algebra su(3) have important applications, for example, in elementary particle physics, nuclear physics, and quantum optics [1-3]. The code presented is particularly relevant for the last two fields. Clebsch-Gordan (CG) coefficients are required whenever the symmetries of many-body systems are used for the evaluation of matrix elements of tensor operators. Moreover, the construction of CG coefficients for SU(3) serves as a nontrivial prototype for larger compact semi-simple Lie algebras and even for non semi-simple Lie algebras. It is the simplest Lie algebra to have multiplicity in its outer products and a non-canonical subalgebra, i.e., SO(3). Method of solution: Vector coherent state theory is first used to construct bases for the products of two irreducible representations (irreps) [4]. The bases are SU(2)-coupled so that SU(2)-reduced CG (or isoscalar factors) can be constructed naturally. The CG coefficients in the SO(3) bases are constructed subsequently from the overlaps between the SU(2) and SO(3) bases. Restriction on the complexity of the problem: The programs are limited by computer memory and the maximum size of variable arrays. As dimension overflow conditions are possible, they are flagged and can be fixed by following the directions given as part of the error message. Typical running time: The calculation time for a single SU(3) CG coefficient is very different for SU(2) and SO(3) bases. It varies between 7.3-54.1 ns in SGI Origin 2000, 0.81-5.48 ms in HP Apollo 9000, or 0.055-0.373 ms in Intel Pentium 4 for SU(2) bases while it is between 0.027-0.255s in Intel Pentium 4 for SO(3) bases. Unusual features of the program: Intrinsic bit functions: and, or, and shift, called iand, ior, and ishft, respectively, in FORTRAN, are used for packing and unpacking the labels for the irreps. Intrinsic logical btest is used to test the bit for the phase factor. © 2004 Elsevier B.V. All rights reserved.