Model

struct Model{T<:Real}

A high-level data structure containing necessary parameters and matrices for constructing Wannier functions (WFs), or called, Wannierization.

In general, the problem of Wannierization is to find a set of unitary matrices $U_{\bm{k}}$ that gives a localized representation of the Bloch states $\psi_{n \bm{k}}$. Depending on the inputs, the Wannierization problem can be categorized into two classes:

• isolated manifold: when number of bands = number of Wannier functions
• entangled manifold: when number of bands > number of Wannier functions

Fields

Using these accronyms, - n_atoms: number of atoms - n_kpoints: number of kpoints - n_bvectors: number of b-vectors - n_bands: number of bands - n_wannier: number of Wannier functions the fields are defined as follows:

• lattice: unit cell, 3 * 3, each column is a lattice vector in Å unit

• atom_positions: atomic positions, length-n_atoms vector, each element is a Vec3 of fractional coordinates

• atom_labels: atomic labels, length-n_atoms vector of string

• kstencil: stencil for finite differences on the kpoint grid, also called $\mathbf{b}$-vectors. Should satisfy completeness condition, see KspaceStencil

• overlaps: overlap matrices between neighboring wavefunctions, $M_{\bm{k},\bm{b}}$. Length-n_kpoints vector, each element is a length-n_bvectors vector, then each element is a n_bands * n_bands matrix. Also called mmn matrices in wannier90

• gauges: unitary or semi-unitary gauge transformation matrices, $U_{\bm{k}}$, or called the gauge matrices. Length-n_kpoints vector, each element is a n_bands * n_wannier matrix

• eigenvalues: energy eigenvalues, $\varepsilon_{n \bm{k}}$, length-n_kpoints vector, each element is a length-n_bands vector, in eV unit

• frozen_bands: mask for frozen bands. Length-n_kpoints vector, each element is a length-n_bands BitVector. If true the the state at that kpoint and band index is kept unchanged during the disentanglement procedure.

• entangled_bands: mask for bands taking part in disentanglement. Length-n_kpoints vector, each element is a length-n_bands BitVector. If true the the state at that kpoint and band index participates the disentanglement procedure.

isapprox(a, b; kwargs...)

Compare two Model objects.

isentangled(model)

Is entangled manifold?

isisolated(model)

Is isolated manifold?

Penalty for minimizing the spread as well as maximizing the "closeness" to the atoms.

struct Spread

The Marzari-Vanderbilt (MV) spread functional.

From MV:

• $\Omega = \sum_n \langle r^2 \rangle_n - | \langle r \rangle_n |^2$
• $\langle r \rangle_n = -\frac{1}{N} \sum_{\bm{k},\bm{b}} w_{\bm{b}} \bm{b} \Im \log M_{nn}^{\bm{k},\bm{b}}$
• $\langle r^2 \rangle_n = \frac{1}{N} \sum_{\bm{k},\bm{b}} w_{\bm{b}} \bm{b} \left[ \left( 1 - | M_{nn}^{\bm{k},\bm{b}} |^2 \right) + \left( \Im \log M_{nn}^{\bm{k},\bm{b}} \right)^2 \right]$

Fields

• Ω: total spread, unit Ų
• ΩI: gauge-invarient part, unit Ų
• ΩOD: off-diagonal part, unit Ų
• ΩD: diagonal part, unit Ų
• Ω̃: Ω̃ = ΩOD + ΩD, unit Ų
• ω: Ω of each WF, unit Ų, length(ω) = n_wann
• r: WF center, Cartesian coordinates, unit Å, 3 * n_wann

struct SpreadCenter

A struct containing both Spread and WF center penalty.

Standard penalty for minimizing the total spread.

center(bvectors, M, U)

Compute WF center in reciprocal space.

Arguments

• bvectors: bvecoters
• M: n_bands * n_bands * * n_bvecs * n_kpts overlap array
• U: n_wann * n_wann * n_kpts array

center(model)

Compute WF center in reciprocal space for Model.

center(bvectors, M, U)

Compute WF center in reciprocal space.

Arguments

• bvectors: bvecoters
• M: n_bands * n_bands * * n_bvecs * n_kpts overlap array
• U: n_wann * n_wann * n_kpts array

center(model, U)

Compute WF center in reciprocal space for Model with given U gauge.

Arguments

• model: the Model
• U: n_wann * n_wann * n_kpts array

compute_berry_connection_kspace(
    kstencil,
    overlaps,
    gauges;
    imlog_diag,
    force_hermiticity
)

Wannier-gauge Berry connection in kspace, WYSV Eq. 44 or MV Eq. C14

Keyword arguments

• imlog_diag: use imaginary part of logrithm for diagonal elements, MV1997 Eq. 31. wannier90 default is true.

omega(model, [U])
omega(bvectors, M, U)

Compute WF spread for a Model, potentially for a given gauge U, or by explicitely giving bvectors and M. In case of the first bvectors = model.bvectors and M = model.M.

omega_center(bvectors, M, U, r₀, λ)

Compute WF spread with center penalty, for maximal localization.

Arguments

• bvectors: bvecoters
• M: n_bands * n_bands * * n_bvecs * n_kpts overlap array
• U: n_wann * n_wann * n_kpts array
• r₀: 3 * n_wann, WF centers in cartesian coordinates
• λ: penalty strength

omega_grad(bvectors, M, U, r)

Compute gradient of WF spread.

Size of output dΩ/dU = n_bands * n_wann * n_kpts.

Arguments

• bvectors: bvecoters
• M: n_bands * n_bands * * n_bvecs * n_kpts overlap array
• U: n_wann * n_wann * n_kpts array
• r: 3 * n_wann, the current WF centers in cartesian coordinates

omega_local(bvectors, M, U)

Local part of the contribution to r^2.

Arguments

• bvectors: bvecoters
• M: n_bands * n_bands * * n_bvecs * n_kpts overlap array
• U: n_wann * n_wann * n_kpts array

position_op(model)

Compute WF postion operator matrix in reciprocal space for Model.

position_op(bvectors, M, U)

Compute WF postion operator matrix in reciprocal space.

Arguments

• bvectors: bvecoters
• M: n_bands * n_bands * * n_bvecs * n_kpts overlap array
• U: n_wann * n_wann * n_kpts array

position_op(model, U)

Compute WF postion operator matrix in reciprocal space for Model with given U gauge.

Arguments

• model: the Model
• U: n_wann * n_wann * n_kpts array