Non-spatial models

As briefly mentioned when discussing parametric models and modifiers, we have a special syntax that allows models to depend on sites directly, instead of on their spatial location. We call these non-spatial models. A simple example, with an onsite energy proportional to the site index

julia> model = @onsite((i; p = 1) --> ind(i) * p)
ParametricModel: model with 1 term
  ParametricOnsiteTerm{ParametricFunction{1}}
    Region            : any
    Sublattices       : any
    Cells             : any
    Coefficient       : 1
    Argument type     : non-spatial
    Parameters        : [:p]

or a modifier that changes a hopping between different cells

julia> modifier = @hopping!((t, i, j; dir = 1) --> (cell(i) - cell(j))[dir])
HoppingModifier{ParametricFunction{3}}:
  Region            : any
  Sublattice pairs  : any
  Cell distances    : any
  Hopping range     : Inf
  Reverse hops      : false
  Argument type     : non-spatial
  Parameters        : [:dir]

Note that we use the special syntax --> instead of ->. This indicates that the positional arguments of the function, here called i and j, are no longer site positions as up to now. These i, j are non-spatial arguments, as noted by the Argument type property shown above. Instead of a position, a non-spatial argument i represent an individual site, whose index is ind(i), its position is pos(i) and the cell it occupies on the lattice is cell(i).

Technically i is of type CellSitePos, which is an internal type not meant for the end user to instantiate. One special property of this type, however, is that it can efficiently index OrbitalSliceArrays. We can use this to build a Hamiltonian that depends on an observable, such as a densitymatrix. A simple example of a four-site chain with onsite energies shifted by a potential proportional to the local density on each site:

julia> h = LP.linear() |> onsite(2) - hopping(1) |> supercell(4) |> supercell;

julia> h(SA[])
4×4 SparseArrays.SparseMatrixCSC{ComplexF64, Int64} with 10 stored entries:
  2.0+0.0im  -1.0+0.0im       ⋅           ⋅
 -1.0+0.0im   2.0+0.0im  -1.0+0.0im       ⋅
      ⋅      -1.0+0.0im   2.0+0.0im  -1.0+0.0im
      ⋅           ⋅      -1.0+0.0im   2.0+0.0im

julia> g = greenfunction(h, GS.Spectrum());

julia> ρ0 = densitymatrix(g[])(0.5, 0) ## density matrix at chemical potential `µ=0.5` and temperature `kBT = 0`  on all sites
4×4 OrbitalSliceMatrix{ComplexF64,Matrix{ComplexF64}}:
 0.138197+0.0im  0.223607+0.0im  0.223607+0.0im  0.138197+0.0im
 0.223607+0.0im  0.361803+0.0im  0.361803+0.0im  0.223607+0.0im
 0.223607+0.0im  0.361803+0.0im  0.361803+0.0im  0.223607+0.0im
 0.138197+0.0im  0.223607+0.0im  0.223607+0.0im  0.138197+0.0im

julia> hρ = h |> @onsite!((o, i; U = 0, ρ = ρ0) --> o + U * ρ[i])
ParametricHamiltonian{Float64,1,0}: Parametric Hamiltonian on a 0D Lattice in 1D space
  Bloch harmonics  : 1
  Harmonic size    : 4 × 4
  Orbitals         : [1]
  Element type     : scalar (ComplexF64)
  Onsites          : 4
  Hoppings         : 6
  Coordination     : 1.5
  Parameters       : [:U, :ρ]

julia> hρ(SA[]; U = 1, ρ = ρ0)
4×4 SparseArrays.SparseMatrixCSC{ComplexF64, Int64} with 10 stored entries:
 2.1382+0.0im    -1.0+0.0im         ⋅             ⋅
   -1.0+0.0im  2.3618+0.0im    -1.0+0.0im         ⋅
        ⋅        -1.0+0.0im  2.3618+0.0im    -1.0+0.0im
        ⋅             ⋅        -1.0+0.0im  2.1382+0.0im

Note the ρ[i] above. This indexes ρ at site i. For a multiorbital hamiltonian, this will be a matrix (the local density matrix on each site i). Here it is just a number, either 0.138197 (sites 1 and 4) or 0.361803 (sites 2 and 3).