Exact dimer ground states for a continuous family of quantum spin chains and zigzag ladders

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.