Spins

This page shows how to declare spin operators, what conventions they follow, and how to build spin Hilbert spaces. A Heisenberg chain then shows how magnetization and total-spin sectors block-diagonalize a spin Hamiltonian, and that its ground state is a singlet.

Spin operators and spaces

@spin T declares one spin. Indexing it with :x, :y, :z gives the spin components, :+ and :- the ladder operators $S^\pm = S^x \pm i S^y$, and T[:] the vector of components. These are spin operators, not Pauli matrices: they satisfy $[S^x, S^y] = i S^z$, and $S^z$ has eigenvalues $-J, \dots, J$.

using FermionicHilbertSpacesusing LinearAlgebra@spin TT[:x] * T[:y] - T[:y] * T[:x] == 1im * T[:z]

true

The spin value $J$ fixes the local dimension $2J+1$. It can be given when declaring the spin, which lets expressions simplify with relations that hold only for that $J$, such as $\vec S^2 = J(J+1)$. Without it, only the commutation relations hold, so a single symbolic Hamiltonian can be reused for any $J$, which is then passed to hilbert_space. The basis states are ordered by increasing $S^z$ eigenvalue.

@spin Shalf 1 // 2sum(s^2 for s in Shalf[:]) == 3 // 4representation(T[:z], hilbert_space(T, 1))

3×3 SparseArrays.SparseMatrixCSC{Float64, Int64} with 2 stored entries:
 -1.0   ⋅    ⋅ 
   ⋅    ⋅    ⋅ 
   ⋅    ⋅   1.0

@spins S declares a field of spins, where S[k] is the spin on site k, and hilbert_space(S, labels) builds the space of the sites in labels (with J as a third argument if the field has no spin value). Constraints act on the basis states of each site through their $S^z$ eigenvalue s.m; for example, AdditiveConstraint(M; maps = s -> s.m) keeps the states with total $S^z$ equal to M.

Heisenberg chain

The spin-1/2 Heisenberg chain $H = \sum_k \vec S_k \cdot \vec S_{k+1}$ commutes with the total magnetization $S^z_\mathrm{tot} = \sum_k S^z_k$, so we can restrict to the sector where it vanishes.

@spins S 1 // 2N = 4ham = sum(S[k][:]'S[k + 1][:] for k in 1:(N - 1))Sz = sum(S[k][:z] for k in 1:N)@assert ham * Sz - Sz * ham == 0Hm = hilbert_space(S, 1:N, AdditiveConstraint(0; maps = s -> s.m))Mham = representation(ham, Hm, :dense);

The chain also conserves the total spin, fixed by the Casimir operator $\vec S_\mathrm{tot}^2 = J(J+1)$. Projecting onto its eigenspaces block-diagonalizes the Hamiltonian, and the blocks together reproduce its spectrum. The lowest energy of each block, printed below, is smallest for $J=0$. Dense matrices are used because nullspace and eigvals need them.

Stot2 = sum(sum(S[k][op] for k in 1:N)^2 for op in (:x, :y, :z))Mtot = representation(Stot2, Hm, :dense)blocks = map(J -> nullspace(Mtot - J * (J + 1) * I), 0:2)spectra = [eigvals(Hermitian(P' * Mham * P)) for P in blocks]@assert sort(reduce(vcat, spectra)) ≈ eigvals(Hermitian(Mham))minimum.(spectra)

3-element Vector{Float64}:
 -1.6160254037844388
 -0.9571067811865472
  0.7500000000000002

For Hamiltonians symmetric under permuting sites, symmetric_sector projects onto permutation sectors; see Spatial symmetries.

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