Two-dimensional semi-Dirac fermions are quasiparticles that disperse linearly in one direction and quadratically in the other. We investigate instabilities of semi-Dirac fermions towards charge, spin-density wave and superconducting orders, driven by short-range interactions. We analyze the critical behavior of the Yukawa theories for the different order parameters using Wilson momentum shell RG. We generalize to a large number N f of fermion flavors to achieve analytic control in 2+1 dimensions and calculate critical exponents at one-loop order, systematically including 1/N f corrections. The latter depend on the specific form of the bosonic infrared propagator in 2+1 dimensions, which needs to be included to regularize divergencies. The 1/N f corrections are surprisingly small, suggesting that the expansion is well controlled in the physical dimension. The order-parameter correlations inherit the electronic anisotropy of the semi-Dirac fermions, leading to correlation lengths that diverge along the spatial directions with distinct exponents, even at the mean-field level. We conjecture that the proximity to the critical point may stabilize novel modulated order phases.

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We investigate the stability of the Néel quantum critical point of two-dimensional quantum antiferromagnets, described by a nonlinear σ model, in the presence of a Kondo coupling to N f flavors of two-component Dirac fermion fields. The long-wavelength order parameter fluctuations are subject to Landau damping by electronic particle-hole fluctuations. Using the momentum-shell renormalization group (RG), we demonstrate that the Landau damping is weakly irrelevant at the Néel quantum critical point, despite the fact that the corresponding self-energy correction dominates over the quadratic gradient terms in the IR limit. In the ordered phase, the Landau damping increases under the RG, indicative of damped spin-wave excitations. Although the Kondo coupling is weakly relevant, sufficiently strong Landau damping renders the Néel quantum critical point quasistable for N f 4 and thermodynamically stable for N f < 4. In the latter case, we identify a multicritical point which describes the transition between the Néel critical and Kondo runaway regimes. The symmetry breaking at this fixed point results in the opening of a gap in the Dirac fermion spectrum. Approaching the multicritical point from the disordered phase, the fermionic quasiparticle residue vanishes, giving rise to non-Fermi-liquid behavior.