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We study theoretically the electron correlation effect in a three-dimensionalDirac fermion system which describes a topologically nontrivial state. It isshown within the mean-field approximation that time-reversal and inversionsymmetries of the system are spontaneously broken in the region where bothspin-orbit coupling and electron correlation are strong. This phase isconsidered as an analog of that in the lattice quantum chromodynamics. It isalso shown that in the presence of magnetic impurities, electron correlationenhances the appearance of the Weyl semimetal phase between the topologicalinsulator phase and the normal insulator phase.

Electron Correlation Induced Spontaneous Symmetry Breaking and Weyl Semimetal Phase in a Strongly Spin-Orbit Coupled System