We recall a simple class of translation invariant states for an infinite quantum spin chain, which was introduced by Accardi. Those states have exponential decay of correlation functions, and a subclass contains the groun states of a certain class of finite-range interactions. We consider, in particular, a family of states for integer spin chains, containing as its simplest member the ground state of a spin-1 Heisenberg antiferromagnet recently studied by Affleck, Kennedy, Lieb and Tasaki. For this family we compute explicitly the correlation functions and other properties.
A finite-cell-scaling analysis of the spin-1 antiferromagnetic Heisenberg-Ising chain, shows that the ground-state properties are completely different than in the spin-\textonehalf{} case. Between the
XY gapless phase and the doublet ground-state N\'eel phase, it appears, in an extended range of the anisotropy ($0\ensuremath{\lesssim}\ensuremath{\lambda}\ensuremath{\lesssim}1.18$) a characteristic Heisenberg-phase with a nonmagnetic singlet ground state, nonzero gap, and exponential decay of the correlation functions.
We study a class of spin-1/2 quantum antiferromagnetic chains using DMRG technique. The exchange interaction in these models decreases linearly as a function of the separation between the spins,
Jij=R−∣i−j∣ for
∣i−j∣≤R. For the separations beyond
R, the interaction is zero. The range parameter
R takes positive integer values. The models corresponding to all the odd values of
R are known to have the same exact doubly degenerate dimer ground state as for the Majumdar-Ghosh (MG) model. In fact, R=3 is the MG model. For even
R, the exact ground state is not known in general, except for R=2 (the Bethe ansatz solvable Heisenberg chain) and in the asymptotic limit of
R where the two MG dimer states again emerge as the exact ground state. In the present work, we numerically investigate the even-
R models whose ground state is not known analytically. In particular, for R=4, 6 and 8, we have computed a number of ground state properties. We find that, unlike R=2, the higher even-
R models are spin-gapped, and show strong dimer-dimer correlations of the MG type. Moreover, the spin-spin correlations decay very rapidly, albeit showing weak periodic revivals.
Inspired by the exact solution of the Majumdar-Ghosh model, a family of one-dimensional, translationally invariant spin Hamiltonians is constructed. The exchange coupling in these models is antiferromagnetic, and decreases linearly with the separation between the spins. The coupling becomes identically zero beyond a certain distance. It is rigorously proved that the dimer configuration is an exact, superstable ground-state configuration of all the members of the family on a periodic chain. The ground state is twofold degenerate, and there exists an energy gap above the ground state. The Majumdar-Ghosh Hamiltonian with a twofold degenerate dimer ground state is just the first member of the family. The scheme of construction is generalized to two and three dimensions, and illustrated with the help of some concrete examples. The first member in two dimensions is the Shastry-Sutherland model. Many of these models have exponentially degenerate, exact dimer ground states.
The authors show that every translation invariant valence bond state on a one-dimensional quantum spin chain arises as the unique ground state of a certain family of finite-range interactions. For each interaction in this family they show the existence of a nonzero spectral gap above the ground state energy. A special example of this structure is a state recently studied by Affleck et al. (1987). For the Hamiltonian studied by these authors they estimate the gap, and prove that it lies between 1/3 and 10/27.
Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.