  Frustration in magnetic interactions can give rise to disordered ground
states with subtle and beautiful properties. The spin ices Ho2Ti2O7 and
Dy2Ti2O7 exemplify this phenomenon, displaying a classical spin liquid state,
with fractionalized magnetic--monopole excitations. Recently there has been
great interest in closely-related "quantum spin ice" materials, following the
realization that anisotropic exchange interactions could convert spin ice into
a massively-entangled, quantum, spin liquid, where magnetic monopoles become
the charges of an emergent quantum electrodynamics. Here we show that even the
simplest model of a quantum spin ice, the XXZ model on the pyrochlore lattice,
can realise a still-richer scenario. Using a combination of classical Monte
Carlo simulation, semi--classical molecular--dynamics simulation, and analytic
field theory, we explore the properties of this model for frustrated transverse
exchange. We find not one, but three, competing forms of spin liquid, as well
as a phase with hidden, spin-nematic, order. We explore the experimental
signatures of each of these different states, making explicit predictions for
inelastic neutron scattering. These results show an intriguing similarity to
experiments on a range of pyrochlore oxides.
