The distribution of 'unbound' states in the Bethe formalism for the linear chain of spin 1/2 atoms with a Heisenbert exchange interaction between nearest neighbors is investigated in the vicinity of the antiferromagnetic ground state, utilizing the spin wave states discovered by des Cloizeaux and Pearson. An upper bound is obtained for the free energy of the anti ferromagnetic chain at very low temperatures. Plausible arguments are presented to show that the 'unbound' states actually make a negligible contribution to the partition function of an infinite chain, at least in the absence of a magnetic field.
The lowest energies of the one-dimensional half-filled Hubbard model as a function of the total spin are computed from the exact solution of one-dimensional interacting fermions. The lowest energies are used to calculate the magnetization curve at zero temperature. In the case of repulsive interaction our work is an extension of Griffiths' theory for the Heisenberg model to the itinerant electron model of antiferromagnet. In the case of attractive interaction it is shown that the magnetic susceptibility in zero field vanishes.
The lowest energies of the one-dimensional half-filled Hubbard model as a function of the total spin are computed from the exact solution of one-dimensional interacting fermions. The lowest energies are used to calculate the magnetization curve at zero temperature. In the case of repulsive interaction our work is an extension of Griffith's theory for the Heisenberg model to the itinerant electron model of antiferromagnet. In the case of attractive interaction it is shown that the magnetic susceptibility in zero field vanishes.
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 $s\ensuremath{\leqslant}5∕2$, 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
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 $3∕2\ensuremath{\leqslant}s\ensuremath{\leqslant}7∕2$. 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
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.