  We study two-dimensional (2D) Dirac fermions in the presence of a periodic
mass term alternating between positive and negative values along one direction.
This scenario could be realized for a graphene monolayer or for the surface
states of topological insulators. The low-energy physics is governed by chiral
Jackiw-Rebbi modes propagating along zero-mass lines, with the energy
dispersion of the Bloch states given by an anisotropic Dirac cone. By means of
the transfer matrix approach, we obtain exact results for a piece-wise constant
mass superlattice. On top of Bloch states, two different classes of boundary
and/or interface modes can exist in a finite-size geometry or in a nonuniform
electrostatic potential, respectively. We compute the dispersion relation for
both types of boundary and interface modes, which originate either from states
close to the superlattice Brillouin zone (BZ) center or, via a Lifshitz
transition, from states near the BZ boundary. In the presence of a potential
step, we predict that the interface modes, the Bloch wave functions, and the
electrical conductance will sensitively depend on the step position relative to
the mass superlattice.
