Topological Mott Insulator in Three-Dimensional Systems with Quadratic Band Touching

We argue that a three-dimensional electronic system with the Fermi level at the quadratic band touching point such as HgTe could be unstable with respect to the spontaneous formation of the (topological) Mott insulator at arbitrary weak long-range Coulomb interaction. The mechanism of the instability can be understood as the collision of Abrikosov's non-Fermi liquid fixed point with another, quantum critical, fixed point, which approaches it in the coupling space as the system's dimensionality $d\ensuremath{\rightarrow}{d}_{\text{low}}+$, with the ``lower critical dimension'' $2<{d}_{\text{low}}<4$. Arguments for the existence of the quantum critical point based on considerations in the large-$N$ limit in $d=3$, as well as close to $d=2$, are given. In the one-loop calculation we find that ${d}_{\text{low}}=3.26$, and thus above, but not far from three dimensions. This translates into a temperature or energy window (${T}_{\mathrm{c}}$, ${T}_{*}$) over which the non-Fermi liquid scaling should still be observable, before the Mott transition finally takes place at the critical temperature ${T}_{\mathrm{c}}\ensuremath{\sim}{T}_{*}\mathrm{exp}[\ensuremath{-}zC/({d}_{\text{low}}\ensuremath{-}d{)}^{1/2}]$. We estimate $C=\ensuremath{\pi}/1.1$, dynamical critical exponent $z\ensuremath{\approx}1.8$, and the temperature scale ${k}_{B}{T}_{*}\ensuremath{\approx}(4m/{m}_{\text{el}}{ϵ}^{2})13.6\text{ }\text{ }\mathrm{eV}$, with $m$ as the band mass and $ϵ$ as the dielectric constant.

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