  We study a model for a quantum critical point in two spatial dimensions
between a semimetallic phase, characterized by a stable quadratic Fermi node,
and an ordered phase, in which the spectrum develops a band gap. The quantum
critical behavior can be computed exactly, and we explicitly derive the scaling
laws of various observables. While the order-parameter correlation function at
criticality satisfies the usual power law with anomalous exponent $\eta_\phi =
2$, the correlation length and the expectation value of the order parameter
exhibit essential singularities upon approaching the quantum critical point
from the insulating side, akin to the Berezinskii-Kosterlitz-Thouless
transition. The susceptibility, on the other hand, has a power-law divergence
with non-mean-field exponent $\gamma = 2$. On the semimetallic side, the
correlation length remains infinite, leading to an emergent scale invariance
throughout this phase.
