Spatial symmetry sectors

Block-diagonalizing a Hamiltonian with a spatial symmetry such as translations shows which symmetry sector holds the ground state and shrinks the matrices to diagonalize. This example builds the sectors of an irreducible representation (irrep) with symmetric_sector and uses them to find the momentum and magnetization of the ground state of a Heisenberg ring.

Permutations and irreducible representations

Suppose a group $G$ acts on the sites of a system by permutations. Each permutation induces an operator $U(g)$ on the Hilbert space. For an irreducible character $\chi$, the projector onto its symmetry sector is

\[P_\chi = \frac{d_\chi}{|G|} \sum_{g \in G} \chi(g)^* U(g), \qquad d_\chi = \chi(e).\]

To calculate the projector, you need the permutation operator and the value $\chi(g)$ for each $g$. FermionicHilbertSpaces.jl handles the permutations, and you supply the characters. With a list of permutations and their weights $\chi(g)^*$ for a particular irrep, symmetric_sector(H, Hs, (perms, weights)) constructs a matrix whose columns form a basis for the sector.

For translations the characters are simple phases, as shown below. For other groups, such as the dihedral group that also includes reflections, character tables can be computed with e.g. Oscar.jl.

Example: a Heisenberg ring

Consider a spin chain with nearest-neighbour coupling and a uniform field:

\[H = J\sum_{k=1}^{N}\mathbf{S}_k\cdot\mathbf{S}_{k+1} - h\sum_{k=1}^{N}S^z_k, \qquad \mathbf{S}_{N+1}=\mathbf{S}_1.\]

We first separate total-magnetization sectors $M=\sum_k S_k^z$, then use the ring's translation symmetry. Let's consider 5 sites, each with spin 3/2.

using FermionicHilbertSpaces, LinearAlgebraN = 5J = 1.0h = 0.04@spins S 3 // 2magnetization(s::SpinState) = s.mspace = hilbert_space(S, 1:N, SectorConstraint(sum; maps=magnetization))mags = quantumnumbers(space)H = J * sum(S[k][op] * S[mod1(k + 1, N)][op] for k in 1:N for op in (:x, :y, :z)) -    h * sum(S[k][:z] for k in 1:N)

Sum with 20 terms: 
-0.04*S[1][:z]-0.04*S[2][:z]-0.04*S[3][:z] + ...

Translations form the cyclic group $\mathbb{Z}_N$ with $N$ elements. We write each as a permutation vector, which is the format FermionicHilbertSpaces.jl takes: translation s sends site i to mod1(i + s, N), for s in 0:N-1.

perms = [[mod1(i + s, N) for i in 1:N] for s in 0:N-1]

5-element Vector{Vector{Int64}}:
 [1, 2, 3, 4, 5]
 [2, 3, 4, 5, 1]
 [3, 4, 5, 1, 2]
 [4, 5, 1, 2, 3]
 [5, 1, 2, 3, 4]

Every irrep of $\mathbb{Z}_N$ is one-dimensional and labelled by a momentum $k$, with character $\chi_k(s) = e^{-2\pi i k s/N}$. The weights are the conjugated characters $\chi_k(s)^* = e^{2\pi i k s/N}$.

momenta = (-(N - 1) ÷ 2):(N ÷ 2)labels = ["k = $k" for k in momenta]weights = [[cis(2π * k * s / N) for s in 0:N-1] for k in momenta];

In each magnetization sector, symmetric_sector constructs a matrix P whose columns form a basis for an irrep's subspace. factors(sec) identifies the tensor factors permuted by the group. We diagonalize $P^\dagger H P$; an absent irrep gets energy Inf.

function sector_energies(M)    sec = sector(M, space)    HM = representation(H, sec)    return map(weights) do w        P = symmetric_sector(sec, factors(sec), (perms, w))        size(P, 2) == 0 && return Inf        eigmin(Hermitian(Matrix(P' * HM * P)))    endendenergies = stack(map(sector_energies, mags))E0, idx = findmin(energies)(; E0, irrep=labels[idx[1]], M=mags[idx[2]])

(E0 = -12.452645032783666, irrep = "k = -1", M = 1/2)

Let's plot the results. The heatmap shows the lowest energy in each (irrep, $M$) block. Green marks the lowest energy for each irrep; red marks the lowest energy for each magnetization.

using Plotsbest_irrep_per_M = map(argmin, eachcol(energies))best_M_per_irrep = map(argmin, eachrow(energies))p = heatmap(labels, mags, permutedims(energies);    xlabel="momentum (units of 2π/N)",    ylabel="magnetization M",    title="Lowest energy per symmetry sector",    c=:viridis, yticks=mags, frame=:box)scatter!(p, labels[best_irrep_per_M], mags;    marker=:vline, markersize=6, markerstrokewidth=5,    color=:red, label="lowest state at fixed M")scatter!(p, labels, mags[best_M_per_irrep];    marker=:hline, markersize=8, markerstrokewidth=5,    color=:green, label="lowest state in fixed irrep")

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