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Time reversal, fermion doubling, and the masses of lattice Dirac fermions in three dimensions

Motivated by recent examples of three-dimensional lattice Hamiltonians with massless Dirac fermions in their (bulk) spectrum, I revisit the problem of fermion doubling on bipartite lattices. The number of components of the Dirac fermion in a time-reversal and parity invariant d-dimensional lattice system is determined by the minimal representation of the Clifford algebra of $d+1$ Hermitian Dirac matrices that allows a construction of the time-reversal operator with the square of unity, and it equals $2^d$ for $d=2,3$. Possible mass-terms for (spinless) Dirac fermions are listed and discussed. In three dimensions there are altogether eight independent masses, out of which four are even, and four are odd under time reversal. A specific violation of time-reversal symmetry that leads to (minimal) four-component massless Dirac fermion in three dimensions at low energies is constructed.