const basicoptions = (
    gauge = "gauge used to perform the Fourier transformation",
    infinitesimal = "infinitesimal added to the diagonal of the matrix representation of the tight-binding Hamiltonian"
)

Basic options of tight-binding actions.

Metric(::Fermionic, hilbert::Hilbert{<:Fock{:f}}) -> OperatorIndexToTuple
Metric(::Bosonic, hilbert::Hilbert{<:Fock{:b}}) -> OperatorIndexToTuple
Metric(::Phononic, hilbert::Hilbert{<:Phonon}) -> OperatorIndexToTuple

Get the index-to-tuple metric for a free fermionic/bosonic/phononic system.

Table(hilbert::Hilbert{Phonon{:}}, by::OperatorIndexToTuple{(kind, :site, :direction)})

Construct a index-sequence table for a phononic system.

BerryCurvature(reciprocalspace::Union{BrillouinZone, ReciprocalZone}, bands::AbstractVector{Int}, abelian::Bool=true)

Construct a BerryCurvature using the Fukui method.

BerryCurvature{B<:ReciprocalSpace, M<:BerryCurvatureMethod} <: Action

Berry curvature of energy bands.

BerryCurvature(reciprocalspace::ReciprocalSpace, μ::Real, d::Real=0.1, kx::T=nothing, ky::T=nothing) where {T<:Union{Nothing, Vector{Float64}}}

Construct a BerryCurvature using the Kubo method.

BerryCurvatureData{R<:ReciprocalSpace, C<:Array{Float64}, N<:Union{Vector{Float64}, Float64, Nothing}} <: Data

Data of Berry curvature, including:

1. reciprocalspace::R: reciprocal space on which the Berry curvature is computed.
2. values::C: Berry curvature for a certain set of bands.
3. chernnumber::N: if not nothing, Chern number of each energy band when it is a vector or total Chern number of all bands when it is a number.

abstract type BerryCurvatureMethod end

Abstract type for calculation of Berry curvature.

Bosonic{K} <: TBAKind{K}

Bosonic quantum lattice system.

CompositeTBA{
    K<:TBAKind,
    L<:Union{AbstractLattice, Nothing},
    S<:Union{OperatorSet{<:Operator}, Generator{<:OperatorSet{<:Operator}}},
    Q<:Quadraticization,
    H<:Union{OperatorSet{<:Quadratic}, Generator{<:OperatorSet{<:Quadratic}}},
    C<:Union{AbstractMatrix, Nothing}
} <:TBA{K, H, C}

Composite tight-binding approximation for quantum lattice systems.

DensityOfStates(brillouinzone::BrillouinZone, bands::Union{Colon, AbstractVector{Int}}=:, orbitals::Union{Colon, AbstractVector{Int}}...=:)
DensityOfStates(brillouinzone::BrillouinZone, bands::Union{Colon, AbstractVector{Int}}, orbitals::Tuple{Vararg{Union{Colon, AbstractVector{Int}}}})

Construct a DensityOfStates.

DensityOfStates{B<:BrillouinZone, A<:Union{Colon, AbstractVector{Int}}, L<:Tuple{Vararg{Union{Colon, AbstractVector{Int}}}}} <: Action

Density of states of a tight-binding system.

DensityOfStatesData <: Data

Data of density of states, including:

1. energies::Vector{Float64}: energy sample points.
2. values::Matrix{Float64}: density of states projected onto several certain sets of orbitals.

EnergyBands(reciprocalspace::ReciprocalSpace, bands::Union{Colon, AbstractVector{Int}}=:, orbitals::AbstractVector{Int}...)
EnergyBands(reciprocalspace::ReciprocalSpace, bands::Union{Colon, AbstractVector{Int}}, orbitals::Tuple{Vararg{AbstractVector{Int}}})

Construct an EnergyBands.

EnergyBands{L<:Tuple{Vararg{AbstractVector{Int}}}, R<:ReciprocalSpace, A<:Union{Colon, AbstractVector{Int}}} <: Action

Energy bands by tight-binding-approximation for quantum lattice systems.

EnergyBandsData{R<:ReciprocalSpace, W<:Union{Array{Float64, 3}, Nothing}} <: Data

Data of energy bands, including:

1. reciprocalspace::R: reciprocal space on which the energy bands are computed.
2. values::Matrix{Float64}: eigen energies of bands with each column storing the values on the same reciprocal point.
3. weights::W: if not nothing, weights of several certain sets of orbitals projected onto the energy bands.

FermiSurface(reciprocalspace::Union{BrillouinZone, ReciprocalZone}, μ::Real=0.0, bands::Union{Colon, AbstractVector{Int}}=:, orbitals::AbstractVector{Int}...)
FermiSurface(reciprocalspace::Union{BrillouinZone, ReciprocalZone}, μ::Real, bands::Union{Colon, AbstractVector{Int}}, orbitals::Tuple{Vararg{AbstractVector{Int}}})

Construct a FermiSurface.

FermiSurface{L<:Tuple{Vararg{AbstractVector{Int}}}, B<:Union{BrillouinZone, ReciprocalZone}, A<:Union{Colon, AbstractVector{Int}}} <: Action

Fermi surface of a free fermionic system.

FermiSurfaceData{S<:ReciprocalScatter} <: Data

Data of Fermi surface, including:

1. values::S: points of the Fermi surface.
2. weights::Matrix{Float64}: weights of several certain sets of orbitals projected onto the Fermi surface.

Fermionic{K} <: TBAKind{K}

Fermionic quantum lattice system.

Fukui <: BerryCurvatureMethod

Fukui method to calculate Berry curvature of energy bands. see Fukui et al, JPSJ 74, 1674 (2005). Hall conductivity (single band) is given by

\[\sigma_{xy} = -{e^2\over h}\sum_n c_n, \nonumber \\ c_n = {1\over 2\pi}\int_{BZ}{dk_x dk_y Ω_{xy}}, \Omega_{xy}=(\nabla\times {\bm A})_z, A_{x}=i\langle u_n|\partial_x|u_n\rangle.\]

InelasticNeutronScatteringSpectra{R<:ReciprocalSpace} <: Action

Inelastic neutron scattering spectra.

InelasticNeutronScatteringSpectraData{R<:ReciprocalSpace} <: Data

Data of inelastic neutron scattering spectra, including:

1. reciprocalspace::R: reciprocal space on which the inelastic neutron scattering spectra are computed
2. energies::Vector{Float64}: energy sample points.
3. values::Matrix{Float64}: rescaled intensity of inelastic neutron scattering spectra.

Kubo{K<:Union{Nothing, Vector{Float64}}} <: BerryCurvatureMethod

Kubo method to calculate the total Berry curvature of occupied energy bands. The Kubo formula is given by

\[\Omega_{ij}(\bm k)=\epsilon_{ijl}\sum_{n}f(\epsilon_n(\bm k))b_n^l(\bm k)=-2{\rm Im}\sum_v\sum_c{V_{vc,i}(\bm k)V_{cv,j}(\bm k)\over [\omega_c(\bm k)-\omega_v(\bm k)]^2},\]

where

\[ V_{cv,j}={\langle u_{c\bm k}|{\partial H\over \partial {\bm k}_j}|u_{v\bm k}\rangle}\]

v and c subscripts denote valence (occupied) and conduction (unoccupied) bands, respectively. Hall conductivity in 2D space is given by

\[\sigma_{xy}=-{e^2\over h}\int_{BZ}{dk_x dk_y\over 2\pi}{\Omega_{xy}}\]

Phononic <: TBAKind{:BdG}

Phononic quantum lattice system.

Quadratic{V<:Number, C<:AbstractVector} <: OperatorPack{V, Tuple{Tuple{Int, Int}, C, C}}

The unified quadratic form in tight-binding approximation.

Quadraticization{K<:TBAKind, T<:Table} <: LinearTransformation

The linear transformation that converts a rank-2 operator to its unified quadratic form.

SimpleTBA{
    K<:TBAKind,
    L<:Union{AbstractLattice, Nothing},
    H<:Union{Formula, OperatorSet{<:Quadratic}, Generator{<:OperatorSet{<:Quadratic}}},
    C<:Union{AbstractMatrix, Nothing}
} <:TBA{K, H, C}

Simple tight-binding approximation for quantum lattice systems.

TBA{K<:TBAKind, H<:Union{Formula, OperatorSet, Generator, Frontend}, C<:Union{Nothing, AbstractMatrix}} <: Frontend

Abstract type for free quantum lattice systems using the tight-binding approximation.

TBA{K}(H::Union{Formula, OperatorSet{<:Quadratic}, Generator{<:OperatorSet{<:Quadratic}}}, commutator::Union{AbstractMatrix, Nothing}=nothing) where {K<:TBAKind}
TBA{K}(lattice::Union{AbstractLattice, Nothing}, H::Union{Formula, OperatorSet{<:Quadratic}, Generator{<:OperatorSet{<:Quadratic}}}, commutator::Union{AbstractMatrix, Nothing}=nothing) where {K<:TBAKind}

TBA{K}(H::Union{OperatorSet{<:Operator}, Generator{<:OperatorSet{<:Operator}}}, q::Quadraticization, commutator::Union{AbstractMatrix, Nothing}=nothing) where {K<:TBAKind}
TBA{K}(lattice::Union{AbstractLattice, Nothing}, H::Union{OperatorSet{<:Operator}, Generator{<:OperatorSet{<:Operator}}}, q::Quadraticization, commutator::Union{AbstractMatrix, Nothing}=nothing) where {K<:TBAKind}

TBA(lattice::AbstractLattice, hilbert::Hilbert, terms::OneOrMore{Term}, boundary::Boundary=plain; neighbors::Union{Int, Neighbors}=nneighbor(terms))
TBA{K}(lattice::AbstractLattice, hilbert::Hilbert, terms::OneOrMore{Term}, boundary::Boundary=plain; neighbors::Union{Int, Neighbors}=nneighbor(terms)) where {K<:TBAKind}

Construct a tight-binding quantum lattice system.

TBAKind{K}

The kind of a free quantum lattice system using the tight-binding approximation.

TBAKind(T::Type{<:Term}, I::Type{<:Internal})

Depending on the kind of a Term type and an Internal type, get the corresponding TBA kind.

TBAMatrix{C<:Union{AbstractMatrix, Nothing}, T, H<:Hermitian{T, <:AbstractMatrix{T}}} <: AbstractMatrix{T}

Matrix representation of a free quantum lattice system using the tight-binding approximation.

TBAMatrixization{D<:Number, V<:Union{AbstractVector{<:Number}, Nothing}} <: Matrixization

Matrixization of the Hamiltonian of a tight-binding system.

eigen(tba::Union{TBA, Algorithm{<:TBA}}, k::Union{AbstractVector{<:Number}, Nothing}=nothing; options...) -> Eigen
eigen(tba::Union{TBA, Algorithm{<:TBA}}, reciprocalspace::ReciprocalSpace; options...) -> Tuple{Vector{<:Vector{<:Number}}, Vector{<:Matrix{<:Number}}}

Get the eigen values and eigen vectors of a free quantum lattice system.

eigen(m::TBAMatrix) -> Eigen

Solve the eigen problem of a free quantum lattice system.

eigvals(tba::Union{TBA, Algorithm{<:TBA}}, k::Union{AbstractVector{<:Number}, Nothing}=nothing; options...) -> Vector{<:Number}
eigvals(tba::Union{TBA, Algorithm{<:TBA}}, reciprocalspace::ReciprocalSpace; options...) -> Vector{<:Vector{<:Number}}

Get the eigen values of a free quantum lattice system.

eigvals(m::TBAMatrix) -> Vector

Get the eigen values of a free quantum lattice system.

eigvecs(tba::Union{TBA, Algorithm{<:TBA}}, k::Union{AbstractVector{<:Number}, Nothing}=nothing; options...) -> Matrix{<:Number}
eigvecs(tba::Union{TBA, Algorithm{<:TBA}}, reciprocalspace::ReciprocalSpace; options...) -> Vector{<:Matrix{<:Number}}

Get the eigen vectors of a free quantum lattice system.

eigvecs(m::TBAMatrix) -> Matrix

Get the eigen vectors of a free quantum lattice system.

matrix(tba::Union{TBA, Algorithm{<:TBA}}, k::Union{AbstractVector{<:Number}, Nothing}=nothing; gauge=:icoordinate, infinitesimal=infinitesimal(kind(tba))) -> TBAMatrix

Get the matrix representation of a free quantum lattice system.

scalartype(::Type{<:TBA{<:TBAKind, H}}) where {H<:Union{Formula, OperatorSet, Generator, Frontend}}
scalartype(::Type{<:Algorithm{F}}) where {F<:TBA}

Get the scalartype of a tight-binding system.

add!(dest::AbstractMatrix, mr::TBAMatrixization, m::Quadratic; infinitesimal=0, kwargs...) -> typeof(dest)

Matrixize a quadratic form and add it to dest.

add!(dest::OperatorSum, q::Quadraticization{<:TBAKind{:TBA}}, m::Operator{<:Number, <:NTuple{2, CoordinatedIndex{<:Index}}}; kwargs...) -> typeof(dest)
add!(dest::OperatorSum, q::Quadraticization{<:TBAKind{:BdG}}, m::Operator{<:Number, <:NTuple{2, CoordinatedIndex{<:Index{<:FockIndex}}}}; kwargs...) -> typeof(dest)
add!(dest::OperatorSum, q::Quadraticization{<:TBAKind{:BdG}}, m::Operator{<:Number, <:NTuple{2, CoordinatedIndex{<:Index{<:PhononIndex}}}}; kwargs...) -> typeof(dest)

Get the unified quadratic form of a rank-2 operator and add it to dest.

dimension(tba::Union{TBA, Algorithm{<:TBA}}) -> Int

Get the dimension of the matrix representation of a free quantum lattice system.

kind(tba::Union{TBA, Algorithm{<:TBA}}) -> TBAKind
kind(::Type{<:TBA{K}}) where K -> TBAKind
kind(::Type{<:Algorithm{F}}) where {F<:TBA} -> TBAKind

Get the kind of a tight-binding system.

berrycurvature(vectors::Matrix{<:Matrix{<:Number}}, commutator::Union{Nothing, AbstractMatrix{<:Number}}, dS::Number, ::Val{true}) -> Array{Float64, 3}
berrycurvature(vectors::Matrix{<:Matrix{<:Number}}, commutator::Union{Nothing, AbstractMatrix{<:Number}}, dS::Number, ::Val{false}) -> Matrix{Float64}

Based on the eigen vectors over a discrete reciprocal space, get the 1) Berry curvature for individual bands and 2) total Berry curvature by the Fukui method.

Here, commutator stands for the commutation relation matrix of the basis operators, dS stands for the area element of the discrete reciprocal space.

berrycurvature(tba::TBA, bc::BerryCurvature{<:ReciprocalSpace, <:Kubo}; options...) -> Matrix{Float64}

Get the Berry curvature by the Kubo method.

berrycurvature(tba::TBA, bc::BerryCurvature{<:Union{BrillouinZone, ReciprocalZone}, <:Fukui{true}}; options...) -> Array{Float64, 3}
berrycurvature(tba::TBA, bc::BerryCurvature{<:Union{BrillouinZone, ReciprocalZone}, <:Fukui{false}}; options...) -> Matrix{Float64}

Get the Berry curvature by the Fukui method.

commutator(k::TBAKind, hilbert::Hilbert{<:Internal}) -> Union{AbstractMatrix, Nothing}

Get the commutation relation of the single-particle operators of a free quantum lattice system using the tight-binding approximation.

infinitesimal(::TBAKind{:TBA}) -> 0
infinitesimal(::TBAKind{:BdG}) -> atol/5
infinitesimal(::Fermionic{:BdG}) -> 0

Infinitesimal used in the matrixization of a tight-binding approximation to be able to capture the Goldstone mode.

SampleNode(reciprocals::AbstractVector{<:AbstractVector{<:Number}}, position::Vector{<:Number}, bands::AbstractVector{Int}, values::Vector{<:Number}, ratio::Number)
SampleNode(reciprocals::AbstractVector{<:AbstractVector{<:Number}}, position::Vector{<:Number}, bands::AbstractVector{Int}, values::Vector{<:Number}, ratios::Vector{<:Number}=ones(length(bands)))

A sample node of a momentum-eigenvalues pair.

deviation(tba::Union{TBA, Algorithm{<:TBA}}, samplenode::SampleNode) -> Float64
deviation(tba::Union{TBA, Algorithm{<:TBA}}, samplesets::Vector{SampleNode}) -> Float64

Get the deviation of the eigenvalues between the sample points and model points.

optimize!(
    tba::Union{TBA, Algorithm{<:TBA}}, samplesets::Vector{SampleNode}, variables=keys(Parameters(tba));
    verbose=false, method=LBFGS(), options=Options()
) -> Tuple{typeof(tba), Optim.MultivariateOptimizationResults}

Optimize the parameters of a tight binding system whose names are specified by variables so that the total deviations of the eigenvalues between the model points and sample points minimize.