(2+1)D topological phases with RT symmetry: many-body invariant, classification, and higher order edge modes

It is common in condensed matter systems for reflection ($R$) and time-reversal ($T$) symmetry to both be broken while the combination $RT$ is preserved. In this paper we study invariants that arise due to $RT$ symmetry. We consider many-body systems of interacting fermions with fermionic symmetry groups $G_f={\mathbb{Z}}_2^f\times {\mathbb{Z}}_2^{RT}$, $U(1{)}^f⋊{\mathbb{Z}}_2^{RT}$, and $U(1{)}^f\times {\mathbb{Z}}_2^{RT}$. We show that (2+1)D invertible fermionic topological phases with these symmetries have a $\mathbb{Z}\times {\mathbb{Z}}_8$, ${\mathbb{Z}}^2\times {\mathbb{Z}}_2$, and ${\mathbb{Z}}^2\times {\mathbb{Z}}_4$ classification, respectively, which we compute using the framework of $G$-crossed braided tensor categories. We provide a many-body $RT$ invariant in terms of a tripartite entanglement measure, and which we show can be understood using an edge conformal field theory computation in terms of vertex states. For $G_f=U(1{)}^f⋊{\mathbb{Z}}_2^{RT}$, which applies to charged fermions in a magnetic field, the non-trivial value of the ${\mathbb{Z}}_2$ invariant requires strong interactions. For symmetry-preserving boundaries, the phases are distinguished by zero modes at the intersection of the reflection axis and the boundary. Additional invariants arise in the presence of translation or rotation symmetry.