  We compute the effects of generic short-range interactions on gapless
electrons residing at the quantum critical point separating a two-dimensional
Dirac semimetal (DSM) and a symmetry-preserving band insulator (BI). The
electronic dispersion at this critical point is anisotropic ($E_{\mathbf k}=\pm
\sqrt{v^2 k^2_x + b^2 k^{2n}_y}$ with $n=2$), which results in unconventional
scaling of physical observables. Due to the vanishing density of states
($\varrho(E) \sim |E|^{1/n}$), this anisotropic semimetal (ASM) is stable
against weak short-range interactions. However, for stronger interactions the
direct DSM-BI transition can either $(i)$ become a first-order transition, or
$(ii)$ get avoided by an intervening broken-symmetry phase (BSP). We perform a
renormalization group analysis by perturbing away from the one-dimensional
limit with the small parameter $\epsilon = 1/n$, augmented with a $1/n$
expansion (parametrically suppressing quantum fluctuations in higher
dimension). We identify charge density wave (CDW), antiferromagnet (AFM) and
singlet s-wave superconductor as the three dominant candidates for the BSP. The
onset of any such order at strong coupling $(\sim \epsilon)$ takes place
through a continuous quantum phase transition across multicritical point. We
also present the phase diagram of an extended Hubbard model for the ASM,
obtained via the controlled deformation of its counterpart in one dimension.
The latter displays spin-charge separation and instabilities to CDW, spin
density wave, and Luther-Emery liquid phases at arbitrarily weak coupling. The
spin density wave and Luther-Emery liquid phases deform into pseudospin
SU(2)-symmetric quantum critical points separating the ASM from the AFM and
superconducting orders, respectively. Our results can be germane for a
uniaxially strained honeycomb lattice or organic compound
$\alpha$-(BEDT-TTF)$_2\text{I}_3$.
