Gauging Spatial Symmetries and the Classification of Topological Crystalline Phases

We put the theory of interacting topological crystalline phases on a systematic footing. These are topological phases protected by space-group symmetries. Our central tool is an elucidation of what it means to “gauge” such symmetries. We introduce the notion of a crystalline topological liquid and argue that most (and perhaps all) phases of interest are likely to satisfy this criterion. We prove a crystalline equivalence principle, which states that in Euclidean space, crystalline topological liquids with symmetry group $G$ are in one-to-one correspondence with topological phases protected by the same symmetry $G$, but acting internally, where if an element of $G$ is orientation reversing, it is realized as an antiunitary symmetry in the internal symmetry group. As an example, we explicitly compute, using group cohomology, a partial classification of bosonic symmetry-protected topological phases protected by crystalline symmetries in ($3+1$) dimensions for 227 of the 230 space groups. For the 65 space groups not containing orientation-reversing elements (Sohncke groups), there are no cobordism invariants that may contribute phases beyond group cohomology, so we conjecture that our classification is complete.

Popular Summary

Matter can exist in many different phases, such as solids and liquids, and these phases are often distinguished by their symmetry. In a solid, atoms are arranged in a regular crystal lattice, which is only symmetric with respect to some discrete set of rotations and translations. Liquids, however, do not appear to be different under rotations or translations. Researchers have recently realized that, regardless of the crystal symmetry, the electrons contained within a solid can be in many different phases, which are distinguished by subtle details of the quantum entanglement between different parts of the system. These phases are known as topological phases. In some cases, the interplay between this entanglement and the crystal symmetries becomes important for distinguishing the phases, a behavior known as crystalline topological phases. We propose a systematic framework to describe crystalline topological phases, based on an assumption that such phases of matter still retain some kind of “fluidity.”

This fluidity allows one to define a consistent coupling to a “crystalline gauge field,” analogous to how one studies internal symmetries by coupling to a gauge field. By considering the possible topological response to the crystalline gauge field, we can separate phases and prove qualitative, universal characteristics about them.

An important result is that crystalline topological phases can be put in a one-to-one correspondence with topological phases with a different kind of symmetry, acting on internal degrees of freedom instead of moving points in space around. This allows us to classify crystalline phases and provides new intuition for crystalline symmetries by comparing them with internal ones.

Figure 1

(a) In a smooth state, the lattice spacing and the correlation length $\xi$ are much less than the unit cell size $a$ and the radius of spatial variation. (b) The topological response of a crystalline topological liquid is captured by a spatially dependent TQFT that captures the spatial dependence within each unit cell but “forgets” about the lattice.

Figure 2

An angular defect of 90 degrees in a vertex-centered square lattice and an angular excess of 120 degrees in a face-centered kagome lattice.

Figure 3

A 90-degree disclination maps discontinuously to the square lattice, as indicated with the colored quadrants. The red line is the branch cut across which the image rotates by 90 degrees. Because the discontinuity is by a rotation in $G$, this map descends to a continuous map from the disclination to the quotient of the square lattice by $G$.

Figure 4

The “patches” picture of a gauge field for an internal symmetry. (a) The manifold $M$ is divided up into patches, and the boundaries between patches are twisted by a group element $g\in G$. (b) The flatness constraint implies that the holonomy around a vertex must be trivial. (c,d) We identify configurations that differ by dividing patches or by acting on a patch with some $g\in G$.

Figure 5

An example of a $C_4$ disclination is shown undergoing a full rotation. Our Hilbert space is a product of $C_4$ spins on each site, and we depict the transformation of a particular basis element. The star indicates the missing quadrant of Sec. 3a, across which spins (green) away from the rotation center (red) are glued by a 90-degree rotation (see also Appendix pp3). For convenience, we have used an orange domain wall to indicate the boundaries between regions of homogeneous spin (compare Fig. 6). The rotation itself is a two-step process which must be performed 3 times. The first step (black arrows) is to simply rotate the picture around the rotation center counterclockwise by 90 degrees. The second step (white arrows) is a gauge transformation (compare Fig. 7) that moves the missing quadrant to its original position. At the end of the process, all green spins have returned to their original configuration, while the spin at the rotation center has been rotated one unit. If there is a charge at the rotation center, the disclination thus picks up that charge as a phase after a full rotation.

Figure 6

The pair of pants as a crystalline gauge background. The branch-cut gluings are indicated with colored arrows. Here, $(x,t)=(0,0)$ is a singularity of the smooth structure but not the continuous structure. It can be smoothed out into a high-curvature region but does not affect our calculations.

Figure 7

A ${\varphi }_2$ flux and a ${\varphi }_1$ flux fuse with a minus sign, computed as a path integral over the pair of pants with an unavoidable domain-wall crossing.

Figure 8

(a) Specifying the homotopy type of the space $\mathrm{\Theta }$ of all TQFTs involves specifying points in this space, paths between arrows (single arrows), deformations between paths (double arrow), and so on. We want these to capture features of the space of quantum ground states. (b) These features can also be interpreted as interfaces. We show a spatial configuration of interfaces in a two-dimensional system, with two one-dimensional interfaces separated by a junction of dimension 0. We can imagine that these interfaces are smoothed out such that the spatial variation occurs on scales that are large compared to the lattice spacing (thus, we have a smooth state as discussed in Secs. 2 and 3b). Traversing a path in ${\mathbb{R}}^2$ from the left half-plane to the right half-plane, the local quantum state goes through the path ${\gamma }_0$ or ${\gamma }_1$ depending on whether the path in ${\mathbb{R}}^2$ goes through the upper one-dimensional interface or the lower one. As one deforms the path in ${\mathbb{R}}^2$ through the zero-dimensional junction (black dot), the corresponding path in the space of quantum states goes through the deformation described by $\mathfrak{d}$.

Figure 9

A reflection acting on a circle. The reflection axis is shown in red, and the fixed points are marked with blue circles. SPT phases of this symmetry setup are classified by a ${\mathbb{Z}}_2$ charge at each fixed point. There are more phases in this geometry than on a line with a single reflection center.

Figure 10

A $p4$ (actually $p4m$) symmetric Majorana Hamiltonian. At each site, there are four Majorana operators (blue dots), which are hybridized along the teal bonds. An arrow from Majorana $c_i$ to Majorana $c_j$ is a term $-(i/2)c_i{c}_j$ in the Hamiltonian. The arrows are chosen to be a $p4$ symmetric Kastelyn orientation, which is necessary for the ground state to be free of fermion parity $\pi$ fluxes (see Refs. [111, 128]). The $C_4$ rotation centers are highlighted in beige. The other sites are $C_2$ rotation centers.

Figure 11

A 90-degree disclination defect in the Majorana lattice above. The green curves denote Majoranas that are equivalent in the projected Hilbert space (see Appendix pp3). In particular, $c_2=c_3=c^{\prime }$ on the defect Hilbert space. This means that the rightward and downward bonds from the singular vertex combine into a single term $-(i/2)(c_0+c_1)c^{\prime }$ in the Hamiltonian, meaning $c_0-c_1$ is an unpaired Majorana zero mode. Likewise, one can show that a 180-degree disclination carries two unpaired Majorana zero modes. These cannot be paired in a symmetric way. However, we expect that a fusion of four 90-degree disclinations will have four unpaired Majorana zero modes, which can be paired in a $C_4$ symmetric way.

Figure 12

In this approach to defining the Hamiltonian coupled to a crystalline gauge field, patches are allowed to overlap to include some vertices. In this particular example, $U_{\text{left}}\cap U_{\text{right}}$ includes vertex 2, which gets duplicated. Hamiltonian terms (denoted by solid edges) lying entirely inside $U_{\text{left}}$ or $U_{\text{right}}$ are taken to act on those Hilbert spaces. Then, spurious degrees of freedom are eliminated by applying a projection operator, which, in a product state basis, identifies the state at 2 with $g$ applied to the state at 2’. This is indicated by the green curve labeled by $g$ cutting the dashed vertical line from 2 to 2’.

Figure 13

When the $g$ domain wall is pulled off of this edge, it is revealed to be an edge from $x\to gy$. Note the similarity with the Hamiltonian coupling procedure in Appendix pp3.

Figure 14

Here, we depict a piece of $M$ (northwest) mapping to a piece of $X$ (southeast). We have given the vertices of $X$ unique labels and labeled the vertices of $M$ with their image vertices in $X$. Note that vertex 2 has two adjacent preimages. This edge of $M$ is mapped to a degenerate edge, and the triangle it lies on (grey) is mapped to a degenerate face 122 in $X$. Note also that vertex $5\in X$ has no preimage, and to map faces to faces, we must refine the lattice of $M$, depicted by the dotted blue lines.

Figure 15

The conservation law for $G$ labels allows us to draw them as $G$ domain walls in $X$. Then, in any contractible patch of $M$, we can describe our local map $M\to X$ by pushing off the domain walls. Then, we look at the northwest picture of our patch in $M$. Note how the vertices have been transformed, as have the edges. Then, we fill in the transformed picture with faces of $X$ as we would in describing an ordinary map $M\to X$. This always requires a choice of base point. Here, our base point is 4, and we have pushed all the domain walls (green) straight to the east. The choice of base point is like a local choice of gauge. It should be compared with the construction for coupling to Hamiltonians in Appendix pp3.

Figure 16

A prism $M\times [0,1]$ mapping to $BG$ means assigning $G$ labels also to the interior edges. These correspond to the vertices of $M$, so we can think of them as a function $g:M^0\to G$, where $M^0$ is the set of vertices of $M$. Then, the conservation law on the internal faces of the prism forces a constraint between corresponding edge labels in each $M$. The constraint is that the top labels are the gauge transformation of the bottom labels by $g$. The direction is fixed by an orientation of the internal prism edges. If we reverse all of them, it takes $g\mapsto g^{-1}$ (and locally as well).

Figure 17

A $p4m$-invariant triangulation and branching structure. The black dots are the vertices of the original $p4m$-invariant cellulation (the simple square lattice), and the red dots are the vertices that had to be added (through the barycentric subdivision) to get a $p4m$-invariant triangulation and branching structure.