We investigate bound states in the continuum (BICs) in dielectric photonic crystal slabs, which occur at the double Dirac point with four-fold degeneracy. The underlying structure consists of a hexagonal cluster of equilateral triangular holes with \(C_{6v}\) symmetry in the unit cell of a honeycomb lattice. The lowest four TE bands intersect at the center of Brillouin zone, exhibiting linear dispersions to form double Dirac cones when the ratio of cluster radius R (the distance from unit cell center to the centroid of each hole) to lattice constant a satisfies the ’degenerate’ condition, where the honeycomb lattice can also be described as a triangular lattice due to symmetry-enforced geometric self-duality. In particular, two resonant states at the double Dirac point are identified as symmetry-protected (SP) BICs with extremely high quality factors. They are high-order vortex polarization singularities (V points) characterized by two irreducible representations ( \(B_1\) and \(B_2\) ) in the \(C_{6v}\) symmetry group, each carrying a topological charge \(q=-2\) . As the ratio R/a deviates from the degenerate condition, the four-fold degeneracy is lifted and a gap is opened between two pairs of doubly degenerate bands. In this situation, Dirac BICs no longer exist and a pair of SP BICs appear on either the upper or lower two bands, with similar polarization patterns and topological charges. By reducing the size of three non-adjacent triangular holes to break \(C_2\) symmetry, while maintaining \(C_{3v}\) symmetry, Dirac BICs are transformed to low-order V points with \(q=+1\) , accompanied by a group of six circularly polarized states (C points) with \(q=-1/2\) that surround the V point, which preserve the total topological charge.
