Anomalous scaling at the quantum critical point

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We show that Hertz ϕ⁴ 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.