Lattices

Models can be defined on different lattices, and several lattices lend themselves to a description in terms of a (quasi-)one-dimensional operator. In order to facilitate this mapping, the combination of the @mpoham macro and the lattices in this package provides an easy interface.

MPSKitModels.AbstractLattice — Type

AbstractLattice{N}

Abstract supertype of all lattices, which are mapped to N-dimensional integer grids.

MPSKitModels.FiniteChain — Type

FiniteChain(length::Integer=1)

A one-dimensional lattice of length L

MPSKitModels.FiniteCylinder — Type

FiniteCylinder(L::Int, N::Int)

A cylinder with circumference L and N sites in total.

MPSKitModels.FiniteStrip — Type

FiniteStrip(L::Int, N::Int)

A finite strip with a width of L and a total number of N sites.

This representes an L by N÷L rectangular patch.

MPSKitModels.InfiniteChain — Type

InfiniteChain(L::Integer=1)

A one dimensional infinite lattice with a unit cell containing L sites.

MPSKitModels.InfiniteCylinder — Type

InfiniteCylinder(L::Int, N::Int)

An infinite cylinder with L sites per rung and N sites per unit cell.

MPSKitModels.InfiniteHelix — Type

InfiniteHelix(L::Integer, N::Integer)

An infinite helix with L sites per rung and N sites per unit cell.

MPSKitModels.InfiniteStrip — Type

InfiniteStrip(L::Int, N::Int)

An infinite strip with L sites per rung and N sites per unit cell.

MPSKitModels.HoneycombYC — Type

HoneycombYC(L::Integer, N::Integer=L)

A honeycomb lattice on an infinite cylinder with L sites per rung and N sites per unit cell. The y-axis is aligned along an edge of the hexagons, and the circumference is $3L/4$.

Having defined a lattice, it is possible to iterate over several points or combinations of points that can be of interest. Such a point is represented as a LatticePoint, which is defined in terms of an integer N-dimensional coordinate system representation, and supports addition and subtraction, both with other points or with tuples. These structures also handle the logic of being mapped to a one-dimensional system.

MPSKitModels.LatticePoint — Type

LatticePoint{N,G}

represents an N-dimensional point on a G lattice.

MPSKitModels.linearize_index — Function

linearize_index(lattice, indices...)

convert a given set of indices into a linear index.

MPSKitModels.vertices — Function

vertices(lattice::AbstractLattice)

construct an iterator over all lattice points.

MPSKitModels.nearest_neighbours — Function

nearest_neighbours(lattice::AbstractLattice)

construct an iterator over all pairs of nearest neighbours.

MPSKitModels.next_nearest_neighbours — Function

next_nearest_neighbours(lattice::AbstractLattice)

construct an iterator over all pairs of next-nearest neighbours.

MPSKitModels.bipartition — Function

bipartition(lattice)

construct two iterators over the vertices of the bipartition of a given lattice.

Sometimes it might be useful to change the order of the linear indices of certain lattices. In this case a wrapper around a lattice can be defined through the following:

MPSKitModels.SnakePattern — Type

SnakePattern(lattice, pattern)

Represents a given lattice with a linear order that is provided by pattern.

Any mapping of linear indices can be used, but the following patterns can be helpful:

MPSKitModels.backandforth_pattern — Function

backandforth_pattern(cylinder)

pattern that alternates directions between different rungs of a cylinder

MPSKitModels.frontandback_pattern — Function

frontandback_pattern(cylinder)

pattern that alternates between a site on the first half of a rung and a site on the second half of a rung.