Excitonic instability of three-dimensional gapless semiconductors: Large-N theory

Three-dimensional gapless semiconductors with quadratic band touching, such as HgTe, $\alpha$-Sn, or Pr$_2$Ir$_2$O$_7$ are believed to display a non-Fermi-liquid ground state due to long-range electron-electron interaction. We argue that this state is inherently unstable towards spontaneous formation of a (topological) excitonic insulator. The instability can be parameterized by a critical fermion number $N_c$. For $N < N_c$ the rotational symmetry is spontaneously broken, the system develops a gap in the spectrum, and features a finite nematic order parameter. To leading order in the 1/N expansion and in the static approximation, the analogy with the problem of dynamical mass generation in (2+1)-dimensional quantum electrodynamics yields $N_c = 16/[3\pi(\pi-2)]$. Taking the important dynamical screening effects into account, we find that $N_c \geq 2.6(2)$ and therefore safely above the physical value of $N = 1$. Some experimental consequences of the nematic ground state are discussed.