Topological Phases Protected by Point Group Symmetry

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 that they 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, which can be applied for arbitrary crystalline point group symmetry and in arbitrary spatial dimensions. 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 ${\mathbb{Z}}_2^P$ (reflection) symmetry, electronic topological crystalline insulators (TCIs) with $\mathrm{U}(1)\times {\mathbb{Z}}_2^P$ 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 ${\mathbb{Z}}_8\times {\mathbb{Z}}_2$ classification, where the ${\mathbb{Z}}_8$ corresponds to known states obtained from noninteracting electrons, and the ${\mathbb{Z}}_2$ 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 phases with crystalline point group symmetry.

Popular Summary

Symmetry-protected topological (SPT) phases are quantum states of matter that are rather boring and featureless on the interior of a material but have unusual properties at surfaces. Examples of these properties are modes that efficiently transport heat or electrical charge. These modes are robust as long as the physical system has a certain symmetry. We understand the most about SPT states and their surface phenomena for so-called internal symmetries, which are rather abstract to visualize but are often related to conservation laws such as the conservation of electric charge. More familiar are the geometrical symmetries arising in art, architecture, and in our everyday experience of nature wherever regular or repeating geometrical patterns are present. Geometrical symmetries are also pervasive in crystalline solids, where the atoms form regular patterns, and they could play a key role in SPT phases formed by electrons in solids. Nonetheless, very little is known about SPT phases with geometrical symmetries.

Here, we develop a general approach to classify and understand SPT phases protected by a type of geometrical symmetry called crystalline point group symmetry. The key result is that such SPT phases are built from simpler quantum states of lower spatial dimensions. This allows us to show that point group SPT phases can be constructed by packing together lower-dimensional states in a regular fashion, an observation that could eventually allow certain SPT phases to be engineered in composite structures.

Looking ahead, these results may help to identify point group SPT phases most likely to occur in solids. The ideas developed here could also be applied to topological phases beyond SPT phases and perhaps to geometrical symmetries beyond point group symmetry.

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) One-dimensional 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 an $n_{E_8}=+1$ state on the mirror plane. A different reduction procedure is illustrated in (b), where states are grouped to give an $n_{E_8}=-1$ 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.

Figure 6

String operators, in the modified toric code model, creating a reflection-symmetric pair of $e$ particles (top) and $m$ particles (bottom). Operators in the $e$ string ($m$ string) are indicated by thick dotted (solid) lines. The $m$ string, whose path is shown by the light gray line, is decorated with a single ${\tau }^z$ operator at the reflection axis (dashed line).

Figure 7

Construction of the gapped surface for the $E_8$ root state. A $\nu =8$ IQH state lies on the mirror plane, while $\nu =\pm 4$ IQH states lie on the surface in regions $R$ and $L$, respectively. Each of these three regions is a half plane, with edges supporting chiral modes indicated by the dark lines. The 1D fermion fields are $c_I$, $d_{Ri}$, $d_{Li}$, with chiralities as indicated. The same ${\mathbb{Z}}_2$ gauge field, which resides on the “T-shaped” lattice formed by the union of the three regions, and connects the regions as indicated by the dashed lines, is coupled to the fermion parity.

Figure 8

In our construction of the three-fermion surface of the $E_8$ root state, before condensing the ${\mathbb{Z}}_2$ gauge flux on the mirror plane, regions $R$ and $L$ of the surface are in the three-fermion state, while the mirror plane is also in a deconfined phase of ${\mathbb{Z}}_2$ gauge theory, with toric code statistics. Here, in the left panel, a ${\mathbb{Z}}_2$ gauge flux ($m_f$) in region $R$ is moved to region $L$, leading to the configuration shown in the right panel. This process leaves behind a ${\mathbb{Z}}_2$ gauge flux excitation on the mirror plane, which can be understood by viewing the ${\mathbb{Z}}_2$ flux excitations as intersection points between the various regions and a flux line in three-dimensional space, as shown. This residual excitation is eliminated upon condensing ${\mathbb{Z}}_2$ fluxes on the mirror plane to produce the $E_8$ root state in the bulk.

Figure 9

Cross section of a 3D system with $C_{2v}$ symmetry, which is generated by the two mirror reflections ${\sigma }_a$ and ${\sigma }_b$. The dashed lines are mirror planes. The shaded regions can be trivialized by applying a symmetry-preserving local unitary, reducing the system to the cross-shaped region near the mirror planes.

Figure 10

(a) $E_8$ root state for $C_{2v}$ symmetry. Solid lines represent an $E_8$ state on each half-plane, with edge chiralities indicated by the arrows. (b) Microscopic construction of the $E_8$ root state in terms of sheets of $\nu =4$ IQH states (dashed lines), with edge chiralities as indicated by the arrows.

Figure 11

(a) Adjoining 2D sheets (solid lines) to a reduced system with $C_{2v}$ symmetry defined on the mirror planes (dashed lines). The arrows indicate one choice of edge chiralities when the adjoined sheets are $E_8$ states. (b)–(d) Adjoining 1D systems, shown as filled circles.