Interplay of quantum criticality and geometric frustration in columbite

CoNb ₂ O ₆ is a material with remarkable properties that are determined by an exciting interplay of quantum mechanics and geometric frustration. On the one hand, weakly coupled ferromagnetic Ising chains of Co ²⁺ ions can be tuned by an applied magnetic field through a quantum critical point to be paramagnetic; on the other hand, the Ising chains must contend with residual interactions on a frustrated triangular lattice in their choice of how to order. Motivated by the material, we theoretically study the phase diagram of quantum ferromagnetic Ising chains coupled antiferromagnetically on a triangular lattice in the plane perpendicular to the chain direction. We combine exact solutions of the quantum criticality in the isolated chains with perturbative approximations for the frustrated interchain couplings. When the triangular lattice has an isosceles distortion, which occurs in the real material, the phase diagram at absolute zero temperature is rich with five different states of matter: ferrimagnetic, Néel, antiferromagnetic, paramagnetic and incommensurate phases, separated by quantum phase transitions. Implications of our results on experiments in CoNb ₂ O ₆ are discussed.