Heisenberg1D — Spin-1/2 AFM Heisenberg Chain

DMIHeisenberg1D(; J::Real = 1.0, D::Real = 0.0) <: AbstractQAtlasModel

Spin-½ Heisenberg chain with Dzyaloshinskii-Moriya interaction along ẑ:

H = J Σ_i Sᵢ · Sᵢ₊₁ + D Σ_i (Sᵢˣ Sᵢ₊₁ʸ − Sᵢʸ Sᵢ₊₁ˣ),    J > 0,  D ∈ ℝ.

Phase 1 implements only the closed-form D = 0 point (pure Heisenberg), delegating to Heisenberg1D. Generic D ≠ 0 introduces a spiral ground state via a gauge mapping to a twisted XXZ chain (Affleck- Oshikawa 1999); closed-form energy is technically Bethe-ansatz-tractable on the gauged model but is deferred to Phase 2, raising DomainError from fetch(..., Energy{:per_site}(), Infinite()).

The default constructor DMIHeisenberg1D() lands on J = 1, D = 0 — the Bethe-Hulthén point, the only Phase-1 closed-form case.

Fields

• J::Float64 — Heisenberg exchange (J > 0 antiferromagnetic).
• D::Float64 — DM coupling along ẑ (any real).

References

• H. Bethe, Z. Physik 71, 205 (1931).
• L. Hulthén, Ark. Mat. Astron. Fys. 26A, No. 11 (1938).
• I. E. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958).
• T. Moriya, Phys. Rev. 120, 91 (1960).
• I. Affleck, M. Oshikawa, Phys. Rev. B 60, 1038 (1999).

fetch(::DMIHeisenberg1D, ::Energy{:per_site}, ::Infinite;
      J=m.J, D=m.D) -> Float64

Ground-state energy density of the spin-½ Heisenberg-DM chain in the thermodynamic limit. Phase 1 supports only the closed-form D = 0 point:

• D = 0 → Bethe-Hulthén: E/N = J · (1/4 − ln 2). Delegates to fetch(Heisenberg1D(), GroundStateEnergyDensity(), Infinite(); J=J).

• D ≠ 0 → DomainError: spiral order via gauge rotation to twisted XXZ (Affleck-Oshikawa 1999); closed-form Bethe-ansatz energy technically available but deferred to Phase 2.

Floating-point tolerance for the D = 0 match is atol = 1e-12.

Note on delegation: Heisenberg1D currently exposes its thermodynamic- limit energy density via the legacy GroundStateEnergyDensity quantity, not the modern Energy{:per_site} axis. This wrapper bridges to the modern axis on the public surface while keeping the closed-form constant single-source-of-truth in the Heisenberg1D delegate.

References

• L. Hulthén, Ark. Mat. Astron. Fys. 26A, No. 11 (1938) — Bethe-Hulthén ground-state energy density.
• I. E. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958).
• T. Moriya, Phys. Rev. 120, 91 (1960).
• I. Affleck, M. Oshikawa, Phys. Rev. B 60, 1038 (1999) — spiral / twisted-XXZ mapping (deferred to Phase 2).

ExactSpectrum

Dispatch tag for the full sorted eigenvalue spectrum of a finite model.

GroundStateEnergyDensity

Dispatch tag for the ground-state energy per site in the thermodynamic limit (N → ∞).

Heisenberg1D

Dispatch tag for the spin-1/2 antiferromagnetic Heisenberg model on a 1D chain (or more generally any finite spin-1/2 cluster). Hamiltonian:

H = J Σ_{⟨i,j⟩} S_i · S_j,   spin-1/2, J > 0 antiferromagnetic

fetch(::Heisenberg1D, ::Energy{:total}, ::OBC; beta, J=1.0) -> Float64

Total thermal energy ⟨H⟩_β for the spin-½ antiferromagnetic Heisenberg OBC chain at finite N (the isotropic point Δ = 1 of XXZ1D). Routes through fetch(::XXZ1D, ::Energy{:total}, ::OBC).

Since Heisenberg1D currently carries no J field, callers must pass J as a kwarg (default J = 1.0). Downstream bridges (e.g. ITensorModels to_qatlas(::Heisenberg1D)) lose J on conversion; use XXZ1D(; J, Δ=1) directly if you need a non-unit coupling.

fetch(::Heisenberg1D, ::ExactSpectrum; N, J=1.0) -> Vector{Float64}

Return the sorted exact spectrum of the spin-1/2 Heisenberg Hamiltonian on an N-site chain or ring with boundary condition bc.

Supported cases

• N=2, bc=:OBC (dimer): [-3J/4, J/4, J/4, J/4] (singlet Es = -3J/4, triplet Et = J/4, three-fold degenerate).
• N=4, bc=:PBC (4-site ring): [-2J, -J×3, 0×7, +J×5] Ground state E₀ = -2J (unique singlet). The ferromagnetic quintet sits at E = +J. The full degeneracy structure is: 1 singlet + 1 triplet + (1 singlet + 2 triplets at E=0) + 1 quintet.

Arguments

• N::Int: number of spin-1/2 sites
• J::Real: Heisenberg coupling constant (default 1.0; J > 0 AFM)
• bc::Symbol: boundary condition, :OBC (default) or :PBC

References

A. Auerbach, "Interacting Electrons and Quantum Magnetism" (1994), §2.
H. Bethe, Z. Physik 71, 205 (1931).

fetch(::Heisenberg1D, ::GroundStateEnergyDensity, ::Infinite; J=1.0) -> Float64

BC-explicit dispatch sister of the legacy fetch(::Heisenberg1D, ::GroundStateEnergyDensity; J=1.0) method. The thermodynamic-limit ground-state energy density is only meaningful at Infinite, so the two methods return the same Hulthén value e₀ = J(1/4 - ln 2). Provided so which(fetch, ::Heisenberg1D, ::GroundStateEnergyDensity, ::Infinite) resolves and the registry drift guard passes.

fetch(::Heisenberg1D, ::GroundStateEnergyDensity; J=1.0) -> Float64

Exact ground-state energy per site of the spin-1/2 antiferromagnetic Heisenberg chain in the thermodynamic limit (N → ∞, PBC):

e₀ = J (1/4 − ln 2) ≈ −0.4431 J

This is one of the earliest and most celebrated results of the Bethe ansatz. The derivation proceeds by solving the Bethe equations for the ground state of

H = J Σᵢ Sᵢ · Sᵢ₊₁

in the limit N → ∞, yielding a linear integral equation for the rapidity distribution whose solution gives the energy via integration.

Finite-size corrections

For a PBC chain of N sites, the ground-state energy density approaches e₀ with corrections of order 1/N² (logarithmic corrections also present):

E₀(N)/N = e₀ + O(1/N²)

See test/verification/test_universality_cross_check.jl for a finite-size extrapolation verification using ED at N = 4, 6, 8.

Arguments

• J::Real: Heisenberg coupling constant (default 1.0; J > 0 AFM)

References

H. Bethe, "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen
  der linearen Atomkette", Z. Physik 71, 205–226 (1931) — original
  Bethe ansatz solution.
L. Hulthén, "Über das Austauschproblem eines Kristalles",
  Ark. Mat. Astron. Fys. 26A, No. 11, 1–106 (1938) — first
  evaluation of e₀ = 1/4 − ln 2 from the Bethe equations.

fetch(::Heisenberg1D, ::LuttingerParameter, ::Infinite; J=1.0) -> Float64

Luttinger-liquid parameter K = 1/2 of the spin-½ Heisenberg antiferromagnetic chain at the SU(2)-symmetric point — the exact Luther–Peschel 1975 / Affleck 1989 result.

This is the Δ → 1 limit of the XXZ Luttinger parameter K_XXZ(Δ) = π / (2 arccos(−Δ)) (Haldane 1980); delegated to XXZ1D(Δ=1.0).

References

• A. Luther, I. Peschel, Phys. Rev. B 12, 3908 (1975).
• I. Affleck, J. Phys. A 22, 1003 (1989).
• F. D. M. Haldane, Phys. Rev. Lett. 45, 1358 (1980).

fetch(m::Heisenberg1D, ::EntanglementGrowthSlope, ::Infinite;
      beta_eff::Real, kwargs...) -> Float64

Calabrese-Cardy 2005 linear-growth slope of post-quench half-system entanglement entropy for the gapless Heisenberg chain (c = 1, v = pi J / 2):

dS_A / dt = pi c v / (3 beta_eff) = pi^2 J / (6 beta_eff).

Delegates to Universality(:Heisenberg).

fetch(::Heisenberg1D, ::EntanglementSaturationDensity, ::Infinite;
      beta_eff::Real, kwargs...) -> Float64

Long-time post-quench saturation of half-system entanglement entropy per unit length, delegated to Universality(:Heisenberg):

S_A(infty) / L = pi / (6 beta_eff).

fetch(::Heisenberg1D, ::FreeEnergy, ::Infinite; beta, J=1.0)

Per-site free energy of the infinite spin-1/2 Heisenberg AF chain via leading-order c = 1 CFT. Returns e₀ - π T² / (6 v_s) for β > 5/J; otherwise NaN + warn. β ≤ 0 raises DomainError.

fetch(m::Heisenberg1D, ::LiebRobinsonVelocity, ::Infinite;
      J = m.J, kwargs...) -> Float64

Maximum group velocity of the des Cloizeaux-Pearson spinon dispersion of the spin-1/2 Heisenberg chain. With epsilon(k) = (pi/2) |J sin k| the spinon velocity is

v_s = pi J / 2,

attained at k = 0 and k = pi. This is the standard CC quasi-particle velocity that enters the entanglement-spreading formulas.

fetch(::Heisenberg1D, ::LogarithmicNegativity, ::Infinite;
      ℓ_A::Real, ℓ_B::Real, kwargs...) -> Float64

Calabrese-Cardy-Tonni 2012 logarithmic negativity of two adjacent intervals, delegated to Universality(:Heisenberg):

E = (1/4) log[ℓ_A * ℓ_B / (ℓ_A + ℓ_B)].

fetch(::Heisenberg1D, ::MutualInformation, ::Infinite;
      ℓ_A::Real, ℓ_B::Real, beta::Real=Inf, kwargs...) -> Float64

Calabrese-Cardy mutual information of two adjacent intervals, delegated to Universality(:Heisenberg) (c = 1). The chain is gapless SU(2)-invariant Luttinger liquid, so the Universality formula applies directly.

fetch(::Heisenberg1D, ::SpecificHeat, ::Infinite; beta, J=1.0)

Per-site heat capacity via leading-order c = 1 CFT: c_v = π T / (3 v_s) = 2T / (3J).

fetch(::Heisenberg1D, ::ThermalEntropy, ::Infinite; beta, J=1.0)

Per-site entropy via leading-order c = 1 CFT: s = π T / (3 v_s) = 2T / (3J).

S1Heisenberg1D(; J::Real = 1.0) <: AbstractQAtlasModel

Spin-1 antiferromagnetic Heisenberg chain (Haldane chain),

H = J Σᵢ Sᵢ · Sᵢ₊₁,    spin = 1,  J > 0 antiferromagnetic.

Distinct from Heisenberg1D (spin-1/2) because the spin representation differs: spin-1 has local dimension 3 and a gapped, topologically non-trivial Haldane phase, while spin-1/2 is gapless and critical (Bethe ansatz).

fetch(model::S1Heisenberg1D, ::Energy{:total}, ::OBC; beta) -> Float64

Total thermal energy ⟨H⟩_β of the spin-1 OBC Heisenberg chain at finite N ≤ 8, computed by dense ED.

    ⟨H⟩_β = Tr(H exp(-βH)) / Tr(exp(-βH))

Convention matches fetch(::TFIM, ::Energy{:total}, ::OBC): 1D finite-size boundary conditions return total energy natively.

fetch(model::S1Heisenberg1D, ::FreeEnergy, ::OBC; beta) -> Float64

Per-site Helmholtz free energy f(β) = -log Z / (Nβ) for the spin-1 OBC chain.

fetch(model::S1Heisenberg1D, ::SpecificHeat, ::OBC; beta) -> Float64

Per-site heat capacity c(β) = β² · Var(H) / N for the spin-1 OBC chain, computed exactly from the energy variance in the eigenbasis (no numerical differentiation).

fetch(model::S1Heisenberg1D, ::ThermalEntropy, ::OBC; beta) -> Float64

Per-site Gibbs entropy s(β) = β · (ε - f) for the spin-1 OBC chain.

fetch(model::S1Heisenberg1D, ::EnergyLocal, ::OBC; beta) -> Vector{Float64}

Site-resolved local energy density ε_i of the OBC spin-1 Heisenberg chain such that Σᵢ ε_i = ⟨H⟩_β. Each bond contribution b_i = J ⟨Sᵢ · Sᵢ₊₁⟩_β is split symmetrically between its two endpoints, with the missing left/right bond at i = 1 / i = N set to zero — i.e.

ε_i = (1/2) (b_{i-1} + b_i),    b_0 ≡ b_N ≡ 0.

fetch(model::S1Heisenberg1D, ::Energy{:per_site}, ::Infinite) -> Float64

Ground-state energy per site of the spin-1 antiferromagnetic Heisenberg chain (Haldane chain) in the thermodynamic limit:

e₀ ≈ -1.40148403897 J

Numerical value from the high-precision DMRG study of S. R. White & D. A. Huse, Phys. Rev. B 48, 3844 (1993). No closed- form expression is known; reliability=:medium in the registry.

fetch(model::S1Heisenberg1D, ::MagnetizationX, ::OBC; beta) -> Float64

Per-site bulk magnetisation ⟨Σᵢ S^x_i⟩_β / N of the spin-1 Heisenberg OBC chain at finite N ≤ 8, computed from the dense thermal density matrix.

Spin-1 convention: S^x carries eigenvalues ±1, 0 (off-diagonal 1/√2 for S^x); see file header. The SU(2)-symmetric AFM ground state has ⟨S^α⟩ = 0 in every direction at any temperature, so this is a non-trivial check only for symmetry- breaking finite-β samples (and for benchmarking sampler bias).

fetch(model::S1Heisenberg1D, ::MagnetizationXLocal, ::OBC; beta) -> Vector{Float64}

Site-resolved [⟨S^x_i⟩_β for i = 1:N] of the spin-1 OBC Heisenberg chain. Sums to N · MagnetizationX/Y/Z (per-site bulk average) by construction.

fetch(model::S1Heisenberg1D, ::MagnetizationY, ::OBC; beta) -> Float64

Per-site bulk magnetisation ⟨Σᵢ S^y_i⟩_β / N of the spin-1 Heisenberg OBC chain at finite N ≤ 8, computed from the dense thermal density matrix.

Spin-1 convention: S^y carries eigenvalues ±1, 0 (off-diagonal 1/√2 for S^x); see file header. The SU(2)-symmetric AFM ground state has ⟨S^α⟩ = 0 in every direction at any temperature, so this is a non-trivial check only for symmetry- breaking finite-β samples (and for benchmarking sampler bias).

fetch(model::S1Heisenberg1D, ::MagnetizationZ, ::OBC; beta) -> Float64

Per-site bulk magnetisation ⟨Σᵢ S^z_i⟩_β / N of the spin-1 Heisenberg OBC chain at finite N ≤ 8, computed from the dense thermal density matrix.

Spin-1 convention: S^z carries eigenvalues ±1, 0 (off-diagonal 1/√2 for S^x); see file header. The SU(2)-symmetric AFM ground state has ⟨S^α⟩ = 0 in every direction at any temperature, so this is a non-trivial check only for symmetry- breaking finite-β samples (and for benchmarking sampler bias).

fetch(model::S1Heisenberg1D, ::MagnetizationZLocal, ::OBC; beta) -> Vector{Float64}

Site-resolved [⟨S^z_i⟩_β for i = 1:N] of the spin-1 OBC Heisenberg chain. Sums to N · MagnetizationX/Y/Z (per-site bulk average) by construction.

fetch(model::S1Heisenberg1D, ::MassGap, ::Infinite) -> Float64

Bulk Haldane gap of the spin-1 antiferromagnetic Heisenberg chain,

Δ_∞ ≈ 0.41048 J

(literature value, S. R. White & D. A. Huse, Phys. Rev. B 48, 3844 (1993)). No closed form; reliability=:medium in the registry.

fetch(model::S1Heisenberg1D, ::MassGap, ::OBC) -> Float64

Single-particle gap of the spin-1 Heisenberg OBC chain at finite N ≤ 8,

Δ = E₁ - E₀

between the lowest two eigenvalues of the dense Hamiltonian. At fixed N this contains finite-size and edge-state corrections relative to the bulk Haldane gap Δ_∞ ≈ 0.41048 J.

fetch(model::S1Heisenberg1D, q::RenyiEntropy, ::OBC;
      ℓ::Int, beta::Real = Inf) -> Float64

Rényi entropy S_α = log Tr ρ_A^α / (1 - α) for the OBC spin-1 Heisenberg chain. See VonNeumannEntropy for the partial-trace convention.

fetch(model::S1Heisenberg1D, ::SusceptibilityXX, ::OBC; beta) -> Float64

Per-site uniform XX susceptibility χ_xx(β) = β · Var(M_x) / N of the spin-1 Heisenberg OBC chain via the dense thermal density matrix.

At infinite temperature each site contributes Tr((Sᵅ)²)/3 = 2/3 so χ_αα → 2β/3 for any axis.

fetch(model::S1Heisenberg1D, ::SusceptibilityYY, ::OBC; beta) -> Float64

Per-site uniform YY susceptibility χ_yy(β) = β · Var(M_y) / N of the spin-1 Heisenberg OBC chain via the dense thermal density matrix.

At infinite temperature each site contributes Tr((Sᵅ)²)/3 = 2/3 so χ_αα → 2β/3 for any axis.

fetch(model::S1Heisenberg1D, ::SusceptibilityZZ, ::OBC; beta) -> Float64

Per-site uniform ZZ susceptibility χ_zz(β) = β · Var(M_z) / N of the spin-1 Heisenberg OBC chain via the dense thermal density matrix.

At infinite temperature each site contributes Tr((Sᵅ)²)/3 = 2/3 so χ_αα → 2β/3 for any axis.

fetch(model::S1Heisenberg1D, ::VonNeumannEntropy{:equilibrium}, ::OBC;
      ℓ::Int, beta::Real = Inf) -> Float64

Von Neumann entanglement entropy S_vN = -Tr ρ_A log ρ_A of the first ℓ sites of the OBC spin-1 Heisenberg chain. The reduced density matrix ρ_A is the partial trace of the thermal density matrix ρ = exp(-βH) / Z (or the GS projector when beta = Inf) over sites ℓ+1 .. N.

Both ℓ and N = bc.N are bounded by _MAX_ED_SITES_S1 because the full 3^N × 3^N ρ is built explicitly.

fetch(model::S1Heisenberg1D, ::XXCorrelation{:connected}, ::OBC;
      beta, i::Int, j::Int) -> Float64

Connected (cumulant) correlator ⟨S^x_i S^x_j⟩ - ⟨S^x_i⟩·⟨S^x_j⟩ for the spin-1 OBC Heisenberg chain.

fetch(model::S1Heisenberg1D, ::XXCorrelation{:static}, ::OBC;
      beta, i::Int, j::Int) -> Float64

Static thermal correlator ⟨S^x_i S^x_j⟩_β of the spin-1 OBC Heisenberg chain. Caller passes both site indices 1 ≤ i, j ≤ N; equal sites give ⟨(S^α)²⟩_β.

fetch(model::S1Heisenberg1D, ::YYCorrelation{:connected}, ::OBC;
      beta, i::Int, j::Int) -> Float64

Connected (cumulant) correlator ⟨S^y_i S^y_j⟩ - ⟨S^y_i⟩·⟨S^y_j⟩ for the spin-1 OBC Heisenberg chain.

fetch(model::S1Heisenberg1D, ::YYCorrelation{:static}, ::OBC;
      beta, i::Int, j::Int) -> Float64

Static thermal correlator ⟨S^y_i S^y_j⟩_β of the spin-1 OBC Heisenberg chain. Caller passes both site indices 1 ≤ i, j ≤ N; equal sites give ⟨(S^α)²⟩_β.

fetch(model::S1Heisenberg1D, ::ZZCorrelation{:connected}, ::OBC;
      beta, i::Int, j::Int) -> Float64

Connected (cumulant) correlator ⟨S^z_i S^z_j⟩ - ⟨S^z_i⟩·⟨S^z_j⟩ for the spin-1 OBC Heisenberg chain.

fetch(model::S1Heisenberg1D, ::ZZCorrelation{:static}, ::OBC;
      beta, i::Int, j::Int) -> Float64

Static thermal correlator ⟨S^z_i S^z_j⟩_β of the spin-1 OBC Heisenberg chain. Caller passes both site indices 1 ≤ i, j ≤ N; equal sites give ⟨(S^α)²⟩_β.

heisenberg_spinon_dispersion(model::Heisenberg1D, k::Real; J::Real = 1.0)
    -> Float64

Single-spinon dispersion of the spin-½ antiferromagnetic Heisenberg chain in the thermodynamic limit (Faddeev–Takhtajan 1981):

ε(k) = (π J / 2) |sin k|,   k ∈ [0, π].

Spinons are massless half-odd-integer-spin excitations and the lower edge of the two-spinon continuum coincides with this dispersion, see heisenberg_two_spinon_lower_edge.

Special values:

• ε(0) = 0 — gapless point.
• ε(π/2) = π J / 2 — band centre maximum.
• ε(π) = 0 — gapless Umklapp point.

J is passed by keyword because Heisenberg1D is parameterless in this codebase (every other quantity threads J through kwargs the same way).

References

L. D. Faddeev, L. A. Takhtajan, "What is the spin of a spin wave?",
  Phys. Lett. A 85, 375 (1981).

heisenberg_two_spinon_lower_edge(model::Heisenberg1D, q::Real;
                                 J::Real = 1.0) -> Float64

Lower edge of the two-spinon continuum at total momentum q for the spin-½ XXX AFM chain (des Cloizeaux–Pearson 1962):

ε_L(q) = (π J / 2) |sin q|.

By construction ε_L(q) ≡ ε(q) where ε is the single-spinon dispersion (heisenberg_spinon_dispersion) — this reflects the kinematic configuration in which one of the two spinons sits at zero energy, so the total energy equals the energy of the other.

Special values:

• ε_L(0) = ε_L(π) = 0 — gapless points (Umklapp included).
• ε_L(π/2) = π J / 2 — maximum of the lower edge.

References

J. des Cloizeaux, J. J. Pearson, "Spin-wave spectrum of the
  antiferromagnetic linear chain", Phys. Rev. 128, 2131 (1962).

heisenberg_two_spinon_upper_edge(model::Heisenberg1D, q::Real;
                                 J::Real = 1.0) -> Float64

Upper edge of the two-spinon continuum at total momentum q for the spin-½ XXX AFM chain (des Cloizeaux–Pearson 1962):

ε_U(q) = π J |sin(q/2)|.

Special values:

• ε_U(0) = 0 — Goldstone-like vanishing of the support window at q = 0.
• ε_U(π) = π J — top of the continuum at the Umklapp point.

For q ∈ (0, π] one has ε_U(q) ≥ ε_L(q) strictly except at the gapless points where the continuum collapses to a line.

References

J. des Cloizeaux, J. J. Pearson, "Spin-wave spectrum of the
  antiferromagnetic linear chain", Phys. Rev. 128, 2131 (1962).

HeisenbergXYZ(; Jx::Real = 1.0, Jy::Real = 1.0, Jz::Real = 1.0)
    <: AbstractQAtlasModel

Spin-½ XYZ chain with three independent exchange couplings

H = Σ_i [ Jx Sˣᵢ Sˣᵢ₊₁ + Jy Sʸᵢ Sʸᵢ₊₁ + Jz Sᶻᵢ Sᶻᵢ₊₁ ].

Most-general 1-D nearest-neighbour spin-½ integrable model (Baxter 1972). This release exposes only the axis-aligned reductions Jx = Jy by delegating to the existing XXZ1D machinery:

HeisenbergXYZ(Jx = J, Jy = J, Jz)   ≡   XXZ1D(J = J, Δ = Jz / J).

General XYZ ground-state energy (Baxter elliptic Bethe ansatz) is tracked as a follow-up phase and raises DomainError here.

Quantities registered:

References

• R. J. Baxter, Annals Phys. 70, 193 (1972).
• L. D. Faddeev, L. A. Takhtajan, J. Soviet Math. 24, 241 (1984).

fetch(m::HeisenbergXYZ, ::CorrelationLength, ::Infinite;
      Jx=m.Jx, Jy=m.Jy, Jz=m.Jz) -> Float64

Asymptotic correlation length of the spin-1/2 XYZ chain at the thermodynamic limit.

References

• E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961).
• B. M. McCoy, T. T. Wu, Phys. Rev. 174, 546 (1968).

fetch(m::HeisenbergXYZ, ::Energy{:per_site}, ::Infinite;
      Jx=m.Jx, Jy=m.Jy, Jz=m.Jz, kwargs...) -> Float64

Ground-state energy per site of the spin-½ XYZ chain in the thermodynamic limit, restricted to the axis-aligned reduction Jx = Jy:

HeisenbergXYZ(Jx = J, Jy = J, Jz)   ⟶   fetch(XXZ1D(J = J, Δ = Jz/J), …).

This routes the call through the XXZ1D Yang-Yang single integral (general -1 < Δ < 1) and the three closed-form points Δ ∈ {-1, 0, 1} already implemented in XXZ1D. General (Jx ≠ Jy) triples require the Baxter elliptic Bethe ansatz and currently raise DomainError — Phase 2.

Jx > 0 is required (so Delta = Jz/Jx is well-defined and the delegation routes into the XXZ1D-tested domain; FM-exchange Jx < 0 is Phase 2).

References

• C. N. Yang, C. P. Yang, Phys. Rev. 150, 327 (1966).
• R. J. Baxter, Annals Phys. 70, 193 (1972).

fetch(m::HeisenbergXYZ, ::GroundStateEnergyDensity, ::Infinite;
      Jx=m.Jx, Jy=m.Jy, Jz=m.Jz) -> Float64

Ground-state energy density (per site) of the spin-1/2 XYZ chain at the thermodynamic limit. Supported parameter regimes:

References

• E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961).
• C. N. Yang, C. P. Yang, Phys. Rev. 150, 327 (1966).
• R. J. Baxter, Ann. Phys. 70, 193 (1972) – generic XYZ via elliptic Bethe ansatz.

fetch(m::HeisenbergXYZ, ::LuttingerParameter, ::Infinite;
      Jx=m.Jx, Jy=m.Jy, Jz=m.Jz) -> Float64

Luttinger-liquid parameter at the isotropic Heisenberg point Jx = Jy = Jz, delegated to XXZ1D(Δ = 1):

K = 1/2          (SU(2)-symmetric AFM, Luther-Peschel 1975)

The delegation chain is HeisenbergXYZ → XXZ1D(Δ=1) (matching the Energy(:per_site) reduction). Once a dedicated fetch(::Heisenberg1D, ::LuttingerParameter, ::Infinite) lands on main (tracked separately), the intermediate Heisenberg1D hop can be reinstated; the final K is unchanged.

For Jx = Jy ≠ Jz (XXZ axial anisotropy), use XXZ1D directly. For generic XYZ (Baxter 1972 elliptic / theta-function regime), defer to a later phase — this Phase-2 path throws DomainError for non-isotropic couplings.

References

• A. Luther, I. Peschel, Phys. Rev. B 12, 3908 (1975).
• R. J. Baxter, Ann. Phys. 70, 193 (1972) — elliptic XYZ.

fetch(m::HeisenbergXYZ, ::MassGap, ::Infinite;
      Jx=m.Jx, Jy=m.Jy, Jz=m.Jz) -> Float64

Single-particle mass gap (smallest excitation energy above the ground state) of the spin-1/2 XYZ chain at the thermodynamic limit.

References

• E. Lieb, T. Schultz, D. Mattis, Ann. Phys. 16, 407 (1961).
• C. N. Yang, C. P. Yang, Phys. Rev. 150, 327 (1966).
• A. Luther, I. Peschel, Phys. Rev. B 12, 3908 (1975).
• R. J. Baxter, Ann. Phys. 70, 193 (1972).

fetch(m::HeisenbergXYZ, ::SpontaneousMagnetization, ::Infinite;
      Jx=m.Jx, Jy=m.Jy, Jz=m.Jz) -> Float64

Spontaneous magnetization (sublattice order parameter) of the spin-1/2 XYZ chain at the thermodynamic limit.

References

• B. M. McCoy, T. T. Wu, Phys. Rev. 174, 546 (1968).
• P. Pfeuty, Ann. Phys. 57, 79 (1970).
• R. J. Baxter, Phys. Rev. Lett. 31, 1294 (1973) – generic XYZ.

J1J2Heisenberg1D(; J1::Real = 1.0, J2::Real = 0.5) <: AbstractQAtlasModel

Spin-½ J₁-J₂ Heisenberg chain:

H = J₁ Σ_i Sᵢ · Sᵢ₊₁ + J₂ Σ_i Sᵢ · Sᵢ₊₂,    J₁ > 0,  J₂ ≥ 0.

Define the frustration ratio j = J₂ / J₁. Phase 1 implements only the two closed-form points:

• j = 0 → Bethe-Hulthén E/N = J₁ (1/4 − ln 2),
• j = 1/2 → Majumdar-Ghosh E/N = −3 J₁ / 8.

Generic j (0 < j < ∞, j ∉ {0, 1/2}) requires DMRG; deferred to Phase 2, raises DomainError from fetch(..., Energy{:per_site}(), Infinite()).

The default constructor J1J2Heisenberg1D() lands on J₁ = 1, J₂ = 1/2 — the Majumdar-Ghosh point, the most physically interesting closed-form case.

Fields

• J1::Float64 — nearest-neighbour antiferromagnetic exchange (J₁ > 0).
• J2::Float64 — next-nearest-neighbour exchange (J₂ ≥ 0).

References

• H. Bethe, Z. Physik 71, 205 (1931).
• L. Hulthén, Ark. Mat. Astron. Fys. 26A, No. 11 (1938).
• C. K. Majumdar, D. K. Ghosh, J. Math. Phys. 10, 1388 (1969).
• S. R. White, I. Affleck, Phys. Rev. B 54, 9862 (1996).

fetch(::J1J2Heisenberg1D, ::Energy{:per_site}, ::Infinite;
      J1=m.J1, J2=m.J2) -> Float64

Ground-state energy density of the spin-½ J₁-J₂ Heisenberg chain in the thermodynamic limit. Phase 1 supports only the two closed-form points:

• j = J₂/J₁ = 0 → Bethe-Hulthén: E/N = J₁ · (1/4 − ln 2). Delegates to fetch(Heisenberg1D(), GroundStateEnergyDensity(), Infinite(); J=J1).

• j = 1/2 → Majumdar-Ghosh dimer GS: E/N = −3 J₁ / 8. Delegates to fetch(MajumdarGhosh(; J=J1), GroundStateEnergyDensity(), Infinite()).

• otherwise → DomainError: no closed form; numerical DMRG required, deferred to Phase 2.

Floating-point tolerance for the j = 0 and j = 1/2 matches is atol = 1e-12.

Note on delegation: Heisenberg1D and MajumdarGhosh currently expose their thermodynamic-limit energy density via the legacy GroundStateEnergyDensity quantity, not the modern Energy{:per_site} axis. This wrapper bridges to the modern axis on the public surface while keeping the closed-form constants in a single source-of-truth location (the delegate model).

References

• L. Hulthén, Ark. Mat. Astron. Fys. 26A, No. 11 (1938) — Bethe-Hulthén ground-state energy density.
• C. K. Majumdar, D. K. Ghosh, J. Math. Phys. 10, 1388 (1969) — exact dimer ground state at j = 1/2.
• S. R. White, I. Affleck, Phys. Rev. B 54, 9862 (1996) — DMRG study of generic j (deferred to Phase 2).