I will describe a subclass of crystalline topological phases known as Higher-Order Topological Phases of Matter (HOTPs). Models in these phases have the defining feature that they necessarily support non-trivial gapless degrees of freedom on hinges or corners of certain spatial manifolds.

In particular I will describe two aspects of HOTPs:

(i) A general procedure that may be used to construct models for weakly interacting fermonic HOTPs in various Altland Zirnbauer (AZ) classes. The procedure involves using spatial symmetries to localize topologically non-trivial mass textures on high symmetry corners and hinges of certain spatial manifolds. Then the classification of topological defects in band insulators can be readily leveraged to predict the existence of gapless modes on these corners/hinges.

(ii) I will generalize the notion of bulk-boundary correspondence for HOTPs by including various possible configurations of symmetric surface topological order.