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Complete spectrum of the infinite-$U$ Hubbard ring using group theory

We present a full analytical solution of the multiconfigurational strongly-correlated mixed-valence problem corresponding to the $N$-Hubbard ring filled with $N-1$ electrons, and infinite on-site repulsion. While the eigenvalues and the eigenstates of the model are known already, analytical determination of their degeneracy is presented here for the first time. The full solution, including degeneracy count, is achieved for each spin configuration by mapping the Hubbard model into a set of Huckel-annulene problems for rings of variable size. The number and size of these effective Huckel annulenes, both crucial to obtain Hubbard states and their degeneracy, are determined by solving a well-known combinatorial enumeration problem, the necklace problem for $N-1$ beads and two colors, within each subgroup of the $C_{N-1}$ permutation group. Symmetry-adapted solution of the necklace enumeration problem is finally achieved by means of the subduction of coset representation technique [S. Fujita, Theor. Chim. Acta 76, 247 (1989)], which provides a general and elegant strategy to solve the one-hole infinite-$U$ Hubbard problem, including degeneracy count, for any ring size. The proposed group theoretical strategy to solve the infinite-$U$ Hubbard problem for $N-1$ electrons, is easily generalized to the case of arbitrary electron count $L$, by analyzing the permutation group $C_L$ and all its subgroups.