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

Abstract

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 $G_f = \mathbb{Z}_2^f \times \mathbb{Z}_2^{RT}$, $U(1)^f \rtimes \mathbb{Z}_2^{RT}$, 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 $G_f = U(1)^f \rtimes \mathbb{Z}_2^{RT}$, 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.

Figures (4)

• Figure 1: (a) Placement of Region A, B, C. Partial $RT$ only acts on A. (b) The teal region represent the 3-vertex state $\ket{\psi_{\text{ABC}}}$. (c) $\ket{\psi_{\text{ABC}}}$ is topologically equivalent to a 3-punctured sphere. (d) $\ket{\psi_{\text{ABC}}}$ after performing the conformal transformation $z\rightarrow z^{3/2}$ where boundaries of the same color are identified.
• Figure 2: The phase of $RT$ invariant $I_{RT}$ for the $H_{1}$ (single layer) and $H_2$ (bilayer) in units of $\pi/8$. The numerics are done on a $16 \times 16$ square lattice with periodic boundary condition. Different saturation of the data points represent different system sizes of the $A\cup B$ region.
• Figure 3: (a) The 3-vertex state $\ket{\psi_{\text{ABC}}}$ is topologically equivalent to a pair of pants. (b) $\rho_{\text{AB}}= \text{Tr}_{\text{C}}(\ket{\psi_{\text{ABC}}}\bra{\psi_{\text{ABC}}})$ is constructed by gluing two 3-vertex states with opposite orientations along $C$. (c) $Z_{RT}$ is constructed by gluing 2 $\rho_{\text{AB}}$ with the top half of the first $\rho_{\text{AB}}$ (tinted yellow) twisted by $U_{\mathbb{Z}_2}T_{\pi}$. We could cut at the red dashed line by inserting ground states, and fictitiously cut through the magenta and green circle, which result in (d): two copies of Ishibashi state $\ket{\mathcal{I}}$, with one of them twisted by $U_{\mathbb{Z}_2}T_{\pi}$.
• Figure 4: (a): $\rho_{\text{AB}}$ is constructed by gluing a 3-vertex state and its inverse. (b) One can cut the top and bottom of $\rho_{\text{AB}}$ by inserting the projector onto a ground state. The width of the ground state is $4\beta$. We also fictitiously cut along the equator. (c) $\rho_{\text{AB}}$ is topologically equivalent to a pair of annuli glued along the inner Boundary.(d) One can express the surface for $Z_{RT}$ as a pair of cylinders with two punctures, which are glued along each pair of punctures of the same color. One of the punctures is acted by $U_{\mathbb{Z}_2}T_{\pi}$ when gluing.