Elementary band representations

calc_bandreps(sgnum::Integer, Dᵛ::Val{D}=Val(3);
              timereversal::Bool=true,
              allpaths::Bool=false,
              explicitly_real::Bool=timereversal)

Compute the band representations of space group sgnum in dimension D, returning a BandRepSet.

Keyword arguments

• timereversal (default, true): whether the irreps used to induce the band representations are assumed to be time-reversal invariant (i.e., are coreps, see realify).
• allpaths (default, false): whether the band representations are projected to all distinct k-points returned by lgirreps (allpaths = false), including high-symmetry k-lines and -plane, or only to the maximal k-points (allpaths = true), i.e., just to high-symmetry points.
• explicitly_real (default, timereversal): whether, if timereversal = true, to ensure that the site symmetry irreps accompanying the band representations are chosen to be explicitly real (or "physically" real; see physical_realify). This is helpful for subsequent analysis of the action of time-reversal symmetry.

Notes

All band representations associated with maximal Wyckoff positions are returned, irregardless of whether they are elementary (i.e., no regard is made to whether the band representation is "composite"). As such, the returned band representations generally are a superset of the set of elementary band representations (and so contain all elementary band representations).

Implementation

The implementation is based on Cano, Bradlyn, Wang, Elcoro, et al., Phys. Rev. B 97, 035139 (2018), Sections II.C-D.

bandreps(sgnum::Integer, D::Integer=3; 
         allpaths::Bool=false, spinful::Bool=false, timereversal::Bool=true)

Returns the elementary band representations (EBRs) as a BandRepSet for space group sgnum and dimension D.

Keyword arguments

• allpaths: include a minimal sufficient set (false, default) or all (true) k-vectors.
• spinful: single- (false, default) or double-valued (true) irreps, as appropriate for spinless and spinful particles, respectively. Only available for D=3.
• timereversal: assume presence (true, default) or absence (false) of time-reversal symmetry.

References

3D EBRs are obtained from the Bilbao Crystallographic Server's BANDREP program; please reference the original research papers noted there if used in published work.

indicator_group(F::Smith) -> Any

Return the symmetry indicator group $X^{\text{BS}}$ associated with an input set of band representations brs (or Smith decomposition thereof, F), i.e., return the the nontrivial (i.e., ≠ {0,1}) elementary factors of the Smith normal form of the band representation matrix. The return value is a Vector{Int} containing the nontrivial factors. If no nontrivial factors exists, the return value is an empty Vector{Int}.

See also indicator_group_as_string for a formatted string version.

Understanding

The symmetry indicator group answers the question "what direct product of $\mathbb{Z}_n$ groups is the the quotient group $X^{\text{BS}} = \{\text{BS}\}/\{\text{AI}\}$ isomorphic to?" (see e.g., Po, Watanabe, & Vishwanath, Nature Commun. 8, 50 (2017) for more information).

Example

julia> brs = calc_bandreps(2, Val(3));

julia> indicator_group(brs)
4-element Vector{Int64}:
 2
 2
 2
 4

indicator_group_as_string(
    nontriv_Λ::AbstractVector{<:Integer}
)

Return the symmetry indicator group $X^{\text{BS}}$ as a formatted string (i.e., as "Zᵢ×Zⱼ×…"). See also indicator_group for a vector representation.

Example

julia> brs = calc_bandreps(2, Val(3));

julia> indicator_group_as_string(brs)
"Z₂×Z₂×Z₂×Z₄"

basisdim(brs::BandRepSet) --> Int

Return the dimension of the (linearly independent parts) of a band representation set. This is $d^{\text{bs}} = d^{\text{ai}}$ in the notation of Po, Watanabe, & Vishwanath, Nature Commun. 8, 50 (2017), or equivalently, the rank of stack(brs) over the ring of integers. This is the number of linearly independent basis vectors that span the expansions of a band structure viewed as symmetry data.