A possible framework of the Lipkin model obeying the su(n)-algebra in arbitrary fermion number. II --- Two subalgebras in the su(n)-Lipkin model and an approach to the construction of linearly independent basis ---

TL;DR

This paper proposes a method to construct linearly independent bases for the su(n)-Lipkin model by utilizing multiple su(2) subalgebras and spherical tensor representations, extending previous results on minimum weight states.

Contribution

It introduces a novel approach to basis construction in the su(n)-Lipkin model using multiple su(2) subalgebras and spherical tensor techniques, providing concrete results for n=2 to 5.

Findings

Constructed linearly independent bases for n=2, 3, 4, 5 cases.

Re-formed the su(n) generators using spherical tensors based on su(2)-subalgebras.

Identified the su(m)-subalgebra within the su(n)-Lipkin model.

Abstract

Standing on the results for the minimum weight states obtained in the previous paper (I), an idea how to construct the linearly independent basis is proposed for the su(n)-Lipkin model. This idea starts in setting up m independent su(2)-subalgebras in the cases with n=2m and n=2m+1 (m=2,3,4,...). The original representation is re-formed in terms of the spherical tensors for the su(n)-generators built under the su(2)-subalgebras. Through this re-formation, the su(m)-subalgebra can be found. For constructing the linearly independent basis, not only the su(2)-algebras but also the su(m)-subalgebra play a central role. Some concrete results in the cases with n=2, 3, 4 and 5 are presented.