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Non-spatial models (e.g. mean-field models) (#248)
* non-spatial models * more tests * test fix * fix test * typo spacial => spatial * change syntax to --> * cleanups and test fixes * cleanup, tests and docs * more docs * cleanup * doctest fix
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| # Non-spatial models and self-consistent mean-field problems | ||
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| 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 | ||
| 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 | ||
| 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] | ||
| ``` | ||
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| 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)`. | ||
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| 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 `OrbitalSliceArray`s. 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 | ||
| julia> h = LP.linear() |> onsite(2) - hopping(1) |> supercell(4) |> supercell; | ||
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| 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 | ||
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| julia> g = greenfunction(h, GS.Spectrum()); | ||
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| julia> ρ = densitymatrix(g[])(0.5, 0) ## density matrix at chemical potential `µ=0.5` and temperature `kBT = 0` on all sites | ||
| 4×4 OrbitalSliceMatrix{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 | ||
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| julia> hρ = h |> @onsite!((o, i; U = 0, ρ) --> 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] | ||
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| 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 | ||
| ``` | ||
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| 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). | ||
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| The above provides the tools to implement self-consistent mean field. We just need to find a fixed point `ρf(ρ) = ρ` of the function `ρf` that produces the density matrix of the system. | ||
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| In the following example we use the FixedPoint.jl package. It provides a simple utility `afps(f, x0)` that computes the solution of `f(x) = x` with initial condition `x0`. The package requires `x0` to be a (real) `AbstractArray`. Note that any other fixed-point search routine that work with `AbstractArray`s should also work. | ||
| ```julia | ||
| julia> using FixedPoint | ||
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| julia> ρf(hρ; µ = 0.5, kBT = 0, U = 0.1) = ρ -> densitymatrix(greenfunction(hρ(; U, ρ), GS.Spectrum())[])(µ, kBT) | ||
| ρf (generic function with 1 method) | ||
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| julia> (ρsol, ρerror, iters) = @time afps(ρf(hρ; U = 0.4), real(ρ0), tol = 1e-8, ep = 0.95, vel = 0.0); @show iters, ρerror; ρsol | ||
| 0.000627 seconds (1.91 k allocations: 255.664 KiB) | ||
| (iters, ρerror) = (8, 2.0632459629688071e-10) | ||
| 4×4 OrbitalSliceMatrix{Matrix{ComplexF64}}: | ||
| 0.145836+0.0im 0.227266+0.0im 0.227266+0.0im 0.145836+0.0im | ||
| 0.227266+0.0im 0.354164+0.0im 0.354164+0.0im 0.227266+0.0im | ||
| 0.227266+0.0im 0.354164+0.0im 0.354164+0.0im 0.227266+0.0im | ||
| 0.145836+0.0im 0.227266+0.0im 0.227266+0.0im 0.145836+0.0im | ||
| ``` | ||
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| !!! note "Bring your own fixed-point solution!" | ||
| Note that fixed-point calculations can be tricky, and the search algorithm can have a huge impact in convergence (if the problem converges at all!). For this reason, Quantica.jl does not provide built-in fixed-point routines, only the functionality to write functions such as `ρf` above. | ||
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| !!! tip "Sparse mean fields" | ||
| The method explained above to build a Hamiltonian supports all the `SiteSelector` and `HopSelector` functionality of conventional models. Therefore, although the density matrix computed above is dense, its application to the Hamiltonian is sparse: it only touches the onsite matrix elements. Likewise, we could for example use `@hopping!` with a finite `range` to apply a Fock mean field within a finite range. In the future we will support built-in Hartree-Fock model presets with adjustable sparsity. |
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