Gradient terms in quantum-critical theories of itinerant fermions

We investigate the origin and renormalization of the gradient (Q²) term in the propagator of soft bosonic fluctuations in theories of itinerant fermions near a quantum critical point (QCP) of the nematic type. In principle, the gradient term come from the two distinct energy scales. The first one is the high-energy (HE) scale, ranging from the upper cutoff of the effective low-energy theory and to the bandwidth. The second, low-energy (LE), scale is of the order v_FQ. We calculated the HE contribution to the Q² term for the model of a Fermi gas with Coulomb interaction and for the Hubbard model and found that the HE contribution is of the same order but numerically much smaller than the LE one. The numerical smallness is especially pronounced in 2D. We argue that if the LE part of the gradient term is the dominant one, its renormalized value has to be calculated self consistently, which may give rise to a novel quantum-critical behavior. Following up on these results, we discuss two possible ways of formulating the theory of a nematic QCP.