Exact momentum-space analysis of small spin-1/2 J₁-J₂ rings

This paper considers an N-site spin-1/2 J₁-J₂ ring with N=6 and 8. With the help of a set of exact few-magnon Bloch states, we obtain the block-diagonalized Hamiltonian consisting of block matrices of at most four dimensions. Partial of the eigenstates are analytically solved. For the six-site anisotropic ring, we reveal a subset of eigenstates that are simultaneous eigenstates of the Hamiltonian and the total angular momentum operator, even though the latter is not conserved. For both the six- and eight-site isotropic rings, we achieve momentum-space manifestations of several important states, including the famous Majumdar-Ghosh (MG) ground states and the Hamada-Kane-Nakagawa-Natsume (HKNN) ground state. The equivalence of these states with their real-space counterparts is explicitly shown for N=6. The structure of the HKNN ground state for small rings suggests that for any even number N this state might behave like a “bound state” with N/2 successive down spins binding together.