In this thesis we investigate interaction driven quantum phase transitions
of semimetals with a point-like Fermi surface. The most famous example
of a member of this family is graphene, which at half-filling hosts gapless
fermionic excitations that are Dirac like, i.e. disperse linearly. Unlike in con-
ventional metals the density of states vanishes at the Fermi level, which in
turn promises novel correlations due to the reduced phase space available to
fluctuations. When subject to strong short-ranged interactions these systems
undergo a phase transition into a broken symmetry phase where the excita-
tions become gapped. However due to the gapless nature of the fermionic
excitations in the semimetallic phase it is not possible to describe the crit-
icality using a Ginzburg-Landau type theory which only contains bosonic
order parameter degrees of freedom. The low-energy theory best equipped
for these so-called fermionic quantum criticality problems is a Yukawa-type
effective field theory which couples the dynamical order parameter field to
the fermions.
With the use of Renormalisation Group (RG) we study the critical phe-
nomena of the quantum phase transition from a nodal-point semimetal to
charge density wave (CDW) insulator. We show that the screening of or-
der parameter fluctuations by particle-hole excitations is crucial. Without
inclusion of this non-perturbative effect the RG flows contain non-universal
dependence on the momentum shell cutoff scheme. We compute the exact
critical exponents for the case of Dirac and semi-Dirac fermions in two spatial
dimensions up to linear order in 1/N f where N f is the number of fermionic
flavours. Lastly we consider the effects of non-magnetic disorder on a Dirac
semimetal to CDW phase transition, and find a new disordered interacting
fixed point which gives rise to non-Fermi liquid behaviour. We investigate
the scaling of physical observables at this critical fixed point.