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Nematic quantum criticality in three-dimensional Fermi system with quadratic band touching

We construct and discuss the field theory for tensorial nematic order parameter coupled to gapless four-component fermions at the quadratic band touching point in three (spatial) dimensions. Within a properly formulated epsilon-expansion this theory is found to have a quantum critical point, which describes the (presumably continuous) transition from the semimetal into a (nematic) Mott insulator. The latter phase breaks the rotational, but not the time-reversal symmetry, and may be relevant to materials such as gray tin or mercury telluride at low temperatures. The critical point represents a simple quantum analogue of the familiar classical isotropic-to-nematic transition in liquid crystals. The properties and the consequences of this quantum critical point are discussed. Its existence supports the scenario of the "fixed-point collision", according to which three-dimensional Fermi systems with quadratic band touching and long-range Coulomb interactions are unstable towards the gapped nematic ground state at low temperatures.