Theory of interacting topological crystalline insulators

We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group $Z_8$. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U\left(1\right)\times Z_2$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.

Figure 1

Periodic array of 1D domain wall fermions, generated by spatially alternating Dirac masses to eight flavors of 2D Dirac fermions [see Eqs. (18) and (19)]. 1D chiral fermion modes in flavors $1,...,4$ (red arrows) and flavors $5,...,8$ (blue arrows) propagate in opposite directions along a domain wall. Counterpropagating chiral fermions have opposite mirror eigenvalues $\pm 1$. Each domain wall becomes gapped under the interaction (10), (13), thus leading to a gapped and mirror-symmetric 2D phase.

Figure 2

(a) A mass domain wall setup on a 3D TCI surface consisting of $n$ flavors of Dirac fermions. One-dimensional chiral fermions reside at the domain wall at $x=0$, with $n_{+}$ ($n_{-}=n-n_{-}$) modes moving in the $+y$ ($-y$) direction, depending on the signs of Dirac masses $m_1,...,m_n$. (b) The domain wall in (a) is expanded to a wide region, sandwiched between semi-infinite regions on the left and on the right, with opposite Dirac masses. Importantly, (a) and (b) are both symmetric under mirror $x\to -x$ and topologically equivalent. For $n\ne 8k$, the domain wall in (a), hence the middle region in (b) as well, cannot be gapped and mirror symmetric.