Robust Hund rule without Coulomb repulsion and exclusion principle in quantum antiferromagnetic chains of composite half spins

Quantum spin chains with composite spins have been used to approximate conventional chains with higher spins. For instance, a spin 1 (or 3/2) chain was sometimes approximated by a chain with two (or three) spin 1/2's per site. However, little examination has been given as to whether this approximation, effectively assuming the first Hund rule per site, is valid and why. In this paper, the validity of this approximation is investigated numerically. We diagonalize the Hamiltonians of spin chains with a spin 1 and 3/2 per site and with two and three spin 1/2's per site. The low energy excitation spectrum for the spin chain with M spin 1/2's per site is found to coincide with that of the corresponding conventional chain with one spin M/2 per site. In particular, we find that as the system size increases, an increasingly larger block of consecutive lowest energy states with maximal spin per site is observed, robustly supporting the first Hund rule even though the exclusion principle does not apply and the system does not possess Coulomb repulsion. As for why this approximation works, we show that this effective Hund rule emerges as a plausible consequence when applying to composite spin systems the Lieb-Mattis theorem, which is originally for the ground state of ferrimagnetic and antiferromagnetic spin systems.