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 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.
Using the matrix product formalism, we define a multiparameter family of spin models on one-dimensional chains, with nearest- and next-nearest-neighbor interactions for which exact analytical expressions can be found for its doubly degenerate ground states. The family of Hamiltonians depends on five continuous parameters and the Majumdar-Ghosh model is a particular point in this parameter space. The model can also be represented as a zigzag spin ladder in which there are extra three-particle couplings between spins on vertices of triangles. The doubly degenerate ground states of these models models have a very simple structure, they are the product of entangled (dimer) states on adjacent sites (or consecutive rungs of the ladder in the zigzag ladder picture). At the Majumdar-Ghosh point, these entangled states become the spin-singlets pertaining to this model. We will also calculate, in closed form, the two point correlation functions, both for finite sizes of the chain and in the thermodynamic limit.
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 investigate the isotropic two-leg
S=21 ladder with general bilinear and biquadratic exchange interactions between spins on neighboring rungs, and determine the Hamiltonians that have a matrix-product wave function as an exact ground state. We demonstrate that a smooth change of parameters leads one from the
S=21 dimer and Majumdar-Ghosh chains to the
S=1 chain with biquadratic exchange, which proves that these model systems are in the same phase. The family of Hamiltonians also includes a set of models of isotropic
S=21 chains with only bilinear nearest- and next-nearest-neighbor interactions. The ground state for these models becomes unstable at a line of first-order phase transitions to the ferromagnetic state.
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 discuss spin-
S antiferromagnetic Heisenberg chains with three-spin interactions, next-nearest-neighbor interactions, and bond alternation. First, we prove rigorously that there exist parameter regions of the exact dimerized ground state in this system. This is a generalization of the Majumdar-Ghosh model to arbitrary
S. Next, we discuss the ground-state phase diagram of the models by introducing several effective field theories and the universality classes of the transitions are described by the level-
2S SU(2) Wess-Zumino-Witten model and the Gaussian model. Finally, we determine the phase diagrams of
S=1 and
S=3/2 systems by using exact diagonalization and level spectroscopy.