The construction of spin eigenfunctions for fermion systems using modular tensor diagram

The construction of spin eigenfunctions for multi-spin systems can become highly nontrivial as the system size increases. Recently, a modular tensor diagram approach was proposed to hierarchically decompose the state space in terms of tensorial modules based on spin pairs, resulting in an effectively organized state space and efficiently constructed symmetry-adapted basis states. Here, it is generalized to treat fermion systems with odd numbers of spins. Elementary modules made of primitive spin-pair modules augmented by an odd-spin tag module are classified into various module classes and further mapped to geometric building blocks of various sizes and shapes to illustrate the hierarchical structure and symmetry of the state space. Spin eigenfunctions are generated from linear combinations of single-spin-tagged or triple-spin-tagged elementary modules using only symmetry and orthogonality conditions, and universal recursive relations for systems with arbitrary odd numbers of spins can be obtained. This work explores the structure and symmetry of the state space of fermion systems that complement previous studies, which may provide new insights into general quantum many-body systems and spin dynamics.