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arXiv:2403.18887 · Physical review. B./Physical review. B

( 2 + 1 )-dimensional topological phases with R T symmetry: Many-body invariant, classification, and higher-order edge modes

Ryohei Kobayashi, Yuxuan Zhang, Yanqi Wang, Maissam Barkeshli

Published Nov 17, 2025• 1 citation
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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 ${G}_{f}={\mathbb{Z}}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{RT}$, $\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{RT}$, and $\mathrm{U}{(1)}^{f}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{RT}$. We show that (2+1)D invertible fermionic topological phases with these symmetries have a $\mathbb{Z}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{8}$, ${\mathbb{Z}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$, and ${\mathbb{Z}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}$ 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 ${G}_{f}=\mathrm{U}{(1)}^{f}\ensuremath{\rtimes}{\mathbb{Z}}_{2}^{RT}$, which applies to charged fermions in a magnetic field, the nontrivial 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.

DOI
10.1103/d6h9-rkkr

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