Quantum spin models with exact dimer ground states

TL;DR

This paper constructs a family of one-dimensional and higher-dimensional quantum spin models with exact dimer ground states, generalizing the Majumdar-Ghosh and Shastry-Sutherland models, and proves their ground state properties.

Contribution

It introduces a new family of quantum spin Hamiltonians with exact dimer ground states, extending known models to higher dimensions with rigorous proofs.

Findings

Ground states are exact and superstable in the constructed models.

The models exhibit energy gaps above the dimer ground states.

Many models have exponentially degenerate ground states.

Abstract

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 two-fold degenerate, and there exists an energy gap above the ground state. The Majumdar-Ghosh hamiltonian with two-fold 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.