Classification of crystalline topological insulators through K-theory

Abstract

Topological phases for free fermions in systems with crystal symmetry are classified by the topology of the valence band viewed as a vector bundle over the Brillouin zone. Additional symmetries, such as crystal symmetries which act non-trivially on the Brillouin zone, or time-reversal symmetry, endow the vector bundle with extra structure. These vector bundles are classified by a suitable version of K-theory. While relatively easy to define, these K-theory groups are notoriously hard to compute in explicit examples. In this paper we describe in detail how one can compute these K-theory groups starting with a decomposition of the Brillouin zone in terms of simple submanifolds on which the symmetries act nicely. The main mathematical tool is the Atiyah-Hirzebruch spectral sequence associated to such a decomposition, which will not only yield the explicit result for several crystal symmetries, but also sheds light on the origin of the topological invariants. This extends results that have appeared in the literature so far. We also describe examples in which this approach fails to directly yield a conclusive answer, and discuss various open problems and directions for future research.

Topological invariants beyond symmetry indicators: Boundary diagnostics for twofold rotationally symmetric superconductors

“…In addition to the symmetries in the AZ class, the presence of crystalline symmetries leads to finer separation of phases. A formalism called twisted equivariant K theory has been developed to accommodate crystalline symmetries in the classification of gapped noninteracting fermionic systems 18,57,58 . In this formalism, different phases in the AZ class n form an Abelian group φ K (τ,c),−n G (BZ), where the symmetry action of the group G is further specified by φ, τ , and c. The group homomorphism φ : G → Z 2 denotes whether an element g ∈ G is unitary (φ(g) = 1) or anti-unitary (φ(g) = −1).…”

“…When the system lives on a d-dimensional lattice with translational symmetries, the BZ is a torus T d . One can make use of the mathematical tool of AHSS to construct an approximation to φ K (τ,c),−n G (BZ) 18,57,58 , which is otherwise hard to compute.…”

“…The first step is to study the classification in the momentum space by calculating the K group for the considered superconductors, and to derive calculable expressions for the momentumspace topological invariants accordingly. Given that the K group is generally difficult to compute in the presence of crystalline symmetries, we approximate the K group using a mathematical tool called Atiyah-Hirzebruch spectral sequence (AHSS) 57,58 . The key idea of the AHSS method is to decompose the full C 2 -symmetric BZ into different subspaces with only effective internal symmetry such that one can directly compute the contribution from each of the subspaces to the total K group.…”

“…In Sec. II, we specify the symmetry class of our case study and introduce key concepts about the equivariant K theory 35,57,58 . In Sec.…”

Dynamic bulk-boundary correspondence for anomalous Floquet topology

“…Each cell now has an emergent AZ class depending on which symmetries leave the cell invariant, resulting in the first-order approximation of the topological classification, known as E 1 -pages. Then, the connections between cells, i.e., how the topological phase of one cell affects its adjacent higher-dimensional (or lower-dimensional in real space AHSS) cells, are considered iteratively until the classification converges after at most d + 1 iterations [42][43][44] .…”