Anomalous scaling at the quantum critical point

We show that Hertz $\phi^4$ theory of quantum criticality is incomplete as it misses anomalous non-local contributions to the interaction vertices. For antiferromagnetic quantum transitions, we found that the theory is renormalizable only if the dynamical exponent $z=2$. The upper critical dimension is still $d= 4-z =2$, however the number of marginal vertices at $d=2$ is infinite. As a result, the theory has a finite anomalous exponent already at the upper critical dimension. We show that for $d<2$ the Gaussian fixed point splits into two non-Gaussian fixed points. For both fixed points, the dynamical exponent remains $z=2$.