Invariants for (2+1)D bosonic crystalline topological insulators for all 17 wallpaper groups

TL;DR

The paper develops a comprehensive framework to characterize (2+1)D bosonic SPTs with wallpaper-group space symmetries and internal K, introducing a complete set of many-body invariants derived from partial rotations and reflections (including twisted and relative variants). By leveraging the crystalline equivalence principle and TQFT intuition, it connects these invariants to the group cohomology classification, providing explicit formulas and topological actions for pure crystalline and mixed invariants, with detailed treatments of the p4m and p4g groups. The results demonstrate that partial symmetry measurements on a single ground state (and, when needed, on torus geometries with twisted boundary conditions) can fully detect the predicted SPT classes, subject to certain filling invariants. The work also lays out a general reading protocol for the 17 wallpaper groups, discusses the interplay with filling and TOR terms, and outlines extensions to fermionic systems and SETs, highlighting practical routes for numerical and experimental exploration.

Abstract

We study bosonic symmetry-protected topological (SPT) phases in (2+1) dimensions with symmetry $G = G_{\text{space}}\times K$, where $G_{\text{space}}$ is a general wallpaper group and $K=\text{U}(1),\mathbb{Z}_N, \text{SO}(3)$ is an internal symmetry. In each case we propose a set of many-body invariants that can detect all the different phases predicted from real space constructions and group cohomology classifications. They are obtained by applying partial rotations and reflections to a given ground state, combined with suitable operations in $K$. The reflection symmetry invariants that we introduce include `double partial reflections', `weak partial reflections' and their `relative' or `twisted' versions which also depend on $K$. We verify our proposal through exact calculations on ground states constructed using real space constructions. We demonstrate our method in detail for the groups p4m and p4g, and in the case of p4m also derive a topological effective action involving gauge fields for orientation-reversing symmetries. Our results provide a concrete method to fully characterize (2+1)D crystalline topological invariants in bosonic SPT ground states.

Figures (13)

• Figure 1: Definition of the decomposition of the disk $D$ into three regions $D_l$, $D_c$ and $D_r$ used in the definition of $\Sigma_{\mathrm{o},l}$.
• Figure 2: Setup to evaluate $\Upsilon_{l}(\mathbf{g};\mathbf{j})$. The red and blue lines denote the identification of sides of the rectangular space into a torus. $D$ is the region in light blue. The dashed purple line denotes boundary conditions twisted by the group element $\mathbf{g}$. The reflection line $l$ is shown in orange.
• Figure 3: Unit cell for space group pmm.
• Figure 4: Unit cell conventions for (a) p4m and (b) p4g. Maximal Wyckoff positions are labeled by early Greek letters ($\alpha,\beta,\gamma)$. Orange and dashed purple lines correspond to reflection and glide axes, respectively. These lines are labeled by mid-range Greek letters ($\lambda,\mu,\nu$). We use labels with the same subscripts to denote positions or lines that are related by a point group symmetry.
• Figure 5: The state on the region $D = D_l \cup D_c \cup D_r$ is represented by a solid sphere, whose boundary is a sphere. The northern hemisphere is the 'ket' part of the state, which we triangulate as shown in panel (a). We have broken region $D_c$ into four smaller regions to accommodate the triangulation. Panel (b) shows the triangulation of the 'bra' part. Note that the boundaries of the regions in panels (a) and (b) are the same because it corresponds to the equator of the sphere representing the state.