Constructing BSs from AIs

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

To compute *X*_{BS}, the natural first step was to identify *X*_{BS} for any MSG M without deriving

To this end, we first revisited the relevant aspects of the theory for an SG G. By definition, *et al*. (*15*) found that *b* = ∑_{i}*q*_{i}*a*_{i} with rational coefficients

On the basis of this result for SGs, we will now prove the same statement, namely, *d*_{BS} = *d*_{AI}, for any MSG M. This result enabled us to efficiently compute {BS} and *X*_{BS} for all MSGs and MLGs using only information contained in {AI}, which could be readily extracted from the tabulated Wyckoff positions (*46*–*48*).

Our proof was centered on the following observation: Recall *g*_{0}. Note that, for type II MSGs,

In the momentum space, a similar symmetrization procedure could be performed on the representation content. Suppose that {|*k*, *i*〉} is a basis of an irrep *b* represents a BS that contains

We are now ready to prove the statement. Observe that any *B* ∈ {BS} also belongs to *B* is M-invariant, *B* ∈ {BS} can be expanded in terms of {AI} (using rational coefficients), an equivalent statement of *d*_{AI} = *d*_{BS}. Furthermore, it implies that the quotient group *X*_{BS} = {BS}/{AI} does not contain any Z-factor and hence is a finite abelian group of the form

To summarize, we showed that the set of AIs and BSs was identical as far as their dimensionality goes. This is a powerful statement, since it means that we could simply focus on AIs, study their symmetry representations in momentum space, and then take rational combinations to generate all BSs. However, we caution that one has to properly rescale the entries of *n* when an irrep is paired with another copy of itself according to the Herring rule. For full details of the treatment, we refer interested readers to (*15*).

Following this strategy, we performed the first calculation of *d*_{BS}, *X*_{BS}, and *ν*_{BS} for all of the 1651 MSGs and 528 MLGs. The full list of the computation results are tabulated in tables S1 to S7. For readers’ convenience, we reproduced a few examples from these tables in Tables 1 and 2.

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