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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

To compute XBS, the natural first step was to identify Embedded Image, the extra compatibility relations enforced by the antiunitary symmetries. Contrary to this expectation, we now show that, on the basis of our previous results on SGs, one can directly compute {BS} and XBS for any MSG M without deriving Embedded Image. This served to demonstrate the power of the present approach: Symmetry content and connectivity of BSs could be readily extracted without the large overheads mandated by the conventional approach.

To this end, we first revisited the relevant aspects of the theory for an SG G. By definition, Embedded Image is a subgroup of Embedded Image, and therefore, a priori, it could be the case that Embedded Image strictly. However, by an explicit computation for all the 230 SGs, Po et al. (15) found that Embedded Image always holds. This statement has an important implication: Every Embedded Image can be expanded as b = ∑iqiai with rational coefficients Embedded Image using a basis Embedded Image of Embedded Image. In other words, full knowledge of the group Embedded Image can be obtained from that of Embedded Image.

On the basis of this result for SGs, we will now prove the same statement, namely, dBS = dAI, for any MSG M. This result enabled us to efficiently compute {BS} and XBS for all MSGs and MLGs using only information contained in {AI}, which could be readily extracted from the tabulated Wyckoff positions (4648).

Our proof was centered on the following observation: Recall Embedded Image, where G is unitary and Embedded Image is the antiunitary part generated by Embedded Image for some spatial symmetry g0. Note that, for type II MSGs, Embedded Image and can be chosen to be the identity, whereas for types III and IV, Embedded Image. Now, consider a G-symmetric collection of fully filled local orbitals in real space, which defines an AI Embedded Image. This AI is not generally symmetric under M,; that is, it may not be invariant under the action of Embedded Image. However, if we stack it together with its Embedded Image-transformed copy, we will arrive at an M-symmetric AI. Algebraically, this means Embedded Image.

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 Embedded Image of Embedded Image. Then, the Embedded Image-transformed copy, Embedded Image, forms a basis of an irrep Embedded Image of Embedded Image (section S4). When b represents a BS that contains Embedded Image-times, we denote by Embedded Image a BS that contains Embedded Image the same number of times. For any Embedded Image, Embedded Image satisfies the compatibility conditions Embedded Image and hence belongs to {BS}. We presented the explicit form of Embedded Image in section S6.

We are now ready to prove the statement. Observe that any B ∈ {BS} also belongs to Embedded Image, and as Embedded Image, it can be expanded on the basis of Embedded ImageEmbedded Image(2)Now, we symmetrize both sides of Eq. 2 by Embedded Image. Since B is M-invariant, Embedded Image, soEmbedded Image(3)As argued, Embedded Image (note that linearity was invoked when we extended the argument form Embedded Image to a general element of Embedded Image). This proves that any B ∈ {BS} can be expanded in terms of {AI} (using rational coefficients), an equivalent statement of dAI = dBS. Furthermore, it implies that the quotient group XBS = {BS}/{AI} does not contain any Z-factor and hence is a finite abelian group of the form Embedded Image.

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 dBS, XBS, and νBS 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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