Exact momentum-space analysis of small spin-1/2 $J_1$-$J_2$ rings

TL;DR

This paper provides an exact momentum-space analysis of small spin-1/2 J1-J2 rings, revealing eigenstates, ground states, and their properties for N=6 and 8 sites, including analytical solutions and state structures.

Contribution

It introduces an exact momentum-space approach for small J1-J2 rings, identifying eigenstates and ground states, and explores their properties and structures.

Findings

Identified eigenstates that are simultaneous eigenstates of Hamiltonian and total angular momentum.

Explicitly showed the equivalence of certain ground states with their real-space counterparts.

Suggested the HKNN ground state behaves like a bound state for any even N.

Abstract

This paper considers an $N$-site spin-1/2 $J_1$-$J_2$ 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.