Gauging spatial symmetries and the classification of topological crystalline phases

TL;DR

The paper develops a comprehensive framework for interacting topological crystalline phases by introducing crystalline gauge fields and the crystalline topological liquid concept, then proves the Crystalline Equivalence Principle: in contractible Euclidean space, crystalline topological liquids with symmetry G are classified identically to internal G-symmetric phases, with orientation-reversing spatial elements mapping to anti-unitary internal actions. It presents a bootstrap method to construct solvable crystalline models from internal SPTs, and analyzes topological responses through symmetry-flux fusion and effective actions encoded by twisted equivariant cohomology, exemplified in multiple.dimensional SPTs (e.g., reflections, rotations, translations, and crystalline insulators). The framework extends to non-contractible spaces via the homotopy quotient X//G, explores spatially-dependent TQFTs, anomalies, and generalizations to Floquet systems and fermions, and discusses beyond-crystalline-liquids by addressing non-invertible defects and fracton-like phases. Together, these results provide a unified, computable strategy to classify and realize symmetry-protected crystalline topological phases and their responses, with concrete 2+1D and 3+1D examples and explicit cocycle-based actions. The work has implications for predicting robust boundary phenomena, guiding experimental exploration of crystalline SPTs, and informing the construction of lattice models with spatial symmetries protected by topology.

Abstract

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 anti-unitary symmetry in the internal symmetry group. As an example, we explicitly compute, using group cohomology, a partial classification of bosonic symmetry-protected topological (SPT) phases protected by crystalline symmetries in (3+1)-D for 227 of the 230 space groups. For the 65 space groups not containing orientation-reversing elements (Sohncke groups), there are no cobordism invariants which may contribute phases beyond group cohomology, and so we conjecture our classification is complete.

Figures (17)

• 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) and (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 depicted is the transformation of a particular basis element. The star indicates the "missing quadrant" of section \ref{['sec_spatial_gauge_fields']}, across which spins (green) away from the rotation center (red) are glued by a 90 degree rotation (see also Appendix \ref{['appendix:gaugecoupling']}). 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 three 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.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Theorem 2
• Definition 4
• Definition 5
• Lemma 1
• Definition 6
• Theorem 3