Exact Results in Frustrated Quantum Magnetism

Abstract

Most of the exact results in frustrated spin systems have for a long time been regarded as of purely academic interest, being realized only due to the special geometry of the lattices concerned. However, recent developments in material design offer the genuine possibility of producing such exact states in real materials. In fact, the exact dimer singlet state of the two-dimensional Shastry–Sutherland model has already been found as the ground state of the quasi-two-dimensional material SrCu₂(BO₃)₂. The cooperation between experimentalists and theorists in investigating this material has caused rapid development in the understanding of low-dimensional frustrated spin systems in general, due to the extreme utility of cases where the ground state is known exactly. This fact provides information essential to recognizing novel magnetic behavior in external magnetic fields, at finite temperatures, and in other regimes. In this chapter, we introduce spin-1 ∕ 2 models which have an exact ground state, considering first exactly solvable spin-1 ∕ 2 Heisenberg models, exemplified by the sawtooth–chain model, the Majumdar–Ghosh model, the two-dimensional Shastry–Sutherland model, and a frustrated ladder model. Such exact states can be realized due to special symmetries on geometrically frustrated lattices. As a second class of examples, we introduce also some exact ground states in spin-1/2 models with multiple-spin interactions.