Edge theories of two-dimensional fermionic symmetry protected topological phases protected by unitary Abelian symmetries

Abelian Chern-Simons theory, characterized by the so-called $K$ matrix, has been quite successful in characterizing and classifying Abelian fractional quantum Hall effect as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT (fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry $G_f$. Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry $G_f$ have three new features: (1) they may support gapless Majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if $G_f$ is a nontrivial extension of bosonic symmetry $G_b$ by ${\mathbb{Z}}_2^f$, and (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge $n-1$ for type-III bosonic SPT phases protected by ${\left({\mathbb{Z}}_n\right)}^3$ symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used in these examples could be generalized to derive the edge theories of fSPT phases with arbitrary unitary Abelian total symmetry $G_f$.

Figure 1

The bulk described by ACS action has a natural Luttinger liquid(s) boundary. (a) The bulk and boundary of one example with $K$ matrix $K={\sigma }_z\oplus {\sigma }_x\oplus {\sigma }_x$; (b) the nonchiral Luttinger liquid boundary has six edge fields, three propagating clockwise and the other three anticlockwise.

Figure 2

Two approaches to detect the symmetry anomaly. (a) If there exist symmetric Higgs term(s) that can drive the edge Luttinger liquid to a symmetric short-range entangled state, the bulk is a trivial SPT; otherwise, it is nontrivial SPT. (b) The symmetric flux is inserted, whose topological spin or (projective) representation of other symmetry can be used to detect whether a SPT is nontrivial or not. If the symmetry $g$ does not permute edge fields, but only shift them by a phase shift $\delta {\varphi }^g$, the symmetry defect on the boundary can be created by applying the “fractionalized” operator ${\mathrm{\Phi }}_{\text{g}}=e^{i{K}^{-1}\delta {\varphi }^g\varphi }$.

Figure 3

The boundary of the bulk with $K$ matrix $K_f={\sigma }_z\oplus K_b$ where $K_b$ is the $K$ matrix for type-III bosonic SPT phases in bosonic systems. The bosonic symmetry $G_b={\left({\mathbb{Z}}_n\right)}^3$ with $n\ge 2$ only acts on the $K_b$ nontrivially. $|{\mathrm{\Phi }}_g\rangle$, necessarily being a multiplet, represents flux of one ${\mathbb{Z}}_n$ symmetry generated by $g$. $U_h,U_k$ are the matrix representations of the elements $h,k$, the generators of the remaining ${\left({\mathbb{Z}}_n\right)}^2$ symmetry, and they form the fundamental projective representation of ${\left({\mathbb{Z}}_n\right)}^2$. In our construction, we conjecture $K_b=\left({\sigma }_x\right)\oplus n-1$ for $n\ge 2$. The shaded regions indicate that the symmetry realization of $h,k$ may permute different edge fields, as seen in (5.4), (5.6), and (5.8), and the creation operator for $g$-symmetry flux involves multiple edge fields.

Figure 4

The $K$ matrix for intrinsically interacting fSPT protected by ${\mathbb{Z}}_4^f\times {\mathbb{Z}}_4\times {\mathbb{Z}}_4$ symmetry is $K={\sigma }_z\oplus {\sigma }_x\oplus {\sigma }_x$. $|{\mathrm{\Phi }}_g\rangle$ represents $\frac{\pi }{2}$ flux of ${\mathbb{Z}}_4^f$ whose generator is denoted as $g$ and $U_h,U_k$ are the matrix representation of the generators $h,k$ of the remaining ${\mathbb{Z}}_4\times {\mathbb{Z}}_4$. The shaded regions indicate that the symmetry realization of $h,k$ may permute different edge fields, including the fermionic ones, as seen in (6.2), and the $\frac{\pi }{2}$ symmetry flux is a multiplet that involves different edge fields.