It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})\ensuremath{\bigoplus}{H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ for all 230 space groups $G$ , where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the $n\mathrm{th}$ twisted group cohomology of $G$ with $\mathbb{Z}$ coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if $g$ reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the ${E}_{8}$ state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in $d$ spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of ${E}_{8}$ states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for $G$ . Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})\ensuremath{\bigoplus}{H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ for all 230 space groups $G$, where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the $n\mathrm{th}$ twisted group cohomology of $G$ with $\mathbb{Z}$ coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if $g$ reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the ${E}_{8}$ state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in $d$ spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of ${E}_{8}$ states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for $G$. Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.

arXiv:1811.06558Physical review. B./Physical review. BApr 22, 202019 citations

Disentangling beyond-cohomology symmetry protected topological phases

Jeongwan Haah, Lukasz Fidkowski, Matthew B. Hastings

Bulletin of the American Physical SocietyMar 6, 20190 citations

Crystallographic interacting topological phases and equivariant cohomology: to assume or not to assume

Daniel Sheinbaum, Omar Antolín Camarena

Abstract For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the P-equivariant cohomology of a d-dimensional torus gives rise to different interacting phases, where P denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases.Abstract For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the P-equivariant cohomology of a d-dimensional torus gives rise to different interacting phases, where P denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases.

arXiv:2007.06595Journal of High Energy PhysicsJul 1, 20211 citation

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

T. R. Liu, Zheng Zhang, Y. X. Zhao

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.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.

cond-mat.mes-hallCross-listedarXiv:2512.21844Dec 29, 2025

Classification of topological states with symmetries

Antônio Sérgio Teixeira Pires

Institute of Physics eBooksNov 1, 20210 citations

Non-Abelian topological phases

Michael Hartley Freedman

Bulletin of the American Physical SocietyMar 15, 20060 citations

G

, where ${H}_{\ensuremath{\phi}}^{n}(G;\mathbb{Z})$ denotes the

n\mathrm{th}

twisted group cohomology of

G

with

\mathbb{Z}

coefficients, and $\ensuremath{\phi}$ indicates that $g\ensuremath{\in}G$ acts nontrivially on coefficients by sending them to their inverses if

g

reverses space-time orientation and acts trivially otherwise. The previously known summand ${H}_{\ensuremath{\phi}}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the

{E}_{8}

state or its multiples on 2-cells of space. It is the crystalline analog of the ``group cohomology proposal'' for classifying bosonic symmetry protected topological (SPT) phases, which takes the form ${H}_{\ensuremath{\phi}}^{d+2}(G;\mathbb{Z})\ensuremath{\cong}{H}_{\ensuremath{\phi}}^{d+1}(G;\text{U}(1))$ for finite internal symmetry groups in

d

spatial dimensions. The new summand ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ classifies possible configurations of

{E}_{8}

states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal. The completeness of our classification and the physical meaning of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ are established through a combination of dimensional reduction, surface topological order, and explicit cellular construction. The value of ${H}_{\ensuremath{\phi}}^{1}(G;\mathbb{Z})$ can be easily read off from the international symbol for

G

. Our classification agrees with the prediction of the ``generalized cohomology hypothesis,'' which concerns the general structure of the classification of SPT phases, and therefore provides strong evidence for the validity of the said hypothesis in the realm of crystalline symmetries.

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.