Stability of quantum critical points in the presence of competing orders

We investigate the stability of quantum critical points (QCPs) in the presence of two competing phases. These phases near QCPs are assumed to be either classical or quantum and assumed to repulsively interact via square-square interactions. We find that for any dynamical exponents and for any dimensionality strong enough interaction renders QCPs unstable and drives transitions to become first order. We propose that this instability and the onset of first-order transitions lead to spatially inhomogeneous states in practical materials near putative QCPs. Our analysis also leads us to suggest that there is a breakdown of conformal field theory scaling in the Anti de Sitter models, and in fact these models contain first-order transitions in the strong-coupling limit.