Internals

CarteCoord

Cartesian coordinates. Vector{Float64}.

FractCoord

Fractional coordinates.

Members

• whole ::Vector{Int}: Integer part of fractional coordinates
• fraction ::Vector{Float64}: [0,1) part of fractional coordinates

UnitCell{O}

Parameters

• O: type of "orbital". Any type can be used, but we recommend using String or tuple of String and Int for compatibility with JSON.

Members

• latticevectors ::Array{Float64, 2}: Lattice vectors
• reducedreciprocallatticevectors ::Array{Float64, 2}: Reduced reciprocal lattice vectors (transpose of inverse of latticevectors)
• reciprocallatticevectors ::Array{Float64, 2}: Reciprocal lattice vectors. 2π * reducedreciprocallatticevectors
• orbitals ::Vector{Tuple{T, FractCoord}}: List of orbitals within unit cell
• orbitalindices ::Dict{T, Int}: Indices of orbitals

addorbital!

Add an orbital to the unit cell.

Arguments

• uc ::UnitCell{T}
• orbitalname ::{T}
• orbitalcoord ::FractCoord

carte2fract

Arguments

• latticevectors ::AbstractArray{<:AbstractFloat, 2}: square matrix whose columns are lattice vectors.
• cc ::CarteCoord: cartesian coordinates
• tol ::Real=sqrt(eps(Float64)): tolerance

carte2fract

Arguments

• latticevectors ::Array{Float64, 2}
• cc ::CarteCoord

dimension

Dimension of the fractional coordinates

Arguments

• fc ::FractCoord: Fractional coordinates.

dimension

Spatial dimension of the unit cell.

fract2carte

Arguments

• latticevectors ::AbstractArray{<:AbstractFloat, 2}: square matrix whose columns are lattice vectors.
• fc ::FractCoord: fractional coordinates

fract2carte

Arguments

• latticevectors ::Array{Float64, 2}
• fc ::FractCoord

getorbital

Arguments

• uc ::UnitCell{T}
• index ::Integer

Return

• (orbitalname, fractcoord)

getorbital

Get the orbital (its orbital name and its fractional coordinates) with the given name.

Arguments

• uc ::UnitCell{O}
• name ::O

Return

• (orbitalname, fractcoord)

getorbitalcoord

Arguments

• uc ::UnitCell
• idx ::Integer

Return

• fractcoord

getorbitalcoord

Get the fractional coordinates of the orbital with the given name.

Arguments

• uc ::UnitCell{O}
• name ::O

Return

• fractcoord

getorbitalindex

Get index of the given orbital.

Arguments

• uc ::UnitCell{O}
• name ::O

getorbitalindexcoord

Arguments

• uc ::UnitCell{T}
• name ::T

Return

• (index, fractcoord)

getorbitalname

Arguments

• uc ::UnitCell
• index ::Integer

Return

• orbitalname

hasorbital{T}

Test whether the unit cell contains the orbital of given name.

Arguments

• uc ::UnitCell{O}
• name ::O

momentumpath

Generate a list of momenta

Arguments

• anchorpoints
• (Optional) nseg - number of points in each segment

UnitCell

Construct an n-dimensional lattice.

Arguments

• latticevectors ::AbstractArray{<:AbstractFloat, 2}: Lattice vectors
• OrbitalType::DataType

Optional Arguments

• tol=sqrt(eps(Float64)): Epsilon

UnitCell

Construct a one-dimensional lattice.

Arguments

• latticeconstant ::Float64: Lattice constant
• OrbitalType: List of orbitals

Optional Arguments

• tol=sqrt(eps(Float64)): Tolerance

momentumgrid

Generate an n-dimensional grid of momenta of given shape

momentumpath

The anchorpoints are given in units of the reciprocal lattice vectors.

numorbital

Number of orbitals of the unit cell.

Arguments

• uc ::UnitCell

whichunitcell

Return

• R ::Vector{Int}: which unit cell the specificied orbital/cartesian coordinates belongs to.

whichunitcell

Return

• R ::Vector{Int}: which unit cell the specificied orbital/cartesian coordinates belongs to.

isapprox(x, y; rtol::Real=atol>0 ? 0 : √eps, atol::Real=0, nans::Bool=false, norm::Function)

Submodule `Spec`

Hopping

Interaction

FullHamiltonian

Members

• unitcell ::UnitCell
• hoppings ::Vector{Hopping}
• interactions ::Vector{Interaction}

Hamiltonian

Create an empty Hamiltonian

Arguments

• unitcell ::UnitCell

HoppingDiagonal{R<:Real}

Represents

Members

• amplitude ::R
• i ::Int: index of orbital
• Ri ::Vector{Int}: which unit cell? (indexed by a1, and a2)

HoppingOffdiagonal{C<:Number}

Represents

t, i, j and the unitcell-coordinates Ri and Rj are stored. Require that (i, Ri) <= (j, Rj)

Members

• amplitude ::C
• i, j ::T
• Ri, Rj ::Vector{Int}

InteractionDiagonal{R<:Real}

Represents

Members

• amplitude ::R
• i, j ::Int
• Ri, Rj ::Vector{Int}

InteractionOffdiagonal{C<:Number}

i < j, k < l, i < k or (i == k and j < l)

Represents

Only keep the first term (and require i < j, k < l, i <= k)

Ordering of orbitals by (i, Ri)

Members

• amplitude ::C
• i, j, k, l ::Int
• Ri, Rj, Rk, Rl ::Vector{Int}

addhopping!

Arguments

• hamiltonian ::Hamiltonian
• hopping ::HoppingOffdiagonal

addhopping!

Arguments

• hamiltonian ::Hamiltonian
• hopping ::HoppingDiagonal

addinteraction!

Arguments

• hamiltonian ::Hamiltonian
• interaction ::InteractionOffdiagonal

addinteraction!

Arguments

• hamiltonian ::Hamiltonian
• interaction ::InteractionDiagonal

hoppingbycarte{T}

Make a hopping element with cartesian coordinates.

Arguments

• uc ::UnitCell{T}
• amplitude ::Number
• i ::T
• j ::T
• ri ::CarteCoord
• rj ::CarteCoord
• tol ::Real : Optional. Defaults to sqrt(eps(Float64))

hoppingbycarte{T}

Make a hopping element with cartesian coordinates.

Arguments

• uc ::UnitCell{T}
• amplitude ::Real
• i ::T
• ri ::CarteCoord
• tol ::Real : Optional. Defaults to sqrt(eps(Float64))

interactionbycarte{T}

Make an interaction element with cartesian coordinates.

Arguments

* `uc ::UnitCell{T}`
* `amplitude ::Number`
* `i ::T`
* `j ::T`
* `k ::T`
* `l ::T`
* `ri ::CarteCoord`
* `rj ::CarteCoord`
* `rk ::CarteCoord`
* `rl ::CarteCoord`
* `tol ::Real` : Optional. Defaults to `sqrt(eps(Float64))`

interactionbycarte{T}

Make an interaction element with cartesian coordinates.

Arguments

* `uc ::UnitCell{T}`
* `amplitude ::Number`
* `i ::T`
* `j ::T`
* `ri ::CarteCoord`
* `rj ::CarteCoord`
* `tol ::Real` : Optional. Defaults to `sqrt(eps(Float64))`

islocal

Check if the hopping element is local (i.e. Ri is zero)

localized

Return a hopping element that is local (i.e. Ri is zero)

Generator submodule

hopping_inplace

hopping_inplace

hopping_inplace

hopping_inplace

hopping_inplace

hopping_inplace

hopping_inplace

squarify

Arguments

• uc::Spec.FullHamiltonian{O}

squarify

Arguments

• uc::HFB.HFBHamiltonian{O}

squarify

In order to make the Hamiltoinian a periodic function of momentum, bring all the sites to the origin. In addition, make the unitcell into a square.

Arguments

• uc::Lattice.UnitCell{O}

chernnumber

Compute chern number of the band structure defined by the hoppings and the selected bands.

z2index

Compute Z₂ index of time-reversal-invariant Hamiltonian.

Arguments

• uc::UnitCell{O}
• hops::AbstractVector{Hopping}
• timereversal::AbstractMatrix
• n1 ::Integer
• n2 ::Integer
• selectpairs::AbstractVector{<:Integer}

Optional Arguments

• tol ::Real = sqrt(eps(Float64))

Returns

(The Z2 index, max| Hₖ - T⁻¹HₖT | for k in TRIMs)

HFBConmputer is a type holding the ρ, t and Γ, Δ of a Hartree-Fock-Bogoliubov Hamiltonian.

Add offdiagonal interaction

addinteraction!

Add diagonal interaction

Compute Γ and Δ from ρ and t.

loop

Perform selfconsistency loop a number of times with the given precondition and given update functions.

Arguments

• solver ::HFBSolver{T}
• sol::HFBAmplitude
• run::Integer

Optional Arguments

• update::Function=simpleupdate
• precondition::Function=identity: on target field
• callback::Function=_noop: Function called after every update as

callback(i, run)

loop

Perform selfconsistency loop a number of times with the given precondition and given update functions.

Arguments

• solver ::HFBSolver{T}
• sf::HFBAmplitude
• run::Integer

Optional Arguments

• update::Function=simpleupdate
• precondition::Function=identity
• callback::Function=_noop: Function called after every update as

callback(i, run)

Return a generator of Δ matrix (which is a function of momentum)

Return a generator of Γ matrix (which is a function of momentum)

make_greencollector

Returns a function which has the following signature

collector(k, eigenvalues, eigenvectors, ρout, tout)

func : (idx, i, j, r) -> val

func : (idx, i, j, r) -> 0

Check if hint contains ρ

Compute Γ and Δ from ρ and t.

Compute Γ and Δ from ρ and t.

Return a generator of hopping matrix (which is a function of momentum)

Randomize a hfbamplitude

CollectRow is holds info on how to compute ρ or t. Its elements are:

1. Is diagonal? (only for rho)
2. row orbital
3. col orbital
4. displacement r(col) - r(row)

DeployRow is holds info on how to compute Γ or Δ. Its elements are:

1. Is diagonal
2. row orbital
3. col orbital
4. displacement r(col) - r(row)
5. list of sources, each of which is a tuple of
6. index of ρ or t from which to compute this Γ or Δ.
7. amplitude (coefficient to multiply to ρ or t)
8. boolean indicating whether (1) conjugation is needed (for ρ/Γ) or (2) minus sign is needed (for t/Δ).

linearizedpairingkernel

Compute the kernel Γ of the linearized gap equation in the pairing channel which is written as Δ = Γ⋅Δ