Real Gentile Statistical Systems: N-Annulenes.

Gentile statistics, which is famous for its advantages in dealing with composite particle systems, is a kind of fractional statistics. Lately, researchers are concentrating on finding real systems that obey Gentile statistics. Five years ago, we discovered that the cyclic hydrocarbon polyenes called N-annulenes in the Hückel model had certain correspondence with Gentile oscillators. According to their rotation and dihedral symmetry, these two systems had the same energy levels and partition functions. In this paper, we give further discussions. We discuss the transformations of wave functions between N-annulenes and Gentile oscillators, the creation and annihilation operators in site pictures of N-annulenes, the coherent state, and the mathematical proofs of the intermediate commutation relations of operators. All of our works prove that N-annulenes in the Hückel model are real Gentile statistical systems and offer a new algebraic method to deal with the problems of electron-electron and electron-phonon interactions.