Notation for tensor product space

[kbarros]

Continuation of #63, here are some ideas for notation that generalizes to entangled units:

# The common case avoids notational issues
set_onsite_coupling!(sys, S -> S[3]^2, i)
set_pair_coupling!(sys, (Si, Sj) -> Si'*J*Sj, bond)

# Equivalent to above
S = spin_matrices(3/2)
set_onsite_coupling!(sys, S[3]^2, i)
Si, Sj = to_product_space(S, S)
set_pair_coupling!(sys, Si'*J*Sj, bond)

# Working in a local tensor product space S=3/2 ⊗ S=1 for explicit spin-orbit
System(... [SiteInfo(i; ProductSpace(3/2, 1), magnetic_dipole)] ...)
S, L = to_product_space(spin_matrices.([3/2, 1])...)
set_onsite_coupling!(sys, S'*L, i)
Si, Li, Sj, Lj = to_product_space(spin_matrices.([3/2, 1, 3/2, 1])...)
set_pair_coupling!(sys, Si'*Sj, bond)

# Projecting onto multiplets S=1/2 ⊕ S=3/2
System(... [SiteInfo(i; MultipletSpace(1/2, 3/2), magnetic_dipole)] ...)
S, L = to_product_space(spin_matrices.([3/2, 1])...)

# P is a (2*4)x(4*3) matrix that maps from tensor product space to projected
# multiplet space
P = projector_onto_multiplets(; j1=3/2, j2=1, J=(1/2, 3/2))
set_onsite_coupling!(sys, P'*S'*L*P, i)

# Building interactions between multiplets should project only at the final
# step
Si, Li, Sj, Lj = to_product_space(spin_matrices.([3/2, 1, 3/2, 1])...)
PP = kron(P, P)
set_pair_coupling!(sys, PP'*(Si'*Sj + (Si'*Sj)^2)*PP, bond)

# If the Hilbert space gets too large, it's also possible to project earlier,
# but this requires a lot of care.
T = [P'*S[α]*P for α in 1:3]
Ti, Tj = to_product_space(T, T)
set_pair_coupling!(sys, Ti'*Tj, bond) # This is OK
set_pair_coupling!(sys, (Ti'*Tj)^2, bond) # This is wrong, projection happened too early!

[kbarros]

Significant progress with #178, which now supports anonymous function syntax in set_onsite_coupling! and set_pair_coupling!, and general quadrupole-quadrupole couplings in all modes.

The next big task is to teach System about the SU(2) representation via some generalization of SpinInfo

[kbarros]

For tensor product spaces, the current plan is to create a wrapper on top of System. Implementation of these entangled units is underway in @ddahlbom's branch. Support for a direct sum of multiplet spin representations is not currently planned.