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

Abstract

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 ${E}_{N}(S,s)$ 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 $s\ensuremath{\leqslant}5∕2$, data are presented for all even $N\ensuremath{\leqslant}20$, which are particularly relevant for experiments on finite magnetic rings. Furthermore, we find that for $s\ensuremath{\geqslant}3∕2$, the dependence of ${E}_{N}(S,s)$ 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 $3∕2\ensuremath{\leqslant}s\ensuremath{\leqslant}7∕2$. Considering ring sizes in the interval $8\ensuremath{\leqslant}N\ensuremath{\leqslant}50$, we find that the energy gap between the ground state and the first excited state approaches zero proportional to $1∕{N}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}\ensuremath{\approx}0.76$ for $s=3∕2$ and $\ensuremath{\alpha}\ensuremath{\approx}0.84$ for $s=5∕2$. Finally, we demonstrate the usefulness of our present results for ${E}_{N}(S,s)$ by examining the ${\mathrm{Fe}}_{12}$ ring-type magnetic molecule, leading to a more accurate estimate of the exchange constant for this system than has been obtained heretofore.