Topological phases protected by point group symmetry

TL;DR

This work develops a universal dimensional-reduction framework for classifying and constructing 3d pgSPT phases protected by crystalline point-group symmetries, by reducing to lower-dimensional topological states on the mirror planes with on-site symmetry. It shows that all bosonic and fermionic pgSPT phases in 3d can be built as stacks/arrays of 2d (and lower-dimensional) states, enabling concrete classifications such as $ ext{Z}_2 imes ext{Z}_2$ for bosonic $ ext{Z}_2^P$-protected pgSPT, $ ext{Z}_8 imes ext{Z}_2$ for electronic TCIs, and $ ext{Z}_{16}$ or trivial classifications for TCSCs depending on $ ext{sigma}^2$. The paper also analyzes symmetry-preserving surfaces, revealing anomalous SET surface states (e.g., toric code with $ePmP$ fractionalization and three-fermion orders) and connects fermionic TCIs to bosonic root states like the $E_8$ state. Overall, the framework provides a practical path to classify, construct, and understand pgSPT and SET phases with crystalline point-group symmetry and motivates extensions to broader symmetry settings and SET phenomena.

Abstract

We consider symmetry protected topological (SPT) phases with crystalline point group symmetry, dubbed point group SPT (pgSPT) phases. We show that such phases can be understood in terms of lower-dimensional topological phases with on-site symmetry, and can be constructed as stacks and arrays of these lower-dimensional states. This provides the basis for a general framework to classify and characterize bosonic and fermionic pgSPT phases, that can be applied for arbitrary crystalline point group symmetry and in arbitrary spatial dimension. We develop and illustrate this framework by means of a few examples, focusing on three-dimensional states. We classify bosonic pgSPT phases and fermionic topological crystalline superconductors with $Z_2^P$ (reflection) symmetry, electronic topological crystalline insulators (TCIs) with ${\rm U}(1) \times {Z}_2^P$ symmetry, and bosonic pgSPT phases with $C_{2v}$ symmetry, which is generated by two perpendicular mirror reflections. We also study surface properties, with a focus on gapped, topologically ordered surface states. For electronic TCIs we find a $Z_8 \times Z_2$ classification, where the $Z_8$ corresponds to known states obtained from non-interacting electrons, and the $Z_2$ corresponds to a "strongly correlated" TCI that requires strong interactions in the bulk. Our approach may also point the way toward a general theory of symmetry enriched topological (SET) phases with crystalline point group symmetry.

Figures (11)

• Figure 1: Three-dimensional system with periodic boundary conditions and $\mathbb{Z}_2^P$ reflection symmetry. Each point on the solid circle corresponds to a $2d$ plane with periodic boundary conditions. The dashed line intersects the system at the two mirror planes $o$ and $\infty$, which are contained in the shaded regions $r_o$ and $r_{\infty}$, respectively. These regions have thickness $w$. Dotted lines indicate the boundaries of these regions with two other regions, $r_1$ and $\sigma r_1$. The regions are chosen so that $r_o$ and $r_{\infty}$ are invariant under reflection, while $r_1$ and $\sigma r_1$ are exchanged under reflection.
• Figure 2: (a) $1d$ local unitary represented as a finite-depth quantum circuit. The vertical lines represent spins, and each shaded rectangle is a unitary operator acting on a pair of spins. (b) Restriction of a $1d$ local unitary to the region between the two dashed lines. The two-spin unitary operators lying outside this region are simply omitted. The restriction procedure is not uniquely defined near the boundaries of the region, but this freedom does not play a role in our discussion.
• Figure 3: Panel (a) depicts a system with mirror reflection $\sigma$, and discrete translation symmetry generated by $T_x$ normal to the mirror plane. Each point on the line represents a $2d$ plane, and there are two inequivalent types of mirror planes. One type (thick dashed lines) is obtained by translating the $\sigma$-plane, and the other type (thin dotted lines) is obtained by translating the $T_x \sigma$-plane, which is separated from the $\sigma$-plane by half a lattice. In this setting we can have a stack of alternating-chirality $E_8$ states, where $+$ / $-$ represent $E_8$ states with $n_{E_8} = \pm 1$ on the two types of mirror planes. Reduction to $2d$ can be visualized by pairing $E_8$ states away from the mirror plane as shown, leaving a $n_{E_8} = +1$ state on the mirror plane. A different reduction procedure is illustrated in (b), where states are grouped to give a $n_{E_8} = -1$ state on the mirror plane.
• Figure 4: Geometry of a symmetry preserving surface of a $3d$ pgSPT phase protected by $\mathbb{Z}_2^P$ reflection symmetry, ignoring any translation symmetry. In the bulk, the system has been reduced to a $2d$ state lying on the mirror plane. The edge of the mirror plane coincides with the reflection axis of the surface.
• Figure 5: Operators in the modified toric code model at the surface of the $\mathbb{Z}_2$ root state. Three vertex operators $A_v$ are shown, with two adjacent to the reflection axis (dashed line), and one away from it. Each operator is a product of Pauli spin operators on the edges marked by thick solid lines, with $X,Y,Z$ corresponding to $\tau^x, \tau^y, \tau^z$. Plaquette operators $B_p$ are products of four $\tau^z$ operators around the perimeter of a plaquette $p$, as indicated by thick dotted lines.