Systematic properties of the first excited state are presented for various ring sizes and spin quantum numbers which are only partly covered by the theorem of Lieb, Schultz, and Mattis. For odd ring sizes the first excited energy eigenvalue shows unexpected degeneracy and related shift quantum numbers. As a byproduct the ground state energy as well as the energy of the first excited state of infinite chains are calculated by extrapolating the properties of only a few, finite, antiferromagnetically coupled Heisenberg rings using the powerful Levin sequence acceleration method.
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.
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 antiferromagnetic Heisenberg spin chain of odd spin
S is in the Haldane phase with several defining physical properties, such as thermodynamical ground-state degeneracy, symmetry-protected edge states, and nonzero string order parameter. If nonzero hole concentration $\ensuremath{\delta}$ and hole hopping energy
t are considered, the spin chain is replaced by a $\text{spin}\ensuremath{-}S$ $t\ensuremath{-}J$ chain. The motivation of this paper is to generalize the discussions of the Haldane phase to the doped spin chain. The first result of this paper is that, for the model considered here, the
Z2 sign structure in the usual Ising basis can be totally removed by two consecutive unitary transformations consisting of a spatially local one and a nonlocal one. Direct from the sign structure, the second result of this paper is that the Marshall theorem and the Lieb-Mattis theorem for pure spin systems are generalized to the $t\ensuremath{-}J$ chain for arbitrary
S and $\ensuremath{\delta}$. A corollary of the theorem provides us with the ground-state degeneracy in the thermodynamic limit. The third result of this paper is about the phase diagram. We show that the defining properties of the Haldane phase survive in the small
t/J limit. The large
t/J phase supports a gapped spin sector with similar properties (ground-state degeneracy, edge state, and string order parameter) of the Haldane chain, although the charge sector is gapless.