arXiv:1901.06419 · Advances in Theoretical and Mathematical Physics

Invertible phases of matter with spatial symmetry

Daniel S. Freed, Michael J. Hopkins

Published Jan 1, 2020• 21 citations

Abstract

We propose a general formula for the group of invertible topological phases on a space $Y$, possibly equipped with the action of a group $G$. Our formula applies to arbitrary symmetry types. When $Y$ is Euclidean space and $G$ a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.

DOI: 10.4310/atmp.2020.v24.n7.a3

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arXiv:1901.06419 · Advances in Theoretical and Mathematical Physics

Invertible phases of matter with spatial symmetry

Daniel S. Freed, Michael J. Hopkins

Published Jan 1, 2020• 21 citations

arXiv HTML PDF

Abstract

We propose a general formula for the group of invertible topological phases on a space

Y

, possibly equipped with the action of a group

G

. Our formula applies to arbitrary symmetry types. When

Y

is Euclidean space and

G

a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.

DOI: 10.4310/atmp.2020.v24.n7.a3

Related papers

Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle

Arun Debray

Freed-Hopkins give a mathematical ansatz for classifying gapped invertible phases of matter with a spatial symmetry in terms of Borel-equivariant generalized homology. We propose a slight generalization of this ansatz to account for cases where the symmetry type mixes nontrivially with the spatial symmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz, we prove as a theorem a "fermionic crystalline equivalence principle," as predicted in the physics literature. Using this and the Adams spectral sequence, we compute classifications of some classes of phases with a point group symmetry; in cases where these phases have been studied by other methods, our results agree with the literature.Freed-Hopkins give a mathematical ansatz for classifying gapped invertible phases of matter with a spatial symmetry in terms of Borel-equivariant generalized homology. We propose a slight generalization of this ansatz to account for cases where the symmetry type mixes nontrivially with the spatial symmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz, we prove as a theorem a "fermionic crystalline equivalence principle," as predicted in the physics literature. Using this and the Adams spectral sequence, we compute classifications of some classes of phases with a point group symmetry; in cases where these phases have been studied by other methods, our results agree with the literature.

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We propose a general formula for the group of invertible topological phases on a space

Y

, possibly equipped with the action of a group

G

. Our formula applies to arbitrary symmetry types. When

Y

is Euclidean space and

G

a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.

{G}_{f}

have been characterized by a triplet of indices $([(\ensuremath{\nu},\ensuremath{\rho})],[\ensuremath{\mu}])$. Our main result is an elementary derivation of the fermionic stacking rules of one-dimensional IFT phases for any given internal symmetry group

{G}_{f}

from the perspective of the boundary, i.e., we give an explicit operational definition for the boundary representation $([({\ensuremath{\nu}}_{\ensuremath{\wedge}},{\ensuremath{\rho}}_{\ensuremath{\wedge}})],[{\ensuremath{\mu}}_{\ensuremath{\wedge}}])$ obtained from stacking two IFT phases characterized by the triplets of boundary indices $([({\ensuremath{\nu}}_{1},{\ensuremath{\rho}}_{1})],[{\ensuremath{\mu}}_{1}])$ and $([({\ensuremath{\nu}}_{2},{\ensuremath{\rho}}_{2})],[{\ensuremath{\mu}}_{2}])$, respectively.

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.

K

theory.