  Motivated by CoNb2O6 (belonging to the columbite family of minerals), we
theoretically study the physics of quantum ferromagnetic Ising chains coupled
anti-ferromagnetically on a triangular lattice in the plane perpendicular to
the chain direction. We combine exact solutions of the chain physics with
perturbative approximations for the transverse couplings. When the triangular
lattice has an isosceles distortion (which occurs in the real material), the
T=0 phase diagram is rich with five different states of matter: ferrimagnetic,
N\'eel, anti-ferromagnetic, paramagnetic and incommensurate phases, separated
by quantum phase transitions. Implications of our results to experiments on
CoNb2O6 are discussed.
