A Cohomological Framework for Topological Phases from Momentum-Space Crystallographic Groups

TL;DR

This work develops a cohomological framework for crystalline topological phases by promoting momentum-space crystallographic groups (MCGs) as the governing symmetry objects. It shows that the Abelian topological content of crystalline insulators is captured by $H^2(\Gamma_F,\mathbb{Z})$, while twistings of point-group actions are encoded in $H^3(\Gamma_F,\mathbb{Z})$, with an exact algebraic isomorphism to cohomology with functional coefficients, $H^{n+1}(\Gamma_F,\mathbb{Z}) \cong H^{n}(\Gamma_F,\mathcal{F}(\mathbb{R}^d_F,U(1)))$ for $n\ge1$. The approach yields fully algebraic formulations of topological invariants (e.g., $C_{\beta\alpha}$ as Chern-number data for $P1$, $\mathbb{Z}_2$ invariants for glide symmetries, and $\mathbb{Z}_3$ and $\mathbb{Z}_4$ invariants for $I23$) and provides a practical route to twistings via $\mathcal{H}^3_G(T^d_F,\mathbb{Z}) \cong H^3(\Gamma_F,\mathbb{Z})$, enabling systematic computation (e.g., with GAP) across all 2D and 3D MCGs. The framework thus offers an algebraic, scalable alternative to differential-form–based classifications and serves as a technical backbone for analyzing crystalline topological phases under projective symmetries. It also connects to twisted equivariant K-theory as the fuller classification beyond the Abelian approximation, clarifying how momentum-space topology emerges from group cohomology.

Abstract

Crystallographic groups are conventionally studied in real space to characterize crystal symmetries. Recent work has recognized that when these symmetries are realized projectively, momentum space inherently accommodates nonsymmorphic symmetries, thereby evoking the concept of \textit{momentum-space crystallographic groups} (MCGs). Here, we reveal that the cohomology of MCGs encodes fundamental data of crystalline topological band structures. Specifically, the collection of second cohomology groups, $H^2(Γ_F,\mathbb{Z})$, for all MCGs $Γ_F$, provides an exhaustive classification of Abelian crystalline topological insulators, serving as an effective approximation to the full crystalline topological classification. Meanwhile, the third cohomology groups $H^3(Γ_F,\mathbb{Z})$ across all MCGs exhaustively classify all possible twistings of point-group actions on the Brillouin torus, essential data for twisted equivariant K-theory. Furthermore, we establish the isomorphism $H^{n+1}(Γ_F,\mathbb{Z})\cong H^n\big(Γ_F,\operatorname{\mathcal{F}}(\mathbb{R}^d_F,U(1))\big)$ for $ n\ge 1$, where $\operatorname{\mathcal{F}}(\mathbb{R}^d_F,U(1))$ denotes the space of continuous $U(1)$-valued functions on the $d$D momentum space $\mathbb{R}^d_F$. The case $n=1$ yields a complete set of topological invariants formulated in purely algebraic terms, which differs fundamentally from the conventional formulation in terms of differential forms. The case $n=2$, analogously, provides a fully algebraic description for all such twistings. Thus, the cohomological theory of MCGs serves as a key technical framework for analyzing crystalline topological phases within the general setting of projective symmetry.

Figures (6)

• Figure 1: Illustration of the isomorphism Eq. \ref{['eq:e-Isomorphism']} for $n=1$ and $n=2$ in (a) and (b), respectively. $n+1$ ordered group elements form an $(n+1)$-simplex, whose boundary consists of $n+2$ oriented $n$-simplices. Compared with the previous simplexes of group cohomology dijkgraaf1990topologicalSPT_Wen, here the vertices are the orbit of $\bm{k}$ under the consecutive action of the $n+1$ group elements.
• Figure 2: Illustration of $\bm{k}$-orbits associated with the two sides of algebraic relations used to formulate topological invariants. (a), (b) and (d) depict orbits for generic $\bm{k}$, while (c) highlights the orbit for the high-symmetry momentum $\bm{K}$.The surfaces spanned by the orbits of each side are shaded in dark and light blue, respectively.
• Figure S1: Wilson loop $\gamma(k_y)$ of the valence bands for the given parameters. When $\gamma(k_y)$ crosses $\pi$ an odd number of times on $k_y\in[-\pi,0)$, the $\mathbb{Z}_2$ topological number is nontrivial.
• Figure S2: Continuity of phase functions along symmetry lines: (a) $\phi_{l_x}(k)$ (abbreviated as $\phi_x$) evaluated along $k_x = \pi$; (b) $\phi_{g_x}(k)$ (abbreviated as $\phi_g$) evaluated along $k_y = 0$. A discontinuity appears in panel (b), which can be resolved by shifting the right segment of the curve upward by $2\pi$, thereby rendering $\phi_{g_x}(k)$ continuous. Accordingly, the value $\phi_{g_x}(\pi, 0)$ is taken to exceed $2\pi$ in the main text.
• Figure S3: Brillouin zone of momentum-space crystallographic group $P2_1/c$. The region $T = [-\pi, -\pi, -\pi] \times [\pi, -\pi, \pi]$ forms a two-dimensional torus topologically, and the integral of the Berry curvature over $T$ yields the Chern number $\nu_1$. The segment $S_1$ is topologically equivalent to a Klein bottle and contributes to the invariant $\nu_4$. Its boundaries, highlighted in red, are identified under the screw rotation symmetry $s_y$. Similarly, the segment $S_2$ is also a Klein bottle, with boundaries related by either $s_y$ or the glide reflection $g_y$.