Low temperature magnetization and the excitation spectrum of antiferromagnetic quantum Heisenberg rings

We have performed quantum Monte Carlo calculations to obtain the low temperature magnetization and differential susceptibility for finite, antiferromagnetic Heisenberg rings of intrinsic spins $s = 1/2, 1,3/2,2,5/2,3,7/2$. From these data we have determined the level-crossing fields as well as the dependence of the minimal excitation energies on the total spin quantum number $S$. For large intrinsic spins ($s \geq 3/2$) we find that the data exhibit scaling behavior, approaching the classical limit proportional to $s^{-1.05}$. Since this limit is approached so slowly, even $s=7/2$ spins are distinctly non-classical. We have also found for large $s$ that as the number of spins $N$ increases, the energy gap between the ground state and the first excited state approaches zero proportional to $1/N^\alpha$, where $\alpha \approx 0.76$ for $s=3/2$ and $\alpha \approx 0.84$ for $s=5/2$. Finally, we demonstrate the usefulness of our results by examining the Fe$_{12}$ molecular ring, leading to a new, more accurate estimate of the exchange constant for this system than has been obtained heretofore.