Parametric bandstructures with cuts

[pablosanjose]

Closes #28

This implements the bandstructure(::ParametricHamiltonian, mesh) and the bandstructure(::Union{Hamiltonian,ParametricHamiltonian}, mesh; cut =...) functionality.

This new feature is quite powerful. It allows to treat parameters in a ParametricHamiltonian as equivalent to Bloch phases when computing bandstructure(ph::ParametricHamiltonian, mesh). The mesh should be a discretization of the parameter space ⊗ Brillouin zone for ph. Each vertex v of mesh is interpreted as v = (p₁,..., pᵢ, ϕ₁,..., ϕⱼ), with parameter values p coming before Bloch phases ϕ.

A possible application: obtain the bandstructure of a planar Josephson junction as a function of parallel momentum and superconductor phase difference. In general, this is useful when needing to compute the bandstructure of a system (or any associated observable) as a function of external parameters.

When providing a cut of the form cut = (mesh_coordinates...) -> v, with mesh_coordinates a Tuple of coordinates of mesh and v the corresponding Bloch phases in the full Brillouin zone v = (ϕ₁,..., ϕⱼ) (or v = (p₁,..., pᵢ, ϕ₁,..., ϕⱼ) in parameter space ⊗ Brillouin zone for ParametricHamiltonians), a section of the bandstructure in the target space of cut is constructed.

The machinery of degeneracy resolution and band reconnection has been generalized to work with cuts and parametric spaces, resulting in clean reconnected bands even when mixing Bloch phases and parameters in an arbitrary way.

There is a rather small performance penalty for using cuts or parameter sampling. Essentially, the user should not have to worry about performance when using this feature. Example:

julia> h = LatticePresets.square() |> hamiltonian(hopping(1)+onsite(0)) |> unitcell(2);

julia> ph = parametric(h, @onsite!((o; μ) -> o - μ));

julia> mesh1D = marchingmesh(0:0.1:1);

julia> cut12 = k -> (k, k)

julia> cut13 = k -> (k, k, k);

julia> @btime bandstructure(h, mesh1D, cut = cut12);
  144.948 μs (876 allocations: 130.44 KiB)

julia> @btime bandstructure(ph, mesh1D, cut = cut13);
  184.907 μs (1161 allocations: 141.66 KiB)

[pablosanjose]

A note on implementation: this introduced a new bloch!(matrix, ph::ParametricHamiltonian, pϕs, [axis]) where pϕs = (p₁,..., pᵢ, ϕ₁,..., ϕⱼ), which injects parameters p into ph, producing h=ph(pnames = p...) (where pnames = parameters(ph)), and then calls bloch!(matrix, ph(pnames = ps...), ϕs, [axis]). The bloch! generalization also applies to bloch.