Bosonic symmetry-protected topological phases with reflection symmetry

We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [$Z_N⋊R, Z_N\times R$, U(1)$⋊R$, or U(1)$\times R$], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer-$S$ spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by $Z_N⋊R$ symmetry are classified as ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ for even $N$ and 0 (no SPT phase) for odd $N$, while those protected by U(1)$⋊R$ symmetry are ${\mathbb{Z}}_2$. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of $S=2$ spins on the square lattice is a ${\mathbb{Z}}_2$ SPT phase protected by reflection and $\pi$-rotation symmetries.

Figure 1

Schematic picture of 2D systems with reflection symmetry and spin-rotation symmetry. The reflection planes (dashed green rectangles) are perpendicular to the 2D systems (blue planes). The spin-rotation axes are represented by black arrows. The symmetry group is $Z_N⋊R$ when the rotation axis is parallel to the reflection plane (left panel), while the symmetry group is $Z_N\times R$ when the rotation axis is perpendicular to the reflection plane (right panel).

Figure 2

Schematic picture of the ground state of the $S=2$ AKLT model on the square lattice. Four blue circles on each site represent four $S=1/2$ spins that form an $S=2$ spin by symmetrization. Solid blue lines denote spin singlets formed by $S=1/2$ spins from neighboring sites. The dashed red line denotes a reflection plane that is perpendicular to the $x$ direction. The unpaired $S=1/2$ spins on the edge form a helical gapless edge mode protected by the symmetry $Z_N⋊R$.