Anomalous scaling at the quantum critical point

Abstract

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 \(z=2\).