The structure and symmetry of modular state space for complex quantum systems

Understanding the state space structure of complex quantum systems can help to effectively characterize the system properties and explore underlying mechanisms. The structure of the state space could be quite complicated for quantum many-body systems, and the systematic decomposition of the state space is normally involved. Recently, a modular tensor diagram approach was proposed to reorganize the state space hierarchically based on a modular basis. Here, we review the construction of spin eigenfunctions for multiple exciton systems and further develop modular tensor diagrams to exemplify the hierarchical symmetry of the state space. The newly constructed spin eigenfunctions for quadruple excitons, along with the results for triple excitons, are used to demonstrate the effective decomposition of the state space into hierarchical tensorial structures. A universal recursive relation is derived to determine the coefficients of spin eigenfunctions exhibiting transformation symmetry between different classes of elementary modules for an arbitrary number of exciton units. Interestingly, different coupling schemes mapped to quantum many-body interactions lead to different spin adapted basis states, which may correspond to different realistic systems upon the breakdown of spin degeneracy. This work highlights the hierarchical symmetry of the tensorial structure of quantum many-body systems, which may facilitate a better understanding of the structure property relationship toward the object-oriented materials design.