Classification and construction of interacting fractonic higher-order topological phases

Symmetry-protected topological (SPT) phases greatly expand our knowledge of quantum phases of matter beyond the conventional Landau symmetry-breaking paradigm. A common feature of SPT phases is the gapless boundary states due to symmetry protection. However, in contrast to the ordinary topological insulator, recently a new class of SPT phases is shown to exist where symmetry-protected gapless modes only show up on certain low-dimensional submanifolds on the boundary while the majority of the boundary can be gapped without breaking the symmetry. These features defined a new class of SPT phases dubbed higher-order topological phases. We show the notion of higher-order topological phases can have interesting generalizations to systems with subsystem symmetries that exhibit fractonic dynamics for charged excitations. We systematically study the higher-order topological phases protected by a combination of subsystem symmetries and ordinary global symmetries in two and three-dimensional interacting boson systems. From the general classification, we establish a few interesting facts. For instance, Abelian subsystem symmetry has no nontrivial 2-foliated higher-order SSPT phase in (2+1)d systems without the aid of global symmetries. In addition, we prove that for inhomogeneous subsystem symmetries, there is no nontrivial higher-order subsystem SPT phase. Besides the general classification, we also explicitly construct models of such subsystem SPT states in 2 and 3 spatial dimensions.