This section contains the documentation for the internal structure of LatticeModels.jl.

Advanced features

These features can be useful in non-trivial cases, but are not necessary for basic usage.

AbstractLattice interface

The base of LatticeModels.jl is its interfaces, allowing to define lattices with arbitrary geometry, topology and boundary conditions. The LatticeModels.AbstractLattice interface is the main interface for defining lattices.

Generally speaking, a lattice is a set of sites. Each site, in turn, has its spatial coordinates in its coords field and maybe some additional properties. It also must be a subtype of LatticeModels.AbstractSite.

Note that the bonds between sites and the boundary conditions are initially not part of the lattice, but are added to its metadata later.

Basic functions

Site lookup

Mutable lattices

Site properties

Lattice metadata

Shapes

AbstractBonds interface

The LatticeModels.AbstractBonds interface is used to define different types of bonds between sites. Most generally speaking, such object is a mapping that decides if the sites are connected for each pair.

Note that there are three basic types of bonds in LatticeModels.jl:

• LatticeModels.AbstractBonds: a most general interface. Basically, it is just a mapping from site pairs to boolean values.
• LatticeModels.DirectedBonds: this type of bonds defines a set of bonds that has a defined direction. The whole topology can be defined by the "destination" sites for each site. Since the bonds are usually sparse, the general performance of this type of bonds is much higher.
• LatticeModels.AbstractTranslation: this is a subtype of DirectedBonds, where every site has one or zero "destination" sites. This allows to increase the performance even more, and also to transform the sites in a convenient manner:

site1 = lat[!, x = 1, y = 1]    # Get the site at [1, 1]
T = Translation(lat, [1, 0])    # Translate the site by [1, 0] vector
site2 = site1 + T               # `site2` is at [2, 1]

Adapting bonds to the lattice

Boundary conditions

Diagonalizing the Hamiltonian

It is very easy to diagonalize a matrix in Julia. However, problems can arise when the matrix is of some custom type (e. g. sparse or a GPU array). By default LatticeModels.jl makes use of KrylovKit.jl to solve the eigenproblem using the Lanczos algorithm for non-trivial matrix types. However, sometimes it is necessary to use a different algorithm. The LatticeModels.diagonalize_routine is a simple way to add a new algorithm to the default toolchain.

EvolutionSolvers

The LatticeModels.EvolutionSolver interface is used to solve the time-dependent Schrödinger equation. It is used in the Evolution struct to perform unitary evolution. As with the diagonalization problem, one can add a new algorithm to the default toolchain by creating a new EvolutionSolver type.

Currents

The LatticeModels.AbstractCurrents interface allows to define different types of currents on the lattice. This allows it to be a lazy object, which computes the currents only when needed.

To implement basic currents semantics, you need to define the following methods:

• LatticeModels.lattice(your_currents): returns the lattice, on which the currents are defined.
• Base.getindex(your_currents, i::Int, j::Int): returns the current between sites with numbers i and j. This is done in such a manner, because you do not usually need the site properties to calculate the currents.