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.
We suggest a general rule for the shift quantum numbers k of the relative ground states of antiferromagnetic Heisenberg spin rings. This rule generalizes the well-known results of Marshall, Peierls, Lieb, Schultz, and Mattis for even rings. Our rule is confirmed by numerical investigations and rigorous proofs for special cases, including systems with a Haldane gap for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}.$ Implications for the total spin quantum number S of relative ground states are discussed as well as generalizations to the XXZ model.
Abstract An efficient projector Monte Carlo method has been used to calculate the energies of the ground state and the first excited state, and hence the gap between them, of the spin-1 Heisenberg antiferromagnetic chains with linear sizes up to <?CDATA
N=40?> . We have computed them on chains up to <?CDATA
N=40?> . We have found that the size-dependence of the ground state energy per site has the form <?CDATA
[e0(N)−e0]∼1/N5/2s?> , which implies that there is a gap between the ground state and the first excited state. The value of the ground state energy per site as <?CDATA
N→∞?> is <?CDATA
e0=−1.3996?> . The direct extrapolation of <?CDATA
ΔE(N)?> also shows that a gap of <?CDATA
ΔE=0.36?> exists as <?CDATA
N→∞?> .
A very powerful method for the numerical study of the ground state of quantum systems is proposed using the quantum transfer matrix as a projection operator onto the ground state. We calculate the ground-state correlation functions of antiferromagnetic Heisenberg rings of S=(1/2) and 1 with spin sizes up to 20 and 12, respectively. The size dependence of the correlations is reexamined. For S=1 the ground-state energy is found to show a quite different size dependence according to whether the ring has even or odd spins. The difference is interpreted by considering a dimer suggested by Hubbard.
The spin pair-correlation function of finite bipartite antiferromagnetic Heisenberg quantum spin rings was studied numerically by means of exact diagonalization techniques. For large spins, the spectrum of the spin pair-correlation function is of a remarkably simple structure: It consists of few characteristic peaks at low, and a broad featureless signal at high temperatures. This arises as the energy spectrum exhibits a set of parallel rotational bands at low energies emerging into a quasi-continuum of states, with transitions from the lowest rotational band to the quasi-continuum being highly suppressed. The energies of the rotational bands can be accurately described by a generalized dispersion relation that depends on the total spin quantum number and, as usual, on the shift quantum number. These regularities are better fulfilled the larger the spin length but the smaller the ring size. It will be shown that all these features are associated with the underlying sublattice structure of the spin rings, and it is argued that they are valid for a more general class of finite Heisenberg systems than rings.
The sequence-transformation method due to Vanden Broeck and Schwartz is used to accelerate the convergence of numerical finite-lattice data on ground-state energies to their infinite-lattice limit. Highly accurate estimates of the ground-state energy are obtained for the
S=21 antiferromagnetic Heisenberg and
XY rings. The ground-state energy of the
S=1 antiferromagnetic Heisenberg ring is also estimated accurately.