Duality in low dimensional quantum field theories

Abstract

In some strongly correlated electronic materials Landau's quasiparticle concept appears to break down, suggesting the possibility of new quantum ground states which support particle-like excitations carrying fractional quantum numbers. Theoretical descriptions of such exotic ground states can be greatly aided by the use of duality transforma-tions which exchange the electronic operators for new quantum fields. This chapter gives a brief and self-contained introduction to duality transformations in the simplest possible context -lattice quantum field theories in one and two spatial dimensions with a global Ising or XY symmetry. The duality transformations are expressed as exact oper-ator change of variables performed on simple lattice Hamiltonians. A Hamiltonian version of the Z2 gauge theory approach to electron frac-tionalization is also reviewed. Several experimental systems of current interest for which the ideas of duality might be beneficial are briefly discussed.