Band topology

This protocol is extracted from research article:

Structure and topology of band structures in the 1651 magnetic space groups

**
Sci Adv**,
Aug 3, 2018;
DOI:
10.1126/sciadv.aat8685

Structure and topology of band structures in the 1651 magnetic space groups

Procedure

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 *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,

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 *K*-theory analysis. {BS}_{phys} then becomes an abelian group *14*, *15*, *35*). In other words, there are *d*_{BS} basis “vectors” *15*). As we will see shortly, the quotient group is always a finite abelian group and hence must be a product of the form

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