Glossary

This is a summary of the type of objects you will be studying.

  • Sublat: a sublattice, representing a number of identical sites within the unit cell of a bounded or unbounded lattice. Each site has a position in an E-dimensional space (E is called the embedding dimension). All sites in a given Sublat will be able to hold the same number of orbitals, and they can be thought of as identical atoms. Each Sublat in a Lattice can be given a unique name, by default :A, :B, etc.

  • Lattice: a collection of Sublats plus a collection of L Bravais vectors that define the periodicity of the lattice. A bounded lattice has L=0, and no Bravais vectors. A Lattice with L > 0 can be understood as a periodic (unbounded) collection of unit cells, each containing a set of sites, each of which belongs to a different sublattice.

  • SiteSelector: a rule that defines a subset of sites in a Lattice (not necessarily restricted to a single unit cell)

  • HopSelector: a rule that defines a subset of site pairs in a Lattice (not necessarily restricted to the same unit cell)

  • LatticeSlice: a finite subset of sites in a Lattice, defined by their cell index (an L-dimensional integer vector, usually denoted by n or cell) and their site index within the unit cell (an integer). A LatticeSlice an be constructed by combining a Lattice and a (bounded) SiteSelector.

  • AbstractModel: either a TightbindingModel or a ParametricModel

    • TightbindingModel: a set of HoppingTerms and OnsiteTerms
      • OnsiteTerm: a rule that, applied to a single site, produces a scalar or a (square) matrix that represents the intra-site Hamiltonian elements (single or multi-orbital)
      • HoppingTerm: a rule that, applied to a pair of sites, produces a scalar or a matrix that represents the inter-site Hamiltonian elements (single or multi-orbital)
    • ParametricModel: a set of ParametricOnsiteTerms and ParametricHoppingTerms
      • ParametricOnsiteTerm: an OnsiteTerm that depends on a set of free parameters that can be adjusted, and that may or may not have a default value
      • ParametricHoppingTerm: a HoppingTerm that depends on parameters, like ParametricOnsiteTerm above

  • AbstractHamiltonian: either a Hamiltonian or a ParametricHamiltonian

    • Hamiltonian: a Lattice combined with a TightBindingModel.

      It also includes a specification of the number of orbitals in each Sublat in the Lattice. A Hamiltonian represents a tight-binding Hamiltonian sharing the same periodicity as the Lattice (it is translationally invariant under Bravais vector shifts).

    • ParametricHamiltonian: like the above, but using a ParametricModel, which makes it dependent on a set of free parameters that can be efficiently adjusted.

    An h::AbstractHamiltonian can be used to produce a Bloch matrix h(ϕ; params...) of the same size as the number of orbitals per unit cell, where ϕ = [ϕᵢ...] are Bloch phases and params are values for the free parameters, if any.

  • Spectrum: the set of eigenpairs (eigenvalues and corresponding eigenvectors) of a Bloch matrix. It can be computed with a number of EigenSolvers.

  • Bandstructure: a collection of spectra, evaluated over a discrete mesh (typically a discretization of the Brillouin zone), that is connected to its mesh neighbors into a linearly-interpolated approximation of the AbstractHamiltonian's bandstructure.

  • SelfEnergy: an operator Σ(ω) defined to act on a LatticeSlice of an AbstractHamiltonian that depends on energy ω.

  • OpenHamiltonian: an AbstractHamiltonian combined with a set of SelfEnergies

  • GreenFunction: an OpenHamiltonian combined with an AbstractGreenSolver, which is an algorithm that can in general compute the retarded or advanced Green function at any energy between any subset of sites of the underlying lattice.

    • GreenSlice: a GreenFunction evaluated on a specific set of sites, but at an unspecified energy
    • GreenSolution: a GreenFunction evaluated at a specific energy, but on an unspecified set of sites

  • OrbitalSliceArray: an AbstractArray that can be indexed with a SiteSelector, in addition to the usual scalar indexing. Particular cases are OrbitalSliceMatrix and OrbitalSliceVector. This is the most common type obtained from GreenFunctions and observables obtained from them.

  • Observables: Supported observables, obtained from Green functions using various algorithms, include local density of states, density matrices, current densities, transmission probabilities, conductance and Josephson currents