Conserved quantum numbers

The basis states of a Hilbert space can be organized into sectors with different quantum numbers. Only quantum numbers which are diagonal in the fock state basis are supported.

Use hilbert_space(labels, qn), where qn can be e.g.

ParityConservation(): Puts all states with odd parity first, then all states with even parity.
ParityConservation(p::Int): Only contains states with parity p (-1 for odd and 1 for even).
number_conservation(): Sorts basis states by the number of fermions.
number_conservation(sectors::Union{Int,Vector{Int}}): Only contains states with the number(s) in the list sectors.
number_conservation(sectors, weight_function): 'weight_function' is a function that takes a label and returns an integer weight that represents the contribution of that fermion to the total number. The returned states will have total weighted number of fermions contained in sectors.
number_conservation(weight_function): As above, but allowing all sectors.
Products of the above quantum numbers, which sorts states according to each factor in the product.

Using number_conservation with small sectors avoids the exponentially large Hilbert space.

Spin

This package does not know anything about spin, but one can treat spin just as an extra label as follows:

using FermionicHilbertSpaces
labels = Base.product(1:4, (:↑,:↓))
H = hilbert_space(labels)

256-dimensional SimpleFockHilbertSpace:
8 fermions: [(1, :↑), (2, :↑), (3, :↑), (4, :↑), (1, :↓), (2, :↓), (3, :↓), (4, :↓)]

If spin is conserved, one can use

H = hilbert_space(labels, number_conservation(label -> :↑ in label) * number_conservation(label -> :↓ in label))

256-dimensional SymmetricFockHilbertSpace:
8 fermions: [(1, :↑), (2, :↑), (3, :↑), (4, :↑), (1, :↓), (2, :↓), (3, :↓), (4, :↓)]
FockSymmetry(Number conservation for 2 subsets)

to sort states according to the number of fermions with spin up and down. However, this package can't help to sort states into sectors with different total angular momentum, because that requires taking superpositions of different fock states.

To pick out the sector with 2 fermions with spin up and 0 fermions with spin down, one can extract it from the hilbert space defined above using sector, or construct it directly

FermionicHilbertSpaces.sector((2,0), H)
hilbert_space(labels, number_conservation(2, label -> :↑ in label) * number_conservation(0, label -> :↓ in label))

6-dimensional SymmetricFockHilbertSpace:
8 fermions: [(1, :↑), (2, :↑), (3, :↑), (4, :↑), (1, :↓), (2, :↓), (3, :↓), (4, :↓)]
FockSymmetry(Number conservation for 2 subsets)

No double occupation

The following example demonstrates how to restrict the Hilbert space to only allow single occupation of a site, which can be useful for simulating systems where the on-site coulomb interaction is strong enough to prevent double occupation.

using FermionicHilbertSpaces, LinearAlgebra
N = 2 # number of fermions
space = 1:N
spin = (:↑,:↓)
Hs = [hilbert_space([(k, s) for s in spin], number_conservation(0:1)) for k in space]
H = tensor_product(Hs)

9-dimensional FockHilbertSpace:
4 fermions: [(1, :↑), (1, :↓), (2, :↑), (2, :↓)]

dim(H) == 3^N

true

Can also take product of symmetries

qn2 = prod(number_conservation(0:1, label -> k in label) for k in space)
H2 = hilbert_space(keys(H), qn2)

9-dimensional SymmetricFockHilbertSpace:
4 fermions: [(1, :↑), (1, :↓), (2, :↑), (2, :↓)]
FockSymmetry(Number conservation for 2 subsets)

This gives the same states but with a different ordering.

sort(basisstates(H2)) == sort(basisstates(H))

true