Z 2 gauge theory of electron fractionalization in strongly correlated systems

We develop a new theoretical framework for describing and analyzing exotic phases of strongly correlated electrons which support excitations with fractional quantum numbers. Starting with a class of microscopic models believed to capture much of the essential physics of the cuprate superconductors, we derive a new gauge theory---based upon a discrete Ising or ${Z}_{2}$ symmetry---which interpolates naturally between an antiferromagnetic Mott insulator and a conventional d-wave superconductor. We explore the intervening regime, and demonstrate the possible existence of an exotic fractionalized insulator, the nodal liquid, as well as various more conventional insulating phases exhibiting broken lattice symmetries. A crucial role is played by vortex configurations in the ${Z}_{2}$ gauge field. Fractionalization is obtained if they are uncondensed. Within the insulating phases, the dynamics of these ${Z}_{2}$ vortices in two dimensions is described, after a duality transformation, by an Ising model in a transverse field, the Ising spins representing the ${Z}_{2}$ vortices. The presence of an unusual Berry's phase term in the gauge theory leads to a doping-dependent ``frustration'' in the dual Ising model, being fully frustrated at half filling. The ${Z}_{2}$ gauge theory is readily generalized to a variety of different situations, in particular, it can also describe three-dimensional insulators with fractional quantum numbers. We point out that the mechanism of fractionalization for $d>1$ is distinct from the well-known one-dimensional spin--charge separation. Other interesting results include a description of an exotic fractionalized superconductor in two or higher dimensions.