Heisenberg1D — Spin-1/2 AFM Heisenberg Chain

Hamiltonian

\[H = J \sum_{i} \mathbf{S}_i \cdot \mathbf{S}_{i+1}, \qquad \mathbf{S}_i = \tfrac{1}{2}\boldsymbol{\sigma}_i, \qquad J > 0\ \text{(antiferromagnetic)}.\]

The chain is gapless; the low-energy theory is a $c = 1$ Luttinger liquid (free compactified boson). The ground state is a singlet ($S_\text{tot} = 0$) for any finite even $N$ with AFM coupling.

Status: thin delegator over XXZ1D(Δ = 1)

Every OBC observable on Heisenberg1D is implemented by forwarding to XXZ1D(Δ=1.0, J=J):

QAtlas.fetch(Heisenberg1D(), Energy(), OBC(6); beta=1.0, J=1.5)
# == QAtlas.fetch(XXZ1D(J=1.5, Δ=1.0), Energy(), OBC(6); beta=1.0)

For $J = 1$ the kwarg can be omitted (default 1.0).

Coverage Matrix

All OBC rows delegate to XXZ1D(Δ=1) dense-ED ($N \le 12$). The infinite-chain ground-state energy density is the original Hulthén value.

The closed-form $N = 2$ dimer and $N = 4$ ring spectra are exposed through ExactSpectrum and used as harness anchors.

SU(2) Symmetry Identities

is_su2_symmetric(::Heisenberg1D) === true, so the SU(2) row of SYMMETRY_IDENTITIES (PR #133) is automatically applied by the test harness. The identities checked numerically are

\[\chi_{xx} = \chi_{yy} = \chi_{zz}, \qquad m_\alpha = 0\ \ (\alpha \in \{x, y, z\}).\]

These hold to ED precision because the dense-ED kernel diagonalises the full SU(2)-symmetric Hamiltonian without breaking the rotation.

Quick-look code

using QAtlas

m = Heisenberg1D()
β = 1.0
N = 6

QAtlas.fetch(m, Energy(),                OBC(N); beta=β, J=1.0)
QAtlas.fetch(m, SpecificHeat(),          OBC(N); beta=β, J=1.0)
QAtlas.fetch(m, MassGap(),               OBC(N);          J=1.0)
QAtlas.fetch(m, MassGap(),               Infinite();      J=1.0)   # 0
QAtlas.fetch(m, GroundStateEnergyDensity(), Infinite();   J=1.0)   # J(1/4 - ln 2)

# Closed-form spectra (used as test anchors)
QAtlas.fetch(m, ExactSpectrum(); N=2, J=1.0, bc=:OBC)   # dimer
QAtlas.fetch(m, ExactSpectrum(); N=4, J=1.0, bc=:PBC)   # 4-site ring

Closed-form anchors

Dimer ($N = 2$, OBC)

The two-site Hilbert space splits into a singlet and a triplet:

\[\text{Spec}(H) = \left\{-\tfrac{3J}{4},\; \tfrac{J}{4},\; \tfrac{J}{4},\; \tfrac{J}{4}\right\}\]

Singlet–triplet gap $\Delta = J$; full derivation in Heisenberg dimer: singlet–triplet.

4-site PBC ring

The 16-dimensional spectrum decomposes into a unique singlet ground state at $E_0 = -2J$, a triplet at $-J$, mixed states at $0$, and a quintet at $+J$. Used as a finite-size cross-check against the Bethe ansatz.

Thermodynamic-limit ground-state energy

\[e_0 = J\!\left(\tfrac{1}{4} - \ln 2\right) \approx -0.4431\,J\]

Full Bethe-ansatz derivation in Bethe ansatz: Heisenberg $e_0$.

References

• H. Bethe, Z. Physik 71, 205 (1931) — original Bethe ansatz.
• L. Hulthén, Ark. Mat. Astron. Fys. 26A, No. 11 (1938) — $e_0 = J(1/4 - \ln 2)$.
• A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer, 1994).
• T. Giamarchi, Quantum Physics in One Dimension (Oxford, 2004), Ch. 6.

Related

• XXZ1D — anisotropic generalisation; Heisenberg lives at $\Delta = 1$.
• S1Heisenberg1D — spin-1 cousin in the Haldane phase (gapped, topologically non-trivial).
• TFIM — $c = 1/2$ Ising critical line for contrast.

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Verified hubs

In the Verified Atlas, these 5 models register 46 hubs (quantity / BC pair). The badge column shows the R1 assurance level; click a hub link to see the exact verify(...) calls, references, and corroboration mechanism.

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