Having described some generalities about BSs, we now review how knowledge about the real space can inform band topology (9, 15). We defined the trivial class of BSs by the AIs, which were band insulators that were smoothly connected to a limit of vanishing hopping and hence were deformable to product states in real space. Equivalently, an AI admits symmetric, exponentially localized Wannier functions.

To specify an AI, one should choose a position x in real space at which electrons were localized and the type of the orbital put on that site. All inequivalent choices of the position x were classified by Wyckoff positions (51). The orbital can be chosen from the (co-)irreps of the site-symmetry group of x (section S5). Given these choices, an M-invariant AI can be constructed by placing a symmetry-related orbital on each site of the M-symmetric lattice and filling them by electrons. The AI has a specific combination of irreps in the momentum space, which automatically satisfies Embedded Image. We listed up all distinct n’s corresponding to an AI by varying x and the orbital type. We listed up all distinct n’s corresponding to an AI by varying the position x and the orbital type, and we obtained {AI}phys, a subset of {BS}phys. If one replaces M above with G, one gets the set of G-symmetric AIs, Embedded Image.

Now, we are ready to tell which elements of {BS}phys must be topologically nontrivial and which elements can be trivial. This can be judged by contrasting the elements of {BS}phys with those in {AI}phys. Namely, any b ∈ {BS}phys not belonging to {AI}phys necessarily features nontrivial band topology because, by definition, there does not exist any atomic limit of the BS with the same combinations of irreps. This is a sufficient (but not necessary) condition to be topologically nontrivial: Here, we exclusively focused on the band topology that can be diagnosed by the set of irreps at high-symmetry momenta.

The simplest way of exploring the nontrivial elements of {BS}phys is thus to consider the complement of {AI}phys in {BS}phys, as in (15) and (52). However, this set has a complicated mathematical structure. To simplify the analysis, we allowed for the formal subtraction of bands and extended the values of Embedded Image to any integer, including the negative ones, à la a K-theory analysis. {BS}phys then becomes an abelian group Embedded Image (known as a “lattice” in the mathematical nomenclature) (14, 15, 35). In other words, there are dBS basis “vectors” Embedded Image, and {BS} can be expressed as Embedded Image. Similarly, by allowing negative integers when taking superposition of AIs, we got another abelian group Embedded Image, which is a subgroup of {BS}. The band topology we are interested in is now encoded in the quotient groupEmbedded Image(1)dubbed the symmetry-based indicator of band topology (15). As we will see shortly, the quotient group is always a finite abelian group and hence must be a product of the form Embedded Image.

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