Emergent Symmetry and Dimensional Reduction at a Quantum Critical Point

We show that the spatial dimensionality of the quantum critical point associated with Bose--Einstein condensation at T=0 is reduced when the underlying lattice comprises a set of layers coupled by a frustrating interaction. For this purpose, we use an heuristic mean field approach that is complemented and justified by a more rigorous renormalization group analysis. Due to the presence of an emergent symmetry, i.e. a symmetry of the ground state that is absent in the underlying Hamiltonian, a three--dimensional interacting Bose system undergoes a chemical potential tuned quantum phase transition that is strictly two dimensional. Our theoretical predictions for the critical temperature as a function of the chemical potential correspond very well with recent measurements in BaCuSi$_{2}$O$_{6}$.