XXZ1D — Spin-1/2 XXZ Chain

Hamiltonian

\[H = J \sum_{i} \bigl[\, S^x_i S^x_{i+1} + S^y_i S^y_{i+1} + \Delta\, S^z_i S^z_{i+1} \,\bigr]\]

with $\mathbf{S}_i = \tfrac{1}{2}\boldsymbol{\sigma}_i$, exchange coupling $J$ (default 1.0), and anisotropy $\Delta$ (default 0.0, the XX point).

Phases

Coverage Matrix

OBC rows are dense-ED (Hilbert dim $2^N$, cap $N \le 12$). Infinite rows are analytic / Bethe-ansatz closed forms.

SpinWaveVelocity is a type-level alias of LuttingerVelocity.

XX Point (Δ = 0) — Free-Fermion Thermo at Infinite()

After Jordan-Wigner the XX chain is non-interacting; the single-particle dispersion (in the spin convention Sᵅ = σᵅ/2) is

\[\varepsilon(k) = -J \cos k, \quad k \in [-\pi, \pi].\]

QAtlas exposes per-site FreeEnergy, Energy{:per_site}, ThermalEntropy, and SpecificHeat at Infinite() for Δ = 0 via adaptive Gauss-Kronrod quadrature on [0, π]:

\[\begin{aligned} f(\beta) &= -\frac{1}{\pi\beta} \int_0^\pi \log\!\left(2\cosh\tfrac{\beta\varepsilon(k)}{2}\right) dk, \ e(\beta) &= -\frac{1}{2\pi} \int_0^\pi \varepsilon(k)\,\tanh\tfrac{\beta\varepsilon(k)}{2}\,dk, \ s(\beta) &= \beta\,\bigl(e(\beta) - f(\beta)\bigr), \ C(\beta) &= \frac{1}{\pi} \int_0^\pi \bigl(\tfrac{\beta\varepsilon}{2}\bigr)^2 \operatorname{sech}^2\!\tfrac{\beta\varepsilon}{2}\,dk. \end{aligned}\]

m = XXZ1D(J=1.0, Δ=0.0)
QAtlas.fetch(m, Energy(),         Infinite(); beta=10.0)   # → ≈ -1/π = -0.3183
QAtlas.fetch(m, FreeEnergy(),     Infinite(); beta=1.0)
QAtlas.fetch(m, ThermalEntropy(), Infinite(); beta=0.01)   # → ≈ log 2
QAtlas.fetch(m, SpecificHeat(),   Infinite(); beta=1.0)    # → > 0

For any Δ ≠ 0 these four calls emit a @warn and return NaN — the general-Δ thermal Bethe ansatz / NLIE is tracked in issue #108.

v0.17 Highlights — Dense-ED Full Suite

A single _xxz1d_thermal_kernel(model, N, β) performs one eigendecomposition of the $2^N \times 2^N$ Hamiltonian and reuses the spectrum / eigenvectors across every observable on the OBC row. The hard cap is N ≤ 12 (Hilbert dimension 2^12 = 4096).

using QAtlas

m = XXZ1D(J=1.0, Δ=0.5)
β = 1.0

QAtlas.fetch(m, FreeEnergy(),         OBC(6); beta=β)
QAtlas.fetch(m, SpecificHeat(),       OBC(6); beta=β)
QAtlas.fetch(m, MagnetizationZ(),     OBC(6); beta=β)         # = 0  (U(1) conservation)
QAtlas.fetch(m, ZZCorrelation{:static}(), OBC(6); beta=β, i=2, j=4)
QAtlas.fetch(m, RenyiEntropy(2.0),    OBC(6); ℓ=3, beta=Inf)

Δ = 1 isotropic point: SU(2) symmetry identities

At $\Delta = 1$ the chain is SU(2)-symmetric, so every observable satisfies the harness-checked identities

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

m_iso = XXZ1D(J=1.0, Δ=1.0)
QAtlas.fetch(m_iso, SusceptibilityXX(), OBC(6); beta=1.0)  # =
QAtlas.fetch(m_iso, SusceptibilityYY(), OBC(6); beta=1.0)  # =
QAtlas.fetch(m_iso, SusceptibilityZZ(), OBC(6); beta=1.0)

These three calls return the same value to ED precision, and are checked by the SU(2) row of SYMMETRY_IDENTITIES (PR #133).

Critical-regime CFT data

For $-1 < \Delta < 1$ the chain flows to a $c = 1$ compactified-boson CFT with continuously varying compactification radius. QAtlas exposes the standard triplet:

QAtlas.fetch(XXZ1D(; Δ=0.3), CentralCharge(),     Infinite())  # → 1.0
QAtlas.fetch(XXZ1D(; Δ=0.0), LuttingerParameter(), Infinite())  # → 1.0
QAtlas.fetch(XXZ1D(; Δ=1.0), LuttingerVelocity(),  Infinite())  # → π/2
QAtlas.fetch(XXZ1D(; Δ=1.5), CentralCharge(),     Infinite())  # → NaN (+ warn)

Full derivation: XXZ Luttinger parameters from Bethe ansatz.

References

• H. Bethe, Z. Physik 71, 205 (1931).
• L. Hulthén, Ark. Mat. Astron. Fys. 26A, No. 11 (1938) — $\Delta = 1$ value.
• C. N. Yang, C. P. Yang, Phys. Rev. 150, 321 (1966) — general-$\Delta$ Bethe-ansatz integral equation.
• M. Takahashi, Thermodynamics of One-Dimensional Solvable Models (Cambridge UP, 1999), Ch. 4.
• T. Giamarchi, Quantum Physics in One Dimension (Oxford, 2004), Ch. 6.

Related

• Heisenberg1D — isotropic SU(2) point ($\Delta = 1$); a thin delegator to XXZ1D(Δ=1.0).
• S1Heisenberg1D — spin-1 generalisation; gapped Haldane phase, distinct universality.
• TFIM — $c = 1/2$ Ising chain for contrast with the $c = 1$ XXZ critical line.

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Verified hubs

In the Verified Atlas, these 2 models register 28 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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