Internal API

OrbitalOrdering(br::NewBandRep{D}) --> OrbitalOrdering{D}

Establishes a canonical, local ordering for the orbitals associated with a band representation br. This is the default ordering used when associating row/column indices in a tight-binding Hamiltonian block to specific orbitals in the associated band representations.

The canonical orbital ordering is stored in .ordering. The ith element of ordering, ordering[i], is a NamedTuple with two fields: wp and idx:

• wp: stores a Wyckoff position in the orbit of the Wyckoff position associated to br.wp.
• idx: stores the index of the partner function of the site-symmetry irrep associated to br at wp.

I.e., the ith orbital associated with br is located at wp and transforms as the idxth partner function of the site-symmetry irrep of br.siteir. The total number of orbitals associated to a band representation, and hence the length of ordering, is the product of the site-symmetry irrep dimensionality and the number of sites in the Wyckoff position orbit.

SiteInducedSGRepElement{D}(
    ρ::AbstractMatrix,
    positions::Vector{DirectPoint{D}},
    op::SymOperation{D}
)

Represents a matrix-valued element of a site-induced representation of a space group, including a global momentum-dependent phase factor.

This structure behaves like a functor: calling it with a momentum k :: AbstractVector returns the matrix representation at k.

Fields (internal)

• ρ :: Matrix{ComplexF64} : The momentum-independent matrix part of the representation.
• positions :: Vector{DirectPoint{D}}: Real-space positions corresponding to the orbitals in the orbit of the associated site-symmetry group.

Computes and adds the symmetry related partners of a hopping term δ to the δ_orbit.

Build the P matrix for a particular symmetry operation acting on k-space, which permutes the rows of the M matrix.

To obtain the P matrix, we exploit that the action is on exponentials of the type $exp(2π k⋅δ)$, and instead act on δ ∈ h_orbit.orbit rather than on k. Because of this, we need to use the inverse of the rotation part of the symmetry operation.

Poor man's "matrix sparsification" via the reduced row echelon form.

Prune near-zero elements of vectors in vs.

add_reversed_orbits!(h_orbits::Vector{HoppingOrbit{D}}) where {D}

Adds the reversed hopping terms to the hopping orbits in h_orbits. The reversed hopping terms are added to the orbit of the hopping term they are related to; if they are already present in another orbit, the two orbits are merged.

construct_M_matrix(
    h_orbit::HoppingOrbit{D}, br1::NewBandRep{D}, br2::NewBandRep{D},
    [ordering1, ordering2]) 
    --> Array{Int,4}

Construct a set of matrices that encode a Hamiltonian term which resembles the hopping from EBR br1 to EBR br2.

The encoding is stored as a 4D matrix. Its last two axes correspond to elements of the Bloch Hamiltonian H(k); its first axis corresponds to orbit(h_orbit) and the associated complex exponentials stored in v; and its second axis to the elements of the vector t. That is:

Hₛₜ(k) = vᵢ(k) Mᵢⱼₛₜ tⱼ

See devdocs.md for details.

evaluate_tight_binding_momentum_gradient_term!(
    tbt::TightBindingTerm,
    k::ReciprocalPointLike, 
    components::NTuple{N, Int},
    [c::Union{Nothing, <:Number} = nothing],
    [∇Hs::NTuple{N, Matrix{ComplexF64}} = ntuple(zeros(ComplexF64, size(tbt)), N)]
)

Evaluate components of the momentum gradient of the tight-binding term tbt at momentum k over momentum, possibly multiplied by a scalar coefficient c (unity if omitted). The components are specified by the components tuple: a value of i for some components element indicates the momentum gradient along k[i] (components values must be in the range 1:D).

The components[idx] term is added into the scratch space matrix ∇Hs[idx]; if ∇Hs is not provided, it is initialized as a suitable tuple of zero matrices of appropriate size. The function returns the modified ∇Hs matrix tuple.

The function is analogous to evaluate_tight_binding_term!, but computes momentum gradient components rather than the Hamiltonian matrix itself.

evaluate_tight_binding_term!(tbt::TightBindingTerm, k, [c, H])

Evaluate the tight-binding term tbt at momentum k, possibly multiplied by a scalar coefficient c (unity if omitted). The term is added into the scratch space matrix H; if H is not provided, it is initialized as a zero matrix of the appropriate size.

The function returns the modified H matrix.

Note

The two-argument form of the function, i.e., returning the value of tbt at k, can be more simply achieved via tbt(k).

inversion(::Val{D}) --> SymOperation{D}

Return the inversion operation in dimension D.

maybe_add_hoppings!(h_orbits, δ, qₐ, qᵦ, R, ops) --> Vector{HoppingOrbit{D}}

Checks whether a hopping term δ is already in the list of representatives. If not, adds it and its symmetry-related partners. If it is, only the symmetry-related partners are added.

obtain_basis_free_parameters(
    h_orbit::HoppingOrbit{D},
    brₐ::NewBandRep{D}, 
    brᵦ::NewBandRep{D}, 
    [orderingₐ = OrbitalOrdering(brₐ), orderingᵦ = OrbitalOrdering(brᵦ)]
    )                            --> Tuple{Array{Int,4}, Vector{Vector{ComplexF64}}}

Obtain the basis of free parameters for the hopping terms between brₐ and brᵦ associated with the hopping orbit h_orbit.

Note

The presence or absence of time-reversal symmetry is inferred implicitly from brₐ and brᵦ.

obtain_basis_free_parameters_TRS(
    h_orbit::HoppingOrbit{D}, 
    brₐ::NewBandRep{D}, 
    brᵦ::NewBandRep{D}, 
    orderingₐ::OrbitalOrdering{D} = OrbitalOrdering(brₐ),
    orderingᵦ::OrbitalOrdering{D} = OrbitalOrdering(brᵦ),
    Mm::AbstractArray{4, Int} = construct_M_matrix(h_orbit, brₐ, brᵦ, orderingₐ, orderingᵦ)
)                             --> Tuple{Array{Int,4}, Vector{Vector{ComplexF64}}}

Obtain the basis of free parameters for the hopping terms between brₐ and brᵦ associated with the hopping orbit h_orbit under time-reversal symmetry.

Real and imaginary parts of the basis vectors are differentiated explicitly.

obtain_basis_free_parameters_hermiticity(
    h_orbit::HoppingOrbit{D},
    brₐ::NewBandRep{D},
    brᵦ::NewBandRep{D},
    orderingₐ::OrbitalOrdering{D} = OrbitalOrdering(brₐ),
    orderingᵦ::OrbitalOrdering{D} = OrbitalOrdering(brᵦ),
    Mm::AbstractArray{Int, 4} = construct_M_matrix(h_orbit, brₐ, brᵦ, orderingₐ, orderingᵦ);
    antihermitian::Bool = false,
) where {D}

Constructs a basis for the coefficient vectors t⁽ⁿ⁾ that span the space of Hermitian (or antihermitian if true) TB Hamiltonians Hₛₜ(k) = vᵢ(k) Mᵢⱼₛₜ tⱼ = vᵀ(k) M⁽ˢᵗ⁾ t. We do this by assuming that each coefficient vector t is sorted into a vector of the form [tᴿ; itᴵ] so that we can take the complex conjugate by as t* = σ₃t, which can then be moved onto M⁽ˢᵗ⁾ instead of t. The constraint Hₛₜ(k) = (H†)ₛₜ(k) = Hₜₛ*(k) can then be expressed as vᵀ(k) M⁽ˢᵗ⁾ tⱼ = v*ᵀ(k) M⁽ᵗˢ⁾ σ₃ t = vᵀ(k) (Pᵀ M⁽ᵗˢ⁾σ₃) t, which requires that t be a solution to the nullspace M⁽ˢᵗ⁾ - Pᵀ M⁽ᵗˢ⁾ σ₃ = 0. We cast this as Z - Q = 0, with Z = M⁽ˢᵗ⁾ and Q = Pᵀ M⁽ᵗˢ⁾ σ₃.

Notes

For anti-Hermitian symmetry, we require Hₛₜ(k) = -Hₜₛ*(k), which translates to M⁽ˢᵗ⁾ + Pᵀ M⁽ᵗˢ⁾ σ₃ = 0; i.e., simply swaps the sign of Q

pin_free(br::NewBandRep{D}, αβγ::AbstractVector{<:Real}) where D

Pin the free parameters of the Wyckoff position associated with the band representation br to the values in αβγ.

Returns a new band representation with all other properties, apart from the Wyckoff position, identical to (and sharing memory with) br.

Note that the associated orbit of the Wyckoff position will be automatically adjusted to ensure that each position in the orbit lies within the primitive unit cell [0,1)ᴰ. That is, if a choice of αβγ sends a position in the orbit outside the primitive unit cell, the position will be adjusted by integer lattice translations to lie within.

primitivized_orbit(br::NewBandRep{D}) where D

Return the orbit of the Wyckoff position associated with the band representation br. The coordinates of positions in the orbit are given relative to the primitive unit cell.

Positions are returned as a Vector{DirectPoint{D}}.

The following checks are made, producing an error if violated:

1. There are no free parameters associated with the Wyckoff position.
2. For every position, its coordinates, referred to the primitive basis, is in the range [0,1); i.e., every position lies in the parallepiped primitive unit cell [0,1)ᴰ.

reciprocal_constraints_matrices(
    Mm::AbstractArray{Int,4}, 
    gens::AbstractVector{SymOperation{D}}, 
    h_orbit::HoppingOrbit{D}
) --> Vector{Array{Int,4}}

Compute the reciprocal constraints matrices for the generators of the SG. This is done by permuting the rows of the M matrix according to the symmetry operation acting on k-space. See more details in permute_symmetry_related_hoppings_under_symmetry_operation and devdocs.md.

representation_constraints_matrices(
    Mm::AbstractArray{Int,4}, 
    brₐ::NewBandRep{D},
    brᵦ::NewBandRep{D}) --> Vector{Array{ComplexF64,4}}

Build the Q matrix for a particular symmetry operation (or, equivalently, a particular matrix from the site-symmetry representation), acting on the M matrix.

(ρₐₐ)ᵣₛ Hₛₜ (ρᵦᵦ⁻¹)ₜₗ = (ρₐₐ)ᵣₛ vᵢ Mᵢⱼₛₜ tⱼ (ρᵦᵦ⁻¹)ₜₗ = vᵢ (ρₐₐ)ᵣₛ Mᵢⱼₛₜ (ρᵦᵦ⁻¹)ₜₗ tⱼ,

then we can define: Qᵢⱼᵣₗ = (ρₐₐ)ᵣₛ Mᵢⱼₛₜ (ρᵦᵦ⁻¹)ₜₗ

sgrep_induced_by_siteir_excl_phase(br::NewBandRep, op::SymOperation)
sgrep_induced_by_siteir_excl_phase(cbr::CompositeBandRep, op::SymOperation)
    --> Matrix{ComplexF64}

Return the representation matrix of a symmetry operation op induced by the site symmetry group of a band representation br or composite band representation cbr, excluding the global momentum-dependent phase factor.

Note

This function assumes Convention 1 for the Fourier transform, so the momentum dependence is introduced as a global phase factor. This is not true if Convention 2 is used. See /docs/src/theory.md for more details.

split_complex(t::Vector{<:Number}) -> Matrix{Real}

Consider αt where α ∈ ℂ and t ∈ ℂⁿ and build from t a matrix representation T that allows access to the real and imaginary parts of the product αt without using complex numbers by splitting α into a real 2-vector of its real and imaginary parts.

In particular, let $α = αᴿ + iαᴵ$ and $t = tᴿ + itᴵ$ with αᴿ, αᴵ ∈ ℝ and $tᴿ, tᴵ ∈ ℝⁿ$, then $αt$ can be rewritten as

\[αt = (αᴿ + iαᴵ)(tᴿ + itᴵ) = (αᴿtᴿ - αᴵtᴵ) + i(αᴿtᴵ + αᴵtᴿ) = [tᴿ, tᴵ]ᵀ [αᴿ, αᴵ] + i [tᴵ, tᴿ]ᵀ [αᴿ, αᴵ]\]

Then, defining T = [tᴿ -tᴵ; tᴵ tᴿ], the above product can then be reexpressed as: $Re(αt) = αᴿtᴿ - αᴵtᴵ =$ (T * [αᴿ; αᴵ])[1:n] and $Im(αt) = αᴿtᴵ + αᴵtᴿ =$ (T * [αᴿ; αᴵ])[n+1:2n]. I.e., the "upper half" of the product T * [real(α), imag(α)] is real(α * t) and the "lower half" is imag(αt).

This functionality is used to avoid complex numbers in amplitude basis coefficients, which simplifies the application of time-reversal symmetry and hermiticity.

Examples

julia> using SymmetricTightBinding: split_complex

julia> t = [im,0]
2-element Vector{Complex{Int64}}:
 0 + 1im
 0 + 0im

julia> T = split_complex(t)
4×2 Matrix{Int64}:
 0  -1
 0   0
 1   0
 0   0

julia> α = 0.5+0.2im; αv = [real(α), imag(α)];

julia> (T * αv)[1:2] == real(α*t) && (T * αv)[3:4] == imag(α*t)

julia> t = [1,im]
2-element Vector{Complex{Int64}}:
 1 + 0im
 0 + 1im

julia> split_complex(t)
4×2 Matrix{Int64}:
 1   0
 0  -1
 0   1
 1   0

zassenhaus_intersection(U::AbstractArray{<:Number}, W::AbstractArray{<:Number}) 
    --> AbstractArray{<:Number}

Finds the intersection of two bases U and W using the Zassenhaus algorithm. It assumes that the basis are given by columns.

References

• https://en.wikipedia.org/wiki/Zassenhaus_algorithm