We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [${Z}_{N}\ensuremath{\rtimes}R, {Z}_{N}\ifmmode\times\else\texttimes\fi{}R$, U(1)$\ensuremath{\rtimes}R$, or U(1)$\ifmmode\times\else\texttimes\fi{}R$], 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 ${Z}_{N}\ensuremath{\rtimes}R$ symmetry are classified as ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ for even
N and 0 (no SPT phase) for odd
N, while those protected by U(1)$\ensuremath{\rtimes}R$ symmetry are
Z2. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of
S=2 spins on the square lattice is a
Z2 SPT phase protected by reflection and $\ensuremath{\pi}$-rotation symmetries.
Symmetry-protected topological (SPT) phases in quantum spin chains are distinguished by topological indices. Ogata's operator-algebraic indices apply to a broad range of symmetry classes without requiring continuous symmetry, whereas the elementary twist index of Tasaki requires
U(1) symmetry. We prove their equivalence in the common setting of integer-spin chains with on-site
U(1)⋊Z2 symmetry and a symmetric locally-unique gapped ground state, without assuming translation invariance or a matrix product representation. A charge-fluctuation estimate yields a strong-limit construction of half-chain rotations with their phases fixed. An exact group commutator identity then identifies the Ogata index with the limit of local twist expectations and gives an explicit error bound. This provides a computable formula for the index when a lower bound on the gap and the relevant ground-state expectations are available. Combining the equivalence with a recent boundary-field gap theorem of OpenAI and our earlier twist-index calculation, we obtain a spin-rotation-invariant locally-unique gapped ground state of the infinite
S=1 antiferromagnetic Heisenberg chain with both indices equal to
−1, without assuming infinite-volume uniqueness.
The author formulated the problem and proposed the basic strategy, while ChatGPT Astra made substantial contributions to the development of the proof.