# Schwinger bosons for a spin-1/2 Heisenberg dimer

The Schwinger-boson representation replaces a local spin by two bosonic modes.[^arovas][^schuckert]

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

The local constraint

$$
n_{i\uparrow}+n_{i\downarrow}=2S
$$

selects the physical spin-$S$ representation. For $S=1/2$, one boson per site gives two local states and reproduces the ordinary spin-$1/2$ algebra.

For two sites,

$$
H=J\,\mathbf S_0\cdot\mathbf S_1.
$$

## Constrained bosonic Hilbert space

```python
from edinpy import boson as edb

site = edb.DoF(2, name="site", labels=("left", "right"))
component = edb.DoF(2, name="component", labels=("up", "down"))
modes = edb.BosonModes(site, component)

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

The constraint leaves four states, exactly the Hilbert space of two spin-$1/2$ degrees of freedom.

```python
UP, DOWN = 0, 1
Sdot = edb.HeisenbergExchange(
    (0, UP), (0, DOWN),
    (1, UP), (1, DOWN),
    J=1.0,
    modes=modes,
)
```

The analytic spectrum is

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

```{figure} ../../_static/figures/example_boson_exchange_spectrum.svg
:width: 76%
:alt: Singlet and triplet energies of a Schwinger-boson Heisenberg dimer

Schwinger-boson spectrum of the locally constrained Heisenberg dimer. The bosonic representation reproduces the singlet and triplet branches exactly.
```

## Total-spin operator

The total-spin Casimir is

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

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

```{figure} ../../_static/figures/example_boson_exchange_total_spin.svg
:width: 72%
:alt: Total spin squared for the four eigenstates of the Schwinger-boson Heisenberg dimer

Total-spin diagnostic at $J=1$. The ground-state singlet has $S(S+1)=0$ and the triplet manifold has $S(S+1)=2$.
```

The local occupancy constraint removes the unphysical bosonic states and leaves precisely the spin-$1/2$ Hilbert space.

[^arovas]: D. P. Arovas and A. Auerbach, "Functional integral theories of low-dimensional quantum Heisenberg models," Phys. Rev. B 38, 316 (1988), [doi:10.1103/PhysRevB.38.316](https://doi.org/10.1103/PhysRevB.38.316).
[^schuckert]: A. Schuckert, A. Piñeiro Orioli, and J. Berges, "Nonequilibrium quantum spin dynamics from two-particle irreducible functional integral techniques in the Schwinger boson representation," Phys. Rev. B 98, 224304 (2018), [doi:10.1103/PhysRevB.98.224304](https://doi.org/10.1103/PhysRevB.98.224304).