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.

We consider the extended half-filled Hubbard model on the honeycomb lattice for second nearest neighbors interactions. Using a functional integral approach, we find that collective fluctuations suppress topological states and instead favor charge ordering, in agreement with previous numerical studies. However, we show that the critical point is not of the putative semimetal-Mott insulator variety. Due to the frustrated nature of the interactions, the ground state is described by a novel hidden metallic charge order with semi-Dirac excitations. We conjecture that this transition is not in the Gross-Neveu universality class. arXiv:1806.07212v2 [cond-mat.str-el]

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