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.
A method to construct various s= 1/2 isotropic spin models with exact dimer ground states is proposed. Constructed models are usually of much simpler forms than those produced by the method using Lowdin's projection operators. They include a series of one-dimensional Heisenberg models which consist of linearly decayed exchange interactions and are natural extensions of the Majumdar-Ghosh model. A two-dimensional spin model with four spin interactions is also constructed to have the same spatial symmetry as a simple square lattice.
We have obtained an analytic expression for the k-dependence of the excitation energy gap for an arbitrary double S = 1/2 spin chain by using the non-local unitary transformation and the variational method. We check that it explains the gap behaviour of various systems, which include the Haldane system and the dimer system in both extreme limits, and also the ladder model and the Majumdar - Ghosh model. The string order parameter, the dimer order parameter, and the local spin value are also calculated in the ground state. The ground-state energy exhibits a great stabilization by an antiferromagnetic bond dimerization, which might be realized in various new compounds. We also mention the relation of the convergence to the Haldane state with the spin-exchange symmetry of the model. The excited state has one domain wall of a local triplet type except in the vicinity of the Majumdar - Ghosh point, where a local triplet is decomposed into two S = 1/2 free spins moving among the singlet dimers.
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.
We present a new class of exactly solvable spin models. The solutions involve broken translational symmetry, in contrast to the Haldane gap, and constitute a new type of ground state for a spin-one chain. The models have ground states with precipitously short-range spin correlations and exhibit gaps to solitonic spin excitations. We construct a solvable model for each value of the total spin of the underlying atoms, with the solution corresponding to the natural generalization to the Majumdar-Ghosh state, of 'dimer' pairing, to larger spin magnitudes.
We have studied the spectrum of low-spin eigenstates of a spin-(1/2 Heisenberg model on 4\ifmmode\times\else\texttimes\fi{}4 square lattices with up to third-neighbor antiferromagnetic exchange interactions. We have computed the staggered susceptibility, spin-spin correlation function, and the overlap of the exact ground state with various variational spin-density-wave ordered states as well as with the equal-amplitude, nearest-neighbor resonating-valence-bond state (which is the prototypical short-ranged RVB state). We find evidence for a complicated phase diagram with a region in which a short-ranged RVB state may be the ground state.