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

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.

17 pages, 6 figures, to appear in Physics Letters A

References in corpus (10)

• Vector chiral and multipolar orders in the spin-1/2 frustrated ferromagnetic chain in magnetic field
• Emergent multipolar spin correlations in a fluctuating spiral - The frustrated ferromagnetic S=1/2 Heisenberg chain in a magnetic field
• Frustrated ferromagnetic spin-1/2 chain in a magnetic field: The phase diagram and thermodynamic properties
• Ground-state phase diagram of a spin-1/2 frustrated ferromagnetic XXZ chain: Haldane dimer phase and gapped/gapless chiral phases
• Exact one- and two-particle excitation spectra of acute-angle helimagnets above their saturation magnetic field
• Multipolar phase in frustrated spin-1/2 and 1 chains
• Exact solutions of few-magnon problems in the spin-$S$ periodic XXZ chain
• Evolution of two-magnon bound states in a higher-spin ferromagnetic chain with single-ion anisotropy: A complete solution
• One-excitation spin dynamics in homogeneous closed chain governed by XX-Hamiltonian
• Four interacting spins: addition of angular momenta, spin-spin correlation functions, and entanglement