Interacting and fractional topological insulators via the $Z_2$ chiral anomaly

Recently, it was shown that the topological properties of 2D and 3D topological insulators are captured by a $Z_2$ chiral anomaly in the boundary field theory. It remained, however, unclear whether the anomaly survives electron-electron interactions. We show that this is, indeed, the case, thereby providing an alternative formalism for treating topological insulators in the interacting regime. We apply this formalism to fractional topological insulators (FTI) via projective/parton constructions and use it to test the robustness of all fractional topological insulators that can be described in this way. The stability criterion we develop is easy to check and based on the pair switching behavior of the noninteracting partons. In particular, we find that FTIs based on bosonic Laughlin states and the $M=0$ bosonic Read-Rezayi states are fragile and may have a completely gapped and nondegenerate edge spectrum in each topological sector. In contrast, FTIs based on fermionic $Z_k$ Read-Rezayi states with $M=1$ and odd $k$ and the bosonic 3D topological insulator with a $\pi /4$ fractional $\theta$ term are topologically stable.

Figure 1

Spectral motions of the noninteracting low-energy many-body spectrum and occupation of single-particle momentum states as a function of the adiabatically threaded flux $\varphi$. Insertion of half of a flux quantum results in one electron being pumped to the edge and a Kramer's degeneracy corresponding to momentum state occupation shown in Fig. 1. Completing the flux insertion one arrives at an orthogonal state with momentum-state occupation as in Fig. 1. Alternatively, the many-body groundstate of a TI evolves into an orthogonal state following a full-flux insertion and exhibits a level crossing at $\pi$ flux as shown in Fig. 1.