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

Abstract

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.

4+ pages, references added and updated, published version

References in corpus (4)

• Electron fractionalization in two-dimensional graphenelike structures
• Theory of interacting electrons on the honeycomb lattice
• Many-body spin Berry phases emerging from the $\pi$-flux state: antiferromagnetic/valence-bond-solid competition
• Zero-energy states and fragmentation of spin in the easy-plane antiferromagnet on a honeycomb lattice

Cited by in corpus (5)

• Topological Mott insulator in three-dimensional systems with quadratic band touching
• Instabilities of a birefringent semi-metal
• Conserved charges of order-parameter textures in Dirac systems
• Gross-Neveu-Yukawa theory of $\text{SO}(2N)\to \text{SO}(N)\times \text{SO}(N)$ spontaneous symmetry breaking
• Color degeneracy of competing orders near topological defects cores in planar quadratic band touching systems