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Maxwell lattices, where the number of degrees of freedom equals the number ofconstraints, are known to host topologically-protected zero-frequency modes andstates of self stress, characterized by a topological index called topologicalpolarization. In this letter, we show that in addition to these knowntopological modes, with the help of a mirror symmetry, the inherent chiralsymmetry of Maxwell lattices creates another topological index, themirror-graded winding number (MGWN). This MGWN is a higher order topologicalindex, which gives rise to topological zero modes and states of self stress atmirror-invariant domain walls and corners between two systems with differentMGWNs. We further show that two systems with same topological polarization canhave different MGWNs, indicating that these two topological indices arefundamentally distinct.

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