S1Heisenberg1D — Spin-1 AFM Heisenberg Chain (Haldane Chain)
The spin-1 dense-ED observable surface debuted in v0.17. The local Hilbert space is 3-dimensional, so the global space is $3^N$ and the hard cap is $N \le 8$. Method signatures and kwarg names may change in v0.19. MagnetizationYLocal is intentionally omitted (Tier 3 in the observable issue) until a use case appears.
Hamiltonian
\[H = J \sum_{i} \mathbf{S}_i \cdot \mathbf{S}_{i+1}, \qquad S = 1\ (\text{3-dimensional local Hilbert space}).\]
The on-site spin operators are the standard $3 \times 3$ matrices _S1_x, _S1_y, _S1_z with eigenvalues $\{-1, 0, +1\}$.
Phases
The spin-1 antiferromagnet is in the Haldane phase: a gapped, topologically non-trivial phase with hidden $\mathbb{Z}_2 \times \mathbb{Z}_2$ string order and (on OBC) free spin-1/2 edge modes.
| Quantity | Value |
|---|---|
| Bulk gap $\Delta_\infty$ | $\approx 0.41048\,J$ (White–Huse 1993, DMRG) |
| GS energy density $e_0$ | $\approx -1.401484\,J$ (White–Huse 1993, DMRG) |
| Edge | gapless spin-1/2 doublet on each end (OBC) |
There is no closed form for $\Delta_\infty$ or $e_0$ — they are exposed by QAtlas as :literature_value rows.
Coverage Matrix
OBC rows are dense-ED with cap $N \le 8$ ($3^N \le 6561$).
| Quantity | OBC ($N \le 8$) | Infinite |
|---|---|---|
Energy{:total} | dense-ED | — |
Energy{:per_site} | conversion | $\approx -1.40148\,J$ (White–Huse 1993) |
FreeEnergy / ThermalEntropy / SpecificHeat | dense-ED | — |
MagnetizationX / Y / Z | dense-ED | — |
MagnetizationXLocal / MagnetizationZLocal | dense-ED | — |
SusceptibilityXX / YY / ZZ | dense-ED | — |
XXCorrelation / YY / ZZ (:static, :connected) | dense-ED | — |
VonNeumannEntropy / RenyiEntropy | partial trace | — |
MassGap | dense-ED ($E_1 - E_0$) | $\approx 0.41048\,J$ (Haldane gap) |
EnergyLocal | dense-ED (symmetric bond split) | — |
MagnetizationYLocal is not registered (Tier 3, deferred).
Spin-1 Convention
The on-site operators are the spin-1 matrices, so eigenvalues run over $\{-1, 0, +1\}$ (NOT the Pauli $\pm 1$ of TFIM/XXZ1D's $\sigma^\alpha$). All observables on S1Heisenberg1D use the $S^\alpha$ convention directly. When comparing against TFIM-style outputs that use $\sigma^\alpha = 2 S^\alpha$, factors of $2$ (linear) or $4$ (variance, two-point) are required.
SU(2) Symmetry Identities
is_su2_symmetric(::S1Heisenberg1D) === true, so the SU(2) row of SYMMETRY_IDENTITIES (PR #133) is checked automatically:
\[\chi_{xx} = \chi_{yy} = \chi_{zz}, \qquad m_\alpha = 0\ \ (\alpha \in \{x, y, z\}).\]
This holds to ED precision since the dense kernel never breaks the spin rotation.
Code Examples
using QAtlas
m = S1Heisenberg1D(J=1.0)
N = 4
β = 1.0
QAtlas.fetch(m, Energy(), OBC(N); beta=β)
QAtlas.fetch(m, MassGap(), OBC(N)) # finite-N gap
QAtlas.fetch(m, MassGap(), Infinite()) # ≈ 0.41048
QAtlas.fetch(m, ZZCorrelation{:static}(), OBC(N); beta=β, i=1, j=4)
QAtlas.fetch(m, VonNeumannEntropy(), OBC(N); ℓ=2, beta=Inf)For OBC $N \le 4$ the gap $E_1 - E_0$ is dominated by the spin-1/2 edge-mode quasi-degeneracy, so it is much smaller than $\Delta_\infty$. The two values are not directly comparable until $N$ is large enough for bulk physics to dominate (typical DMRG convergence around $N \sim 40$).
References
- F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983) — $\sigma$-model prediction of the integer-spin gap.
- S. R. White, D. A. Huse, Phys. Rev. B 48, 3844 (1993) — state-of-the-art DMRG values for $\Delta_\infty$ and $e_0$.
- T. Kennedy, H. Tasaki, Phys. Rev. B 45, 304 (1992) — string order and hidden $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry.
- I. Affleck, J. Phys. Condens. Matter 1, 3047 (1989) — bosonisation of the spin-1 chain.
Related
- Heisenberg1D — spin-1/2 cousin; gapless Luttinger liquid, topologically trivial — a distinct phase.
- XXZ1D — spin-1/2 anisotropic generalisation; SU(2)-symmetric only at $\Delta = 1$.