Mirror-symmetry-protected higher-order topological zero-frequency edge and corner modes in Maxwell lattices

Higher order topological phases in two dimensions are known to host protected corner modes. However, these corner modes are only robust in the presence of chiral symmetry. Here, we demonstrate mirror symmetry protected higher order topology in a Maxwell lattice consisting of point masses connected by springs where the number of degrees of freedom equals the number of constraints. We show that along mirror symmetric lines in the reciprocal space the compatibility matrix of the Maxwell lattice can be block-diagonalized into even and odd parity sectors, and a winding number can be defined within each sector (mirror graded winding number MGWN). We further show that two systems with the same topological polarization can have different MGWN. Using analytical theory and numerical diagonalization, we prove the existence of edge and corner modes localized at mirror invariant domain walls and corners between two systems with different MGWN. Interestingly, even and odd parity edge/corner modes appear at opposite edges/corners. Furthermore, due to an inherent chiral symmetry of Maxwell lattices, these edge and corner modes are pinned at zero frequency and cannot be removed as long as the bulk spectrum is gapped.