  In this paper, we prove the existence of a bound state in a waveguide that
consists of two semi-infinite periodic structures separated by an interface.
The two periodic structures are perturbed from the same periodic medium with a
Dirac point and they possess a common band gap enclosing the Dirac point. The
bound state, which is called interface mode here, decays exponentially away
from the interface with a frequency located in the common band gap and can be
viewed as a bifurcation from the Dirac point. Using the layer potential
technique and asymptotic analysis, we first characterize the band gap opening
for the two perturbed periodic media and derive the asymptotics of the Bloch
modes near the band gap edges. By formulating the eigenvalue problem for the
waveguide with two semi-infinite structures using a boundary integral equation
over the interface and analyzing the characteristic values of the associated
boundary integral operator, we prove the existence of the interface mode for
the waveguide when the perturbation of the periodic medium is small.
