  Charge, spin, and orbital degrees of freedom underlie the physics of
transition metal compounds. Much work has revealed quantum critical points
associated with spin and charge degrees of freedom in many of these systems.
Here we illustrate that the simplest models that embody the orbital degrees of
freedom - the two- and three-dimensional quantum orbital compass models -
exhibit an exact quantum critical behavior on diluted square and cubic lattices
(with a doping fraction of 1/4 and 1/2 respectively). This raises the
possibility of quantum critical points triggered by the degradation of orbital
order upon doping (or applying pressure to) such transition metal systems. We
prove the existence of an orbital spin glass in several related systems in
which the orbital couplings are made non-uniform. Moreover, a new orbital
Larmor precession (i.e., a periodic change in the orbital state) is predicted
when uniaxial pressure is applied.
