Low-temperature magnetization and the excitation spectrum of antiferromagnetic Heisenberg spin rings

Accurate results are obtained for the low-temperature magnetization vs magnetic field of Heisenberg spin rings consisting of an even number N of intrinsic spins $s=1∕2,1,3∕2,2,5∕2,3,7∕2$ with nearest-neighbor antiferromagnetic exchange by employing a numerically exact quantum Monte Carlo method. A straightforward analysis of this data, in particular, the values of the level-crossing fields, provides accurate results for the lowest-energy eigenvalue EN(S,s) for each value of the total spin quantum number S. In particular, the results are substantially more accurate than those provided by the rotational band approximation. For s5∕2, data are presented for all even N20, which are particularly relevant for experiments on finite magnetic rings. Furthermore, we find that for s3∕2, the dependence of EN(S,s) on s can be described by a scaling relation, and this relation is shown to hold well for ring sizes up to $N=80$ for all intrinsic spins in the range 3∕2s7∕2. Considering ring sizes in the interval 8N50, we find that the energy gap between the ground state and the first excited state approaches zero proportional to 1∕N^α, where α≈0.76 for $s=3∕2$ and α≈0.84 for $s=5∕2$. Finally, we demonstrate the usefulness of our present results for EN(S,s) by examining the Fe₁₂ ring-type magnetic molecule, leading to a more accurate estimate of the exchange constant for this system than has been obtained heretofore.