Anomalous breaking of anisotropic scaling symmetry in the quantum lifshitz model

In this note we investigate the anomalous breaking of anisotropic scaling symmetry (t, x) → (λ z t, λ x) in a non-relativistic field theory with dynamical exponent z = 2. On general grounds, one can show that there exist two possible "central charges" which characterize the breaking of scale invariance. Using heat kernel methods, we compute these two central charges in the quantum Lifshitz model, a free field theory which is second order in time and fourth order in spatial derivatives. We find that one of the two central charges vanishes. Interestingly, this is also true for strongly coupled non-relativistic field theories with a geometric dual described by a metric and a massive vector field.