We study the quantum criticality of the phase transition between the Dirac semimetal and the excitonic insulator in two dimensions. Even though the system has a semimetallic ground state, there are observable effects of excitonic pairing at finite temperatures and/or finite energies, provided that the system is in proximity to the excitonic insulating transition. To determine the quantum critical behavior, we consider three potentially important interactions, including the Yukawa coupling between Dirac fermions and the excitonic order parameter fluctuation, the long-range Coulomb interaction, and the disorder scattering. We employ the renormalization group technique to study how these interactions affect quantum criticality and also how they influence each other. We first investigate the Yukawa coupling in the clean limit, and show that it gives rise to typical non-Fermi liquid behavior. Adding random scalar potential to the system always turns such a non-Fermi liquid into a compressible diffusive metal. In comparison, the non-Fermi liquid behavior is further enhanced by random vector potential, but is nearly unaffected by random mass. Incorporating the Coulomb interaction may change the results qualitatively. In particular, the non-Fermi liquid state is protected by the Coulomb interaction for weak random scalar potential, and it becomes a diffusive metal only when random scalar potential becomes sufficiently strong. When random vector potential or random mass coexists with Yukawa coupling and Coulomb interaction, the system is a stable non-Fermi liquid state, with fermion velocities flowing to constants in the former case and being singularly renormalized in the latter case. These quantum critical phenomena can be probed by measuring observable quantities.
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