Heisenberg dimer in a constrained fermionic Hilbert space#

Exchange Hamiltonians describe the low-energy spin dynamics of Mott insulators, quantum magnets, and Hubbard systems after high-energy charge fluctuations are integrated out. For two spin-\(1/2\) sites,

\[ H=J\,\mathbf S_0\cdot\mathbf S_1. \]

The local spin operators are represented by fermionic bilinears,

\[ S_i^+=c_{i\uparrow}^\dagger c_{i\downarrow},\qquad S_i^-=c_{i\downarrow}^\dagger c_{i\uparrow},\qquad S_i^z=\frac12(n_{i\uparrow}-n_{i\downarrow}). \]

A pure two-spin problem occupies the subspace with exactly one fermion on each site. The local-occupancy constraint reduces the \(N=2\) Fock sector from six states to four.

from edinpy import fermion as edf

UP, DOWN = 0, 1

site = edf.DoF(2, name="site", labels=("left", "right"))
spin = edf.DoF(2, name="spin", labels=("up", "down"))
modes = edf.FermionModes(site, spin)

sector = (
    edf.NParticleSector(modes, N=2)
    .project_particles("site", left=1, right=1)
    .build()
)

Sdot = edf.HeisenbergExchange(
    (0, UP), (0, DOWN),
    (1, UP), (1, DOWN),
    J=1.0,
    modes=modes,
)

The analytic energies follow from

\[ \mathbf S_0\cdot\mathbf S_1 =\frac12\left(\mathbf S_{\mathrm{tot}}^2-\mathbf S_0^2-\mathbf S_1^2\right), \]

so that

\[ E_{S=0}=-\frac{3J}{4},\qquad E_{S=1}=\frac{J}{4}. \]

Singlet and triplet energies of the constrained fermionic Heisenberg dimer — Spectrum of the two-spin Heisenberg dimer. Antiferromagnetic \(J>0\) selects the singlet, while ferromagnetic \(J<0\) selects the triplet manifold.#

Total-spin diagnostic#

The total-spin operator can be constructed literally from the summed raising, lowering, and \(z\) components,

\[ \mathbf S_{\mathrm{tot}}^2 =(S_{\mathrm{tot}}^z)^2 +\frac12\left(S_{\mathrm{tot}}^+S_{\mathrm{tot}}^-+S_{\mathrm{tot}}^-S_{\mathrm{tot}}^+\right). \]

Splus = sum(
    (edf.SpinPlus((i, UP), (i, DOWN), modes) for i in range(2)),
    start=0,
)
Sminus = sum(
    (edf.SpinMinus((i, UP), (i, DOWN), modes) for i in range(2)),
    start=0,
)
Sz = sum(
    (edf.SpinZ((i, UP), (i, DOWN), modes) for i in range(2)),
    start=0,
)
S2 = Sz * Sz + 0.5 * (Splus * Sminus + Sminus * Splus)

Total spin squared for the four eigenstates of the fermionic Heisenberg dimer — Total-spin quantum number at \(J=1\). The singlet has \(\langle S_{\mathrm{tot}}^2\rangle=0\) and the three triplets have \(\langle S_{\mathrm{tot}}^2\rangle=2\).#

The occupancy constraint removes the doublon-holon states and leaves the four-dimensional effective spin Hilbert space.