  We introduce a new set of one dimensional quantum lattice models which we
refer to as The quantum torus chain. These models have discrete global
symmetry, and projective on-site representations. They possess an
integer-valued parameter which controls the presence or absence of frustration.
Depending on whether this parameter is even or odd these models either exhibit
gapped symmetry breaking phases with isolated critical points, or gapped
symmetry breaking phases separated by gapless phases. We discuss the property
of these phases and phase transitions for two special values of the parameter
and point out many open problems
