Fermionic criticality of anisotropic nodal point semimetals away from the upper critical dimension: Exact exponents to leading order in 1/N_{f}

Abstract

We consider the fermionic quantum criticality of anisotropic nodal point semimetals in d=d_{L}+d_{Q} spatial dimensions that disperse linearly in d_{L} dimensions, and quadratically in the remaining d_{Q} dimensions. When subject to strong interactions, these systems are susceptible to semimetal-insulator transitions concurrent with spontaneous symmetry breaking. Such quantum critical points are described by effective field theories of anisotropic nodal fermions coupled to dynamical order parameter fields. We analyze the universal scaling in the physically relevant spatial dimensions, generalizing to a large number N_{f} of fermion flavors for analytic control. Landau damping by gapless fermionic excitations gives rise to nonanalytic self-energy corrections to the bosonic order-parameter propagator that dominate the long-wavelength behavior. We show that perturbative momentum shell RG leads to nonuniversal, cutof-dependent results, as it does not correctly account for this nonanalytic structure. In turn, using a completely general soft cutoff formulation, we demonstrate that the correct IR scaling of the dressed bosonic propagator can be deduced by enforcing that results are independent of the cutoff scheme. Using the soft cutoff RG with the dressed dynamical RPA boson propagator, we compute the exact critical exponents for anisotropic semi-Dirac fermions (d_{L}=1, d_{Q}=1) to leading order in 1/N_{f} and to all loop orders. Applying the same method to relativistic Dirac fermions, we reproduce the critical exponents obtained by other methods, such as conformal bootstrap. Unlike in the relativistic case, where the UV-IR connection is reestablished at the upper critical dimension, nonanalytic IR contributions persist near the upper critical line 2d_{L}+d_{Q}=4 of anisotropic nodal fermions. We present ε expansions in both the number of linear and quadratic dimensions. The corrections to critical exponents are nonanalytic in ε, with a functional form that depends on the starting point on the upper critical line.