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

Abstract

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 $N_c\ge 2.6(2)$ and therefore safely above the physical value of $N=1$. Some experimental consequences of the nematic ground state are discussed.

6 pages, 1 figure, v2: section on experimental implications expanded, references added, published version

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