QAtlas.jl

REGISTRY :: Vector{Implementation}

Module-level mutable vector populated at include-time by @register calls scattered across src/models/.../<Model>_registry.jl files. Public read API: implementation_status.

_MAX_ED_SITES = 12

Hard cap on chain length for dense-ED helpers. 2^12 = 4096 → 4096×4096 Hermitian eigendecomposition takes ~1 s on a laptop; 2^14 ~ 16384 is already several seconds and 64 GB+ of workspace, which is past the "cheap reference data" regime this helper targets.

_MAX_ED_SITES_S1 = 8

Hard cap on chain length for spin-1 dense-ED helpers. 3⁸ = 6561 → a 6561×6561 Hermitian eigendecomposition is ~1 s on a laptop; 3⁹ ≈ 20 k already pushes 5 GB of workspace, past the "cheap reference" regime.

AKLT1D(; J::Real = 1.0) <: AbstractQAtlasModel

Spin-1 bilinear-biquadratic chain at the AKLT point,

H = J Σᵢ [ Sᵢ · Sᵢ₊₁ + (1/3) (Sᵢ · Sᵢ₊₁)² ],

with J > 0 antiferromagnetic. Same local Hilbert space as S1Heisenberg1D but with the biquadratic coefficient tuned to the special AKLT value 1/3 where the ground state is an exact Valence Bond Solid (VBS). The infinite-system observables exposed through fetch are closed-form (energy density, correlation length, string order parameter); the Haldane gap is the García-Saez–Murg–Verstraete (2013) DMRG numerical-exact value.

AKLT2D(; J::Real = 1.0) <: AbstractQAtlasModel

2D honeycomb-lattice Affleck-Kennedy-Lieb-Tasaki (AKLT) spin-3/2 valence-bond-solid model. Frustration-free Hamiltonian whose unique gapped VBS ground state has exact energy density e0/N = 0 (J-independent); the 2D AKLT phase is the prototypical 2D symmetry-protected topological (SPT) state. See file header for full references.

const AbstractModel = AbstractQAtlasModel

Backward-compatible alias. Existing downstream code dispatches on ::AbstractModel; new code should use ::AbstractQAtlasModel directly or — preferably — a concrete model struct.

AbstractQAtlasModel

Abstract parent type for every QAtlas model. Concrete subtypes carry their physics parameters as typed fields (e.g. struct TFIM <: AbstractQAtlasModel; J::Float64; h::Float64; end).

The older Model{S}(params::Dict) phantom-typed wrapper is still an AbstractQAtlasModel (via the deprecated alias below) but new models must use concrete structs.

AbstractQuantity

Abstract parent type for quantities. New code defines concrete structs (e.g. struct MagnetizationX <: AbstractQuantity end) so dispatch is static and naming is explicit (axis, entropy variant, …). The older Quantity{S} phantom-type wrapper is retained for legacy symbol-based dispatch; see the Quantity(::Symbol) shim below.

AnyonStatistics() <: AbstractQuantity

Topological data of the anyon content of a 2D topologically ordered phase. The dispatched fetch method takes a type::Symbol kwarg selecting an anyon (or a mutual-braiding row) and returns a NamedTuple whose shape depends on the requested row — typically (label, statistics, self_phase, quantum_dim, fusion) for individual anyons, and (label, mutual_phase, anyons) for two-anyon braids.

For Abelian theories like the toric code's Z₂ order all quantum dimensions are 1; for non-Abelian theories the schema is unchanged but quantum_dim becomes irrational and additional fields (e.g. F, R matrices) may be added by the implementing method.

Has no boundary-condition argument: anyon statistics are purely topological invariants.

BCFT() <: AbstractQAtlasModel

Boundary Conformal Field Theory with Cardy (1989) boundary states and Affleck-Ludwig (1991) g-functions.

Phase 1 ships only Ising-CFT Cardy boundary states; other CFTs (minimal models, free boson, WZW, …) are tracked as Phase 2.

Quantities registered (Phase 1):

The Cardy state is selected via the state keyword to fetch:

• state=:fixed, :fixed_plus, :fixed_minus ⟹ g = 1/√2, log g = -log(2)/2 (physical fixed-spin boundaries, ≡ (|1⟩ ± |ε⟩)/√2)
• state=:free ⟹ g = 1, log g = 0 (physical free boundary, ≡ the |σ⟩ Cardy state)
• state=:sigma ⟹ g = 1, log g = 0 (|σ⟩ Cardy state — alias of :free)
• state=:identity, :vacuum ⟹ g = 1/√2, log g = -log(2)/2 (|1⟩ Cardy state)
• state=:epsilon, :energy ⟹ g = 1/√2, log g = -log(2)/2 (|ε⟩ Cardy state)

References

• J. L. Cardy, Nucl. Phys. B 324, 581 (1989).
• I. Affleck, A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 (1991).
• D. Friedan, A. Konechny, Phys. Rev. Lett. 93, 030402 (2004).

BoundaryCondition

Abstract parent type. The three concrete subtypes carry system-size information where applicable, so fetch can read it from the BC instead of kwargs.

• Infinite — thermodynamic limit; no size.
• PBC(N::Int) — periodic boundary conditions at finite N.
• OBC(N::Int) — open boundary conditions at finite N.

For backward compatibility, the zero-argument constructors PBC() and OBC() exist and set N = 0, which signals "caller will pass N via kwargs" — legacy fetch methods still look at kwargs[:N]. New fetch methods read bc.N directly.

CasimirEnergyCorrection() <: AbstractQuantity

Universal 1/L finite-size correction to the ground-state energy of a 1+1D conformal field theory.

For a critical 1+1D system with central charge c and CFT velocity v on a system of size L:

• Periodic boundary (PBC): $E_0(L) = L\,\varepsilon_\infty - \dfrac{\pi c v}{6 L} + O(L^{-2})$
• Open boundary (OBC): $E_0(L) = L\,\varepsilon_\infty + \varepsilon_{\mathrm{surf}} - \dfrac{\pi c v}{24 L} + O(L^{-2})$

This quantity returns only the universal $1/L$ correction term ($-\pi c v/(6 L)$ at PBC, $-\pi c v/(24 L)$ at OBC), not the extensive $L \varepsilon_\infty$ piece nor the OBC surface term $\varepsilon_{\mathrm{surf}}$. The PBC-to-OBC ratio is exactly 4, independent of the universality class.

The CFT velocity v is model-dependent (e.g. $v = 2J$ for the TFIM at the critical point, $v = (\pi/2) J$ for the AFM Heisenberg chain, $v = v_F$ for the XXZ Luttinger liquid) and is supplied by the caller as a kwarg. The central charge c is read from the universality class via the same data the Universality{C} entry exposes for CriticalExponents.

References

• J. Cardy, Nucl. Phys. B 270, 186 (1986).
• H. W. J. Blöte, J. L. Cardy, M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986).
• I. Affleck, Phys. Rev. Lett. 56, 746 (1986).

CentralCharge() <: AbstractQuantity

Central charge c of the emergent CFT. For 1D critical systems extracted from the Calabrese–Cardy entanglement formula; universality pages return literature values.

ChargeGap() <: AbstractQuantity

Charge (Mott) gap of an electron system,

Δ_c = E₀(N+1) + E₀(N-1) - 2 E₀(N),

i.e. the energy cost of adding a particle plus the cost of removing one, equivalent to the gap between the half-filled ground state and the lowest charged excitation. Strictly positive in a Mott insulator and exactly zero in a metal / superconductor.

Implemented analytically for Hubbard1D at half filling via the Lieb–Wu (1968) closed-form integral.

ChernSimons3D(; N::Integer = 2, k::Integer = 1) <: AbstractQAtlasModel

3-D SU(N)k Chern-Simons TQFT (Witten 1989). N ≥ 2 is the gauge- group rank-plus-one and `k ∈ ℤ{>0}` is the (integer) Chern-Simons level.

Phase 1 exposed the boundary WZW central charge via the Sugawara construction. Phase 2 adds the closed-form S³ partition function Z(S³; SU(N)_k) = S_{0,0} (Witten 1989 / Verlinde formula). Wilson-loop knot invariants and modular S / T matrices remain tracked for later phases.

Quantities registered:

References

• E. Witten, Comm. Math. Phys. 121, 351 (1989).
• V. G. Knizhnik, A. B. Zamolodchikov, Nucl. Phys. B 247, 83 (1984).

ChiralCondensate() <: AbstractQuantity

Vacuum expectation value ⟨ψ̄ψ⟩ of a fermion bilinear, signalling spontaneous (anomalous) chiral-symmetry breaking. The massless Schwinger model is the canonical 1+1-D example: even though the classical Lagrangian is chirally symmetric, the anomaly forces a non-zero condensate

⟨ψ̄ψ⟩ = − exp(γ_E) · e / (2π^{3/2}),    m_γ = e/√π.

(Schwinger 1962; Coleman-Jackiw-Susskind 1975.)

Cluster1D(; J::Real = 1.0) <: AbstractQAtlasModel

1D Z₂×Z₂ symmetry-protected topological cluster Hamiltonian H = -J Σ_i σ^z_{i-1} σ^x_i σ^z_{i+1}. Ground state is the cluster state (Briegel-Raussendorf 2001).

Quantities registered (Phase 1):

References

• H. J. Briegel, R. Raussendorf, Phys. Rev. Lett. 86, 910 (2001).
• D. V. Else, S. D. Bartlett, A. C. Doherty, New J. Phys. 14, 113016 (2012).
• R. Verresen, R. Moessner, F. Pollmann, Phys. Rev. B 96, 165124 (2017).

Compass1D(; J_x::Real = 1.0, J_y::Real = 1.0) <: AbstractQAtlasModel

1D alternating-bond compass chain

H = -J_x Σ_{i odd}  σ^x_i σ^x_{i+1}
    -J_y Σ_{i even} σ^y_i σ^y_{i+1},

with J_x, J_y > 0. Dual to a dimerised Kitaev / p-wave wire via Jordan–Wigner; exactly solvable.

Phase 1 scope (this release)

Phase 1 exposes the closed-form bulk gap only:

• MassGap at Infinite() → Δ = 2 |J_x − J_y|.

The gap closes at the symmetric point J_x = J_y, which is a first-order quantum phase transition between the X-bond-dimerised and Y-bond-dimerised ground states (Brzezicki–Dziarmaga–Oleś 2007).

References

• Brzezicki–Dziarmaga–Oleś, PRB 75, 134415 (2007).
• Kugel–Khomskii, Sov. Phys. Usp. 25, 231 (1982).
• Kitaev, Annals of Physics 321, 2 (2006).

ConformalBootstrap() <: AbstractQAtlasModel

The 3D Ising critical point with high-precision conformal-bootstrap scaling dimensions. Methodology: Kos-Poland-Simmons-Duffin 2014 (mixed-correlator bootstrap). Precise values: KPSD-Vichi 2016 "Precision Islands" (arXiv:1603.04436). Parameterless: the 3D Ising universality class is unique.

Phase 1 ships only the 3D Ising universality class; other bootstrap-pinned CFTs (O(N) vector models, stress-tensor and higher-spin multiplets, fermionic CFTs) are deferred to Phase 2.

Quantities registered (Phase 1):

Available kwargs for ConformalWeights:

• field = :σ — lowest scalar primary, Δ_σ = 0.51814894.
• field = :ε — lowest energy primary, Δ_ε = 1.41262528.

References

• F. Kos, D. Poland, D. Simmons-Duffin, JHEP 06 (2014) 091 — methodology (mixed-correlator bootstrap).
• F. Kos, D. Poland, D. Simmons-Duffin, A. Vichi, JHEP 08 (2016) 036; arXiv:1603.04436 — "Precision Islands", precise Δσ, Δε.
• D. Simmons-Duffin, JHEP 03 (2017) 086 — lightcone-bootstrap cross-check.
• M. Reehorst, S. Rychkov, D. Simmons-Duffin, B. Sirois, N. Su, B. van Rees, SciPost Phys. 11 (2021) 072.

ConformalWeights() <: AbstractQuantity

Primary scaling dimension h of a 2D rational CFT. For Virasoro MinimalModel this is the Kac-table entry h_{r,s}; for WZWSU2 it is the SU(2)-spin label h_j = j(j+1)/(k+2).

Concrete model fetch methods take additional keyword arguments identifying the primary (r, s for MinimalModel; j for WZWSU2) and return an exact Rational{Int}.

CorrelationLength() <: AbstractQuantity

Two-point correlation length ξ controlling the exponential decay of connected equal-time correlators in a gapped phase,

⟨σ_α(0) σ_α(r)⟩_c ~ e^{-r/ξ}    (r → ∞).

For a critical system ξ = ∞; implementations return Inf in that case. At T = 0 and 1D free-fermion models like TFIM, ξ is set by the inverse mass gap (ξ = 1/(2|h - J|)).

CriticalExponents() <: AbstractQuantity

Standard set of equilibrium critical exponents {α, β, γ, δ, ν, η} of a universality class. Returns a NamedTuple.

For exact values: fields are Rational{Int}. For numerical estimates: fields are Float64 with corresponding _err fields (e.g., β_err) giving the uncertainty.

CriticalTemperature() <: AbstractQuantity

Exact critical temperature of a classical model.

CurieWeissIsing(; J::Real = 1.0, h::Real = 0.0) <: AbstractQAtlasModel

Classical Ising model on the complete graph (mean field) with saddle-point Hamiltonian

H = -(J/N) Σ_{i<j} σ_i σ_j  -  h Σ_i σ_i.

The 1/N normalisation makes the energy extensive and the thermodynamic limit well-defined. At h = 0 and J > 0 the model has zero-field critical temperature T_c = J and mean-field critical exponents (β, γ, δ, ν) = (1/2, 1, 3, 1/2) exposed by Universality(:MeanField).

For h ≠ 0 there is no sharp transition; the equilibrium magnetisation m*(β, J, h) is the unique stable root of the self-consistency equation m = tanh(β(Jm + h)) (same sign as h).

Quantities registered:

References

• L. D. Landau, E. M. Lifshitz, Statistical Physics §149.

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).

E8() <: AbstractQAtlasModel

The E8 integrable field theory — the low-energy continuum theory of the 2D Ising model perturbed by a small longitudinal magnetic field at T = T_c (Zamolodchikov 1989). The only physics parameter is the underlying magnetic field perturbation strength, but QAtlas currently only exposes the universal mass-ratio spectrum, so this struct carries no fields.

E8Spectrum() <: AbstractQuantity

Zamolodchikov E8 mass spectrum (8 stable particles). Concrete implementation lives in src/universalities/E8.jl; the type is defined here so src/core/alias.jl can reference it without circular loads.

EdgeModeEnergy() <: AbstractQuantity

Energy of the lowest-lying boundary mode on an open chain. In a topological 1D superconductor (Kitaev 2001) the OBC chain hosts two Majorana zero modes at the chain ends; their hybridization energy decays exponentially with chain length,

\[E_\text{edge}(N) \sim e^{-N/\xi},\]

where ξ is the bulk correlation length. In the trivial phase the OBC lowest single-particle excitation is set by the bulk gap.

EdgeModeEnergy is the smallest positive BdG eigenvalue of the OBC chain — the same quantity as MassGap at OBC, exposed under a name that makes the boundary-mode interpretation explicit at the call site.

Currently used by Kitaev1D.

Energy{G}() <: AbstractQuantity
Energy()                 # G = :natural — model-and-BC-natural granularity
Energy(:total)           # explicit ⟨H⟩
Energy(:per_site)        # explicit ⟨H⟩ / N

Ground-state / thermal energy expectation. The type parameter G makes the granularity (total vs per-site) a dispatch axis instead of a hidden docstring contract.

Energy() resolves to the model's native granularity via the native_energy_granularity trait — keeping every existing fetch(model, Energy(), bc; ...) call site working unchanged. Use the explicit constructors when the caller needs a specific granularity (e.g. the thermodynamic-identity harness comparing f + T·s against per-site ε).

The non-native granularity is provided automatically by a generic conversion fallback for 1D BCs (OBC / PBC) that uses _bc_size. Models on lattices whose size is not captured by bc.N (e.g. 2D Kitaev with Lx, Ly kwargs) currently support only their declared native granularity.

EnergyLocal() <: AbstractQuantity

Bond-resolved energy density vector, length N_bulk − 1 for a bond Hamiltonian Σ_b h_b.

ExactSpectrum

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

ExtendedHubbard1D(; t::Real = 1.0, U::Real = 4.0, V::Real = 0.0)
    <: AbstractQAtlasModel

1D t-U-V Hubbard chain (nearest-neighbour-density extension of the standard Hubbard model):

H = -t Σ_{i, σ} (c†_{i,σ} c_{i+1,σ} + h.c.)
    + U Σ_i n_{i,↑} n_{i,↓}
    + V Σ_i n_i n_{i+1}.

Convention: t > 0 hopping, U on-site, V nearest-neighbour density-density.

Phase 1 scope (this release)

Phase 1 exposes only the V = 0 limit, at which the model reduces to the standard 1D Hubbard chain. The ChargeGap at Infinite() is delegated to Hubbard1D at half filling (μ = U/2), i.e. the Lieb–Wu (1968) integral

Δ_c = (16 t² / U) ∫_1^∞ dω  √(ω² - 1) / sinh(2π t ω / U).

Any V ≠ 0 raises DomainError. The V ≠ 0 phase diagram (CDW / SDW / BOW / phase separation, Voit 1995; Nakamura 2000) is deferred to Phase 2.

References

• Lieb, Wu, Phys. Rev. Lett. 20, 1445 (1968).
• Voit, Rep. Prog. Phys. 58, 977 (1995).
• Nakamura, Phys. Rev. B 61, 16377 (2000).

FermiVelocity() <: AbstractQuantity

Fermi velocity v_F = ∂ε/∂k |_{k_F}. Meaningful for non-interacting / mean-field fermionic band structures (tight-binding lattices, Bogoliubov-de Gennes diagonalisations). In QAtlas this is the type returned by models like Honeycomb (at the Dirac cones), the other tight-binding lattices, and the TFIM Majorana mode at the critical field.

FibonacciAnyons <: AbstractQAtlasModel

Non-Abelian Fibonacci anyon model.

The anyon spectrum has two charges {1, τ} with the fusion rule

τ × τ = 1 + τ

and quantum dimensions

d_1 = 1,   d_τ = φ = (1 + √5)/2  (golden ratio).

Total quantum dimension

𝒟 = √(d_1² + d_τ²) = √(1 + φ²) = √(φ + 2)

(using φ² = φ + 1).

Fibonacci anyons are universal for topological quantum computation (Freedman-Kitaev-Larsen-Wang 2003) and arise as edge excitations of the Read-Rezayi Z_3 parafermion state (Read-Rezayi 1999).

This model has no continuous parameters — the spectrum and quantum dimensions are fixed by the fusion rules.

References

• M. Freedman, A. Kitaev, M. Larsen, Z. Wang, Bull. Amer. Math. Soc. 40, 31 (2003).
• N. Read, E. Rezayi, Phys. Rev. B 59, 8084 (1999).
• A. Kitaev, J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
• M. Levin, X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).

FidelitySusceptibility() <: AbstractQuantity

Fidelity susceptibility χ_F(λ) = −∂²⟨ψ(λ)|ψ(λ + δλ)⟩/∂δλ².

FractalDimension() <: AbstractQuantity

Hausdorff dimension d_H of the random geometric set associated with a model — e.g. the SLEκ curve's `dH(κ) = min(2, 1 + κ/8)` (Beffara 2008). Real-valued, dimensionless, capped at the ambient space dimension.

FreeEnergy() <: AbstractQuantity

Helmholtz free energy per site, f = -β⁻¹ log Z / N.

GGEValue{Q<:AbstractQuantity}(inner) <: AbstractQuantity

Wrapper quantity carrying an underlying observable inner::Q whose generalised Gibbs ensemble (GGE) stationary value is to be computed — i.e. the t → ∞ long-time average that an integrable (free-fermion) quench reaches.

For an integrable system the ordinary (canonical) Gibbs ensemble does not describe the long-time relaxed state: every mode-occupation n_k = ⟨c_k† c_k⟩ is a separate conserved quantity, so the diagonal ensemble is a generalised Gibbs ensemble fixed by the full distribution {n_k}. See Rigol et al. PRL 98, 050405 (2007) for the foundational argument and Calabrese, Essler, Fagotti J. Stat. Mech. (2012) P07016 / P07022 for the TFIM-specific closed-form expressions.

fetch(model_f, ::GGEValue{Q}, bc; initial::ModelType, kwargs...) returns the GGE expectation of the Q observable in the post-quench Hamiltonian model_f, with the conserved mode occupations frozen by the initial-state (initial) Bogoliubov rotation.

Construction

GGEValue(Energy())                  # ⟨H_f⟩ stationary value
GGEValue(MagnetizationX())          # ⟨σˣ⟩ stationary value

Fetch signature (TFIM)

fetch(TFIM(h = h_f), GGEValue(Energy()), Infinite();
      initial = TFIM(h = h_0)) -> Float64

A no-quench limit h_0 = h_f reduces to the static ground-state value of the inner observable.

const Graphene = Honeycomb

Backward-compatible alias for Honeycomb. New code should prefer the lattice-named model.

GrossNeveu(; N::Integer = 1, g::Real = 0.0) <: AbstractQAtlasModel

1+1-D Gross-Neveu model (Gross-Neveu 1974) with N Dirac flavours (equivalently O(2N) Majorana symmetry) and four-fermion coupling g ∈ ℝ. Asymptotically free for N ≥ 1.

Phase 1 registered the UV free-fermion central charge at g = 0. Phase 2 (#247) adds the large-N dynamical mass m_F = Λ exp(-π/(N g²)) via MassGap. Andrei-Lowenstein exact S-matrix and the chiral condensate remain tracked as Phase 3.

Quantities registered:

References

• D. J. Gross, A. Neveu, Phys. Rev. D 10, 3235 (1974).
• N. Andrei, J. H. Lowenstein, Phys. Rev. Lett. 43, 1698 (1979).

GroundStateDegeneracy() <: AbstractQuantity

Dimension of the ground-state subspace as an Int. In topologically ordered phases this is a robust, lattice-independent invariant determined by the ambient surface (e.g. 4^g on a closed orientable genus-g surface for the toric code) and is set by the kwarg genus on the fetch call. Trivially 1 for any gapped, symmetry-unbroken phase.

GroundStateEnergyDensity

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

GrowthExponents() <: AbstractQuantity

KPZ-type growth / roughness / dynamic exponents. Returns (β_growth, α_rough, z) instead of the equilibrium set.

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

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).

Honeycomb(; t::Real = 1.0, Lx::Int = 0, Ly::Int = 0) <: AbstractQAtlasModel

Nearest-neighbor tight-binding model on the honeycomb lattice (spinless, single-orbital) — historically known as graphene.

Hamiltonian: H = -t Σ_{⟨i,j⟩} (c†_i c_j + h.c.).

Fields:

• t — nearest-neighbor hopping amplitude (default 1.0).
• Lx, Ly — unit-cell counts in the two lattice directions. 0 is a sentinel meaning "caller passes via kwargs at fetch time".

For backward compatibility with the pre-v0.13 call form fetch(Graphene(), TightBindingSpectrum(); Lx, Ly, t=…), the fetch method still accepts Lx, Ly, t as kwargs (they override the struct fields when supplied). The legacy name Graphene is retained as a type alias in src/deprecate/legacy_honeycomb.jl.

Hubbard1D(; t::Real = 1.0, U::Real = 4.0, μ::Real = 2.0) <: AbstractQAtlasModel

1D Hubbard model

H = -t Σ_{i, σ} (c†_{i,σ} c_{i+1,σ} + h.c.)
    + U Σ_i n_{i,↑} n_{i,↓}
    - μ Σ_i n_i

Convention: t > 0 hopping, U > 0 on-site repulsion. Half filling corresponds to μ = U/2 (particle-hole-symmetric point).

Phase 1 scope (this release)

Phase 1 implements the Lieb–Wu (1968) closed-form integrals at half filling only (μ = U/2). The supported Infinite() quantities are

• GroundStateEnergyDensity — E₀/N via the Lieb–Wu integral,
• ChargeGap — Mott gap Δ_c via the Lieb–Wu second integral,
• SpinGap — 0 exactly (rigorous gapless-spinon result).

Off half filling raises a DomainError — the general-filling (coupled-integral-equation) and finite-temperature (QTM) surfaces are deferred to Phase 2.

Implementation

A single (model, quantity, bc) row of the QAtlas implementation registry. See @register for how rows are added and implementation_status for how to query them.

Infinite()

Thermodynamic-limit boundary condition — no finite size.

Ising2D <: AbstractQAtlasModel

Dispatch tag for the two-dimensional Ising universality class. Acts as a convenience wrapper over Universality(:Ising) with the dimension pinned to d = 2, so call sites do not have to repeat d = 2 on every fetch:

QAtlas.fetch(Ising2D(), CriticalExponents())
# ≡ QAtlas.fetch(Universality(:Ising), CriticalExponents(); d = 2)

The struct is kept as a distinct nominal type (rather than const Ising2D = Universality{:Ising}) because the d kwarg is fixed at 2 here — a type-level const alias would defeat that fix. Subtyped to AbstractQAtlasModel so it composes with the canonical fetch(::AbstractQAtlasModel, ::AbstractQuantity, ::BoundaryCondition) signature.

See also: Universality, CriticalExponents.

IsingChain1D(; J::Real = 1.0, h::Real = 0.0) <: AbstractQAtlasModel

Classical 1-D Ising chain with nearest-neighbour exchange J and uniform longitudinal field h:

H = -J Σ_i σ_i σ_{i+1} - h Σ_i σ_i,   σ_i ∈ {-1, +1}.

Solved exactly by the 2×2 transfer matrix (Ising 1925) with eigenvalues

λ_± = e^{β J} cosh(β h) ± √( e^{2 β J} sinh²(β h) + e^{-2 β J} ).

There is no finite-temperature phase transition.

Quantities registered:

References

• E. Ising, Z. Phys. 31, 253 (1925).

IsingSquare(; J::Real = 1.0, Lx::Int = 0, Ly::Int = 0) <: AbstractQAtlasModel

Classical 2D Ising model on a square lattice with periodic boundary conditions (PBC) in both directions.

Hamiltonian: H = -J Σ_{⟨i,j⟩} σᵢ σⱼ, σᵢ ∈ {-1, +1}.

Physics parameters are carried as typed struct fields:

• J — Ising coupling (default 1.0, J > 0 ferromagnetic).
• Lx, Ly — lattice extents. 0 is a legacy sentinel meaning "thermodynamic limit / unspecified"; finite-size quantities like PartitionFunction require both to be positive.

For backward compatibility the fetch methods accept Lx, Ly, J, β as kwargs too — kwargs override the struct fields when both are supplied.

See also: PartitionFunction, CriticalTemperature, SpontaneousMagnetization.

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

Classical 2D Ising model on the triangular lattice with the Wannier 1950 sign convention

H = +J Σ_{⟨i,j⟩} σ_i σ_j,   σ_i ∈ {-1, +1}.

The coupling sign determines the physics:

• J > 0 — antiferromagnetic, frustrated. The triangular plaquette cannot satisfy all three antiferromagnetic bonds simultaneously, so the ground-state manifold is macroscopically degenerate and there is no long-range order at any $T > 0$ (Wannier 1950).
• J < 0 — ferromagnetic. The lattice supports a standard order-disorder transition at $T_c = 4|J| / \ln 3$ (Houtappel 1950).
• J = 0 — non-interacting; degenerate value preserved for symmetry.

Quantities currently registered for this model:

See also: IsingSquare for the square-lattice (non-frustrated) counterpart with the Onsager / Yang closed forms.

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).

KPZ1D <: AbstractQAtlasModel

Dispatch tag for the 1+1-dimensional KPZ (Kardar–Parisi–Zhang) universality class. Acts as a convenience wrapper over Universality(:KPZ) with the dimension pinned to d = 1:

QAtlas.fetch(KPZ1D(), CriticalExponents())
# ≡ QAtlas.fetch(Universality(:KPZ), GrowthExponents(); d = 1)

The struct is kept as a distinct nominal type (rather than const KPZ1D = Universality{:KPZ}) because the dimension is fixed here — a type-level alias would defeat that fix. Subtyped to AbstractQAtlasModel so dispatch composes with the canonical fetch(::AbstractQAtlasModel, ::AbstractQuantity, ::BoundaryCondition) signature.

See also: Universality, GrowthExponents.

Kagome

Dispatch tag for the nearest-neighbor tight-binding model on the kagome lattice (spinless, single-orbital, three sublattices).

Hamiltonian: H = -t Σ{⟨i,j⟩} (c†i c_j + h.c.)

The spectrum exhibits a dispersionless flat band at E = +2t touching the lower dispersive bands at the Γ-point.

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

Spin-½ Heisenberg antiferromagnet on the Kagome lattice (highly frustrated, quantum spin liquid candidate).

Quantities registered (Phase 1, DMRG reference values):

Both registered with reliability :medium (different methods — DMRG vs variational Monte Carlo — give slightly different gap estimates; the precise spin-liquid character is an open question).

References

• S. Yan, D. A. Huse, S. R. White, Science 332, 1173 (2011).
• S. Depenbrock, I. P. McCulloch, U. Schollwöck, Phys. Rev. Lett. 109, 067201 (2012).
• Y. Iqbal, F. Becca, S. Sorella, D. Poilblanc, Phys. Rev. B 87, 060405(R) (2013).

Kitaev1D(; μ = 0.0, t = 1.0, Δ = 1.0) <: AbstractQAtlasModel

Kitaev (2001) one-dimensional p-wave superconducting wire,

\[H = -\mu \sum_i c_i^{\dagger} c_i - t \sum_i (c_i^{\dagger} c_{i+1} + \text{h.c.}) + \Delta \sum_i (c_i c_{i+1} + \text{h.c.}).\]

μ is the chemical potential, t the hopping, Δ the p-wave pairing. |μ| < 2|t| is the topological phase (Majorana edge modes); |μ| > 2|t| is the trivial phase; |μ| = 2|t| is the gapless critical line.

This is the 1D Majorana wire and is distinct from the 2D KitaevHoneycomb spin model.

The TFIM is a special case (μ = -2h, t = J, Δ = J); the BdG spectrum of Kitaev1D(μ=-2h, t=J, Δ=J) agrees exactly with that of TFIM(J=J, h=h) at OBC.

KitaevHeisenberg(; K::Real = 1.0, J::Real = 0.0, Γ::Real = 0.0)
    <: AbstractQAtlasModel

K-J-Γ honeycomb model (α-RuCl₃ family). Three independent exchanges on the three honeycomb bond directions:

H = Σ_{⟨ij⟩_γ} [ K Sᵢ^γ Sⱼ^γ + J Sᵢ · Sⱼ + Γ (Sᵢ^α Sⱼ^β + Sᵢ^β Sⱼ^α) ].

Quantities registered (Phase 1):

Phase 1 exposes only the K-only limit J = Γ = 0, delegated to the existing exactly-solvable KitaevHoneycomb entry. Any non-zero J or Γ raises DomainError pointing to the DMRG / ED phase-2 follow-up.

References

• A. Kitaev, Annals Phys. 321, 2 (2006).
• G. Jackeli, G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).
• J. G. Rau, E. K.-H. Lee, H.-Y. Kee, Phys. Rev. Lett. 112, 077204 (2014).

KitaevHoneycomb(; Kx=1.0, Ky=1.0, Kz=1.0) <: AbstractQAtlasModel

Kitaev honeycomb model with coupling amplitudes on the three bond types. Kx = Ky = Kz is the isotropic gapless point. See module header for the full Hamiltonian and solution outline.

Lieb

Dispatch tag for the nearest-neighbor tight-binding model on the Lieb (line-centred square) lattice (spinless, single-orbital, three sublattices). Hamiltonian:

H = -t Σ_{⟨i,j⟩} (c†_i c_j + h.c.)

The spectrum contains a dispersionless flat band at E = 0 for every momentum, with an additional three-fold band touching at the M-point.

LiouvilleCFT(; b::Real = 1.0) <: AbstractQAtlasModel

Non-compact Liouville CFT (Polyakov 1981) at coupling b > 0. The canonical self-dual point b = 1 (background charge Q = 2, c = 25) is used as the default. The b ↔ 1/b self-duality leaves both Q and c invariant.

Quantities registered:

References

• A. M. Polyakov, Phys. Lett. B 103, 207 (1981).
• H. Dorn, H.-J. Otto, Nucl. Phys. B 429, 375 (1994).
• A. B. Zamolodchikov, A. B. Zamolodchikov, Nucl. Phys. B 477, 577 (1996).

LogarithmicCFT <: AbstractQAtlasModel

The c = 0 logarithmic conformal field theory describing the universality class of self-avoiding polymers (Saleur 1992), critical percolation (Cardy 2001), and dilute / dense polymer phases.

The theory is logarithmic because the dilatation operator L_0 admits indecomposable (non-diagonalisable) Jordan blocks — pairs of primary fields share scaling dimensions but mix under conformal transformations, producing logarithms in correlation functions. The total central charge identically vanishes, yet the theory is non-trivial (Pearce-Rasmussen-Zuber 2006; Vasseur-Jacobsen-Saleur 2011).

Phase 1 exposes only CentralCharge = 0. Logarithmic-operator structure, indecomposable representations, and specific β-coupling parametrisations are deferred to Phase 2.

LongRangeIsing1D(; J::Real = 1.0, h::Real = 1.0, α::Real = Inf)
    <: AbstractQAtlasModel

1D transverse-field Ising chain with power-law-decaying couplings:

H = -J Σ_{i<j} σᶻᵢ σᶻⱼ / |i-j|^α  -  h Σᵢ σˣᵢ.

J > 0 (ferromagnetic), h > 0 (transverse field), α > 0 (decay exponent). The default α = Inf sits at the nearest-neighbour TFIM point, for which the gap is known in closed form.

Quantities registered (Phase 1):

Any finite α raises DomainError — Phase 2 will add DMRG / VMC support for the long-range phase diagram (Koffel-Lewenstein-Tagliacozzo 2012, Gong-Foss-Feig-Michalakis-Gorshkov 2014).

References

• P. Pfeuty, Annals Phys. 57, 79 (1970).
• L. Koffel, M. Lewenstein, L. Tagliacozzo, Phys. Rev. Lett. 109, 267203 (2012).
• Z.-X. Gong, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, Phys. Rev. Lett. 113, 030601 (2014).

LongRangeXY1D(; J::Real = 1.0, h::Real = 0.0, α::Real = Inf)
    <: AbstractQAtlasModel

1D XY chain with power-law-decaying interactions in a transverse field:

H = -J Σ_{i<j} (σᵢˣ σⱼˣ + σᵢʸ σⱼʸ) / |i-j|^α  -  h Σᵢ σᵢᶻ.

Constraints: J > 0, α > 0, h ∈ ℝ.

Quantities registered (Phase 1):

Phase 1 exposes only the α = Inf nearest-neighbour XX-in-transverse-field limit, whose mass gap is the closed form Δ = 2·max(0, |h| - 2J) (Lieb-Schultz-Mattis 1961, isotropic XX limit of the XY chain; Pfeuty 1970 for the TFIM-context derivation). Finite α raises DomainError pending the Phase 2 DMRG follow-up (Maghrebi-Gong-Gorshkov 2017 for the XY chain power-law family).

References

• E. Lieb, T. Schultz, D. Mattis, Annals Phys. 16, 407 (1961).
• P. Pfeuty, Annals Phys. 57, 79 (1970).
• M. F. Maghrebi, Z.-X. Gong, A. V. Gorshkov, Phys. Rev. Lett. 119, 023001 (2017).

LoschmidtEcho{M}() <: AbstractQuantity
LoschmidtEcho(:amplitude)
LoschmidtEcho(:rate)
LoschmidtRateFunction()        # alias for LoschmidtEcho{:rate}

Loschmidt-echo family for sudden-quench dynamics. After preparing |ψ_0⟩ as the ground state of an "initial" model H_0 and quenching to the "final" model H_f (passed as the first positional argument to fetch), the Loschmidt amplitude is

G(t) = ⟨ψ_0 | e^{-i H_f t} | ψ_0⟩,

with the Loschmidt echo L(t) = |G(t)|² ∈ [0, 1] and the rate function

λ(t) = -log L(t) / N         (finite N)
λ(t) = -lim_{N→∞} log L(t)/N (thermodynamic limit / Infinite)

Non-analytic cusps in λ(t) are dynamical quantum phase transitions (DQPT). See Heyl, Polkovnikov, Kehrein, PRL 110, 135704 (2013) and the review Heyl, Rep. Prog. Phys. 81, 054001 (2018).

The mode M::Symbol ∈ (:amplitude, :rate) is a phantom type parameter so that :amplitude (returns L(t)) and :rate (returns λ(t)) dispatch separately. The convenience alias LoschmidtRateFunction is the only flavour defined for Infinite, since the :amplitude itself is identically zero in the thermodynamic limit (extensive cumulants).

The pre-quench Hamiltonian is passed via the initial keyword on fetch, e.g.

fetch(TFIM(J=1.0, h=0.5), LoschmidtRateFunction(), Infinite();
      initial=TFIM(J=1.0, h=2.0), t=1.0)

const LoschmidtRateFunction = LoschmidtEcho{:rate}

Convenience alias for the rate-function flavour λ(t) = -log L(t)/N. See LoschmidtEcho.

LuttingerParameter() <: AbstractQuantity

Luttinger liquid parameter K. Meaningful for critical 1D models with U(1) symmetry (e.g. XXZ in the critical regime |Δ| < 1).

LuttingerVelocity() <: AbstractQuantity

Luttinger-liquid / bosonisation velocity u (a.k.a. v_{LL}) of the low-energy linear-dispersion mode in a 1D critical interacting system. Used by models like XXZ1D in the Luttinger regime |Δ| < 1, the Heisenberg chain at the SU(2) point, and any other bosonised 1D critical theory.

For a free-fermion model this coincides with FermiVelocity; for interacting systems u includes the Luttinger renormalisation.

MagnetizationX() <: AbstractQuantity

Bulk-averaged ⟨σˣ⟩ in Pauli convention (= 2 ⟨Sˣ⟩ in spin-1/2 units). For a spin-1/2 chain H = -J ΣSᶻSᶻ - h ΣSˣ this is the transverse magnetization; the axis-explicit name avoids the "transverse" / "longitudinal" ambiguity that depends on the model's Hamiltonian choice.

MagnetizationXLocal{M}() <: AbstractQuantity
MagnetizationXLocal()                       # M = :equilibrium (default)
MagnetizationXLocal(:equilibrium)           # explicit equilibrium ⟨σˣ_i⟩_β
MagnetizationXLocal(:quench)                # post-quench ⟨σˣ_i⟩(t)

Site-resolved ⟨σˣ_i⟩ quantity. The mode parameter M::Symbol is a phantom type that splits the dispatch into:

• :equilibrium — site-resolved thermal expectation [⟨σˣ_i⟩_β for i = 1:N] (Vector{Float64}). This is the original meaning; the no-argument constructor MagnetizationXLocal() keeps back-compatibility by routing here.

• :quench — time-evolved local transverse magnetisation ⟨σˣ_i⟩(t) = ⟨ψ_0|e^{iH_f t} σˣ_i e^{-iH_f t}|ψ_0⟩ after a sudden quench from the ground state of an initial::AbstractQAtlasModel (H_0) to the post-quench Hamiltonian (the model argument to fetch). Returns a single Float64 for one (i, t) pair.

See docs/src/calc/tfim-sigma-x-quench.md for the closed-form derivation in the TFIM (Calabrese–Essler–Fagotti, J. Stat. Mech. P07016 (2012); Barouch–McCoy–Dresden, PRA 2 (1970)).

MagnetizationY() <: AbstractQuantity

Bulk-averaged ⟨σʸ⟩.

MagnetizationYLocal() <: AbstractQuantity

Site-resolved ⟨σʸ_i⟩ vector of length N_bulk. Identically zero for any real Hermitian Hamiltonian (parity / time-reversal); a model that returns it explicitly does so as an exact baseline against random-sample estimators that fluctuate around zero.

MagnetizationZ() <: AbstractQuantity

Bulk-averaged ⟨σᶻ⟩. For Z₂-symmetric phases on an infinite system this is the order parameter at low temperature; finite-system fetch methods may return the absolute value / the ordered-phase limit as documented.

MagnetizationZLocal() <: AbstractQuantity

Site-resolved ⟨σᶻ_i⟩ vector of length N_bulk.

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

Spin-½ Majumdar–Ghosh chain — the J₁–J₂ Heisenberg chain locked to the special ratio J₂/J₁ = 1/2:

H = J Σ_i Sᵢ · Sᵢ₊₁ + (J/2) Σ_i Sᵢ · Sᵢ₊₂,   S = 1/2.

At this point the ground state is exactly the product of nearest- neighbour singlets (two-fold degenerate), with size-independent ground- state energy density E₀/N = −3J/8. An analytical lower bound on the gap Δ ≈ 0.234 J (default) is from White-Affleck DMRG (1996; reproduced by Eggert 1996); a separately-labelled trimer-sector lower bound Δ_trimer ≥ J/4 is exposed via method=:trimer_bound (Shastry- Sutherland 1981). Note: the SS bound exceeds the actual DMRG gap, so it is best read as a sector-specific bound (see MassGap docstring).

Fields

• J::Float64 — antiferromagnetic exchange coupling (J > 0). The next-nearest-neighbour coupling is locked to J/2 by the model definition.

References

• C. K. Majumdar, D. K. Ghosh, J. Math. Phys. 10, 1388 (1969).
• B. S. Shastry, B. Sutherland, J. Phys. C 14, L765 (1981).
• S. R. White, I. Affleck, Phys. Rev. B 54, 9862 (1996).

MassGap() <: AbstractQuantity

Energy gap between the ground state and the first excited state.

MeanField() <: AbstractQAtlasModel

Mean-field (Landau) universality class. Exact for d ≥ d_c (upper critical dimension). Kept as a top-level alias so that existing fetch(MeanField(), ...) callers keep working after the v0.13 API redesign; the canonical form is Universality(:MeanField).

MeanRatio() <: AbstractQuantity

Mean of the consecutive level-spacing ratio r_n = min(s_n, s_{n+1}) / max(s_n, s_{n+1}), introduced by Oganesyan-Huse (2007) and tabulated for the Wigner-Dyson and Poisson ensembles by Atas-Bogomolny-Giraud-Roux, Phys. Rev. Lett. 110, 084101 (2013):

MinimalModel(p::Int, p_prime::Int) <: AbstractQAtlasModel

Virasoro minimal model M(p, pprime). p and `pprimemust be coprime integers withp > p_prime ≥ 2. Construction validates those conditions and throwsDomainError` otherwise.

Special cases (cross-check):

The Ising special case MinimalModel(4, 3) reproduces the central charge stored in Universality(:Ising)'s CriticalExponents table (c = 1//2).

Use fetch with CentralCharge or ConformalWeights:

fetch(MinimalModel(4, 3), CentralCharge())                  # 1//2
fetch(MinimalModel(4, 3), ConformalWeights(); r=1, s=2)     # 1//16

See also: WZWSU2, Universality, CentralCharge, ConformalWeights, PrimaryFields.

MixedFieldIsing1D(; J = 1.0, h_x = 1.0, h_z = 0.0) <: AbstractQAtlasModel

The 1D mixed-field Ising chain

H = -J Σ_i σᶻ_i σᶻ_{i+1} - h_x Σ_i σˣ_i - h_z Σ_i σᶻ_i,    J > 0.

The default h_z = 0.0 sits on the Phase-1 closed-form point where the model coincides with the standard TFIM (Δ_∞ = 2|h_x − J|, gap closing at the quantum critical point h_x = J).

At h_z ≠ 0 the model is non-integrable — fetch methods will throw DomainError in Phase 1 and route to numerical solvers in Phase 2.

Model{M} <: AbstractQAtlasModel  (deprecated)

Phantom-typed Dict wrapper kept for backward compatibility. The Model(:TFIM; J=1.0, h=1.0) constructor below still works but is routed through the Symbol-dispatch deprecation shim in src/deprecate/legacy_fetch.jl. Prefer concrete model structs for new code.

OBC(N::Int)
OBC(; N::Int = 0)

Open boundary condition. N is the chain length. N = 0 is a legacy sentinel meaning "size unspecified — caller passes it via kwargs"; fetch methods that accept OBC(0) must look up kwargs[:N].

PBC(N::Int)
PBC(; N::Int = 0)

Periodic boundary condition. See OBC for the N = 0 sentinel.

PXP1D(; Ω::Real = 1.0) <: AbstractQAtlasModel

Rydberg-blockade chain (PXP model):

H = Ω Σ_i P_{i-1} σ^x_i P_{i+1},   P_i = (1 - σ^z_i)/2,   Ω > 0.

Acts within the constrained Hilbert space forbidding adjacent excitations. Paradigmatic host of many-body quantum scars (Turner et al. 2018).

Quantities registered (Phase 1):

Reliability :medium (numerical DMRG/ED thermodynamic-limit reference; not analytically closed). Scar-tower observables — revival frequency, OTOC, Z₂-state survival — are deferred to Phase 2.

References

• P. Fendley, K. Sengupta, S. Sachdev, Phys. Rev. B 69, 075106 (2004).
• H. Bernien et al., Nature 551, 579 (2017).
• C. J. Turner et al., Nat. Phys. 14, 745 (2018).
• C. J. Lin, O. I. Motrunich, Phys. Rev. Lett. 122, 173401 (2019).
• T. Iadecola, M. Schecter, S. Xu, Phys. Rev. B 100, 184312 (2019).
• F. M. Surace et al., Phys. Rev. X 10, 021041 (2020).

PartitionFunction() <: AbstractQuantity

Canonical partition function Z = Σ_σ exp(-β H(σ)).

PpIp2DSC(; Δ₀::Real = 1.0, μ::Real = 1.0) <: AbstractQAtlasModel

2-D px + i py chiral superconductor (Read–Green 2000) for spinless fermions, with BCS pairing amplitude Δ₀ > 0 and chemical potential μ > 0 (weak-pairing topological phase).

Phase 1 exposes the parameter-independent bulk topological data of the weak-pairing phase:

The strong-pairing trivial phase (μ ≤ 0) is excluded by the constructor: only the topological branch is exposed in Phase 1.

References

• N. Read, D. Green, Phys. Rev. B 61, 10267 (2000).
• A. Y. Kitaev, Ann. Phys. 321, 2 (2006).

PrimaryFields() <: AbstractQuantity

Full list of primary fields of a 2D rational CFT. For MinimalModel the result is a Vector{NamedTuple{(:r, :s, :h)}} of length (p - 1)(p_prime - 1) / 2, with one entry per Kac-symmetry orbit.

Future CFT classes may return different NamedTuple schemas (e.g. (j, h) for WZW). The return type is therefore a Vector{<:NamedTuple} whose schema depends on the model.

Quantity{Q} <: AbstractQuantity  (deprecated)

Phantom-typed wrapper kept for the legacy symbol API. New code should use concrete quantity structs such as Energy(), MagnetizationX(), ZZCorrelation(; mode=:static).

RFIM(; J::Real = 1.0, Δ::Real = 1.0) <: AbstractQAtlasModel

Random-Field Ising Model: H = -J Σ σσ - Σ h_i σ_i with quenched i.i.d. random fields of variance Δ² (Gaussian or bimodal ±√Δ²). The Imry-Ma (1975) argument gives the lower critical dimension d_l = 2: for d ≤ 2, Δ > 0 there is no long-range ferromagnetic order at any positive temperature.

Quantities registered (Phase 1):

References

• Y. Imry, S. Ma, Phys. Rev. Lett. 35, 1399 (1975).
• J. Imbrie, Comm. Math. Phys. 98, 145 (1985).

RandomBondIsing2D(; J::Real = 1.0, p::Real = 1.0) <: AbstractQAtlasModel

2D square-lattice ±J random-bond Ising model (Edwards-Anderson 1975):

H = -Σ_{⟨i,j⟩} J_{ij} σ^z_i σ^z_j,
J_{ij} = +J  (prob. p)  or  -J  (prob. 1-p).

The default p = 1 is the pure ferromagnet — Onsager's 2D Ising critical point with central charge c = 1/2.

Quantities registered (Phase 1):

Nishimori-line and multicritical Nishimori-point universality (logarithmic CFT, Honecker-Picco-Pujol 2001) are Phase 2.

References

• S. F. Edwards, P. W. Anderson, J. Phys. F 5, 965 (1975).
• H. Nishimori, Prog. Theor. Phys. 66, 1169 (1981).
• A. Honecker, M. Picco, P. Pujol, Phys. Rev. Lett. 87, 047201 (2001).

RenyiEntropy(α) <: AbstractQuantity

Rényi entropy of order α, S_α = (1 − α)⁻¹ log Tr ρ_A^α.

• α = 1 recovers VonNeumannEntropy (implementations may dispatch accordingly).
• α = 2 is the second Rényi entropy, frequently measured experimentally.
• α > 0, α ≠ 1 are the supported generic cases.

The inner constructor rejects α ≤ 0 and α = 1 (use VonNeumannEntropy() explicitly) — this is intentional, to force the call site to be explicit about which entropy it wants.

ResidualEntropy() <: AbstractQuantity

Zero-temperature configurational (residual) entropy density. Real- valued, non-negative; non-zero in the presence of macroscopic ground-state degeneracy (e.g. ice rule, frustrated Ising AFM, Pauling-1935-style models). Distinct from ThermalEntropy: ThermalEntropy is a finite-temperature thermodynamic quantity, while ResidualEntropy is the lim_{T -> 0} S(T) / N residual term. Zero-temperature ground-state entropy per site,

S_residual / (N k_B) = lim_{T → 0⁺} S(T) / N,

i.e. the entropy density of the (possibly degenerate) ground-state manifold. Non-zero for frustrated classical models with extensive ground-state degeneracy — e.g. the antiferromagnetic Ising model on the triangular lattice (Wannier 1950, ≈ 0.3230659669) and on the kagome lattice (Houtappel 1950). Defined as a separate quantity from ThermalEntropy to keep the zero-temperature limit explicit at the dispatch level (avoiding β → ∞ extrapolations of a finite-T fetch).

S1AnisotropicD1D(; J::Real = 1.0, D::Real = 0.0) <: AbstractQAtlasModel

Spin-1 Heisenberg chain with uniaxial single-ion anisotropy,

H = J Σᵢ Sᵢ · Sᵢ₊₁ + D Σᵢ (Sᵢ^z)²,

with J > 0 (antiferromagnetic) and D ∈ ℝ arbitrary single-ion anisotropy strength. Same 3-dimensional local Hilbert space as S1Heisenberg1D but with the on-site easy-axis (D < 0) or easy-plane (D > 0) term added.

Quantities registered (Phase 1):

Phase 1 exposes only the D = 0 Haldane-chain reference point, delegated to the existing S1Heisenberg1D. Any D ≠ 0 raises DomainError — finite-D crosses Gaussian (large-D) and Ising (Néel) transitions whose closed forms are unknown and are deferred to the DMRG-based Phase 2 (Chen-Roncaglia 2008; Tzeng-Yang-Hsu 2017).

References

• F. D. M. Haldane, Phys. Lett. A 93, 464 (1983).
• S. R. White, D. A. Huse, Phys. Rev. B 48, 3844 (1993).
• Y.-C. Chen, R. Roncaglia, J. Stat. Mech. P10024 (2008).
• Y.-D. Tzeng, H.-H. Hung, Y.-C. Chen, M.-F. Yang, Phys. Rev. B 96, 205104 (2017).

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).

S1XXZ1D(; J::Real = 1.0, Δ::Real = 1.0) <: AbstractQAtlasModel

Spin-1 antiferromagnetic XXZ chain,

H = J Σᵢ [Sˣᵢ Sˣᵢ₊₁ + Sʸᵢ Sʸᵢ₊₁ + Δ Sᶻᵢ Sᶻᵢ₊₁],   S = 1, J > 0.

The default Δ = 1.0 places the model at the SU(2)-symmetric Haldane point, where Phase-1 closed-form MassGap is available by delegation to S1Heisenberg1D.

SLEkappa(; κ::Real = 6.0) <: AbstractQAtlasModel

Schramm-Loewner Evolution SLE_κ (random planar conformally-invariant curve) at parameter κ > 0. Default κ = 6 is the percolation cluster-boundary fixed point — the original Schramm prediction that launched the field.

Quantities registered:

References

• O. Schramm, Israel J. Math. 118, 221 (2000).
• V. Beffara, Annals Probab. 36, 1421 (2008).
• M. Bauer, D. Bernard, Phys. Rep. 432, 115 (2006).

SYK(; q::Integer = 4) <: AbstractQAtlasModel

Sachdev-Ye-Kitaev model: N Majorana fermions with all-to-all random q-body coupling. The defining structural parameter for the universal large-N IR is q (which must be even — Majorana antisymmetry — and ≥ 2). The fermion number N and the coupling scale J set the UV / finite-N data but do not enter the leading large-N IR conformal dimension exposed here.

Default q = 4 is the most-studied case: 4-body Majorana coupling, exactly soluble Schwarzian-dressed two-point function, and a maximal Lyapunov exponent saturating the Maldacena- Shenker-Stanford bound.

Quantities registered (Phase 1):

Phase 2 will add composite-operator dimensions (bilinear tower), the Schwarzian zero-T entropy, maximal Lyapunov saturation, and the JT-gravity bulk dual.

References

• S. Sachdev, J. Ye, Phys. Rev. Lett. 70, 3339 (1993).
• A. Kitaev, KITP talks (2015) — "A simple model of quantum holography".
• J. Maldacena, D. Stanford, Phys. Rev. D 94, 106002 (2016).

SchwingerModel(; e::Real = 1.0, m::Real = 0.0) <: AbstractQAtlasModel

1+1-D quantum electrodynamics (Schwinger 1962) with gauge coupling e > 0 and fermion mass m ≥ 0. At the massless point (m = 0) the spectrum is exactly a free massive scalar with the Schwinger mass m_γ = e/√π.

Quantities registered (Phases 1+2):

Massive m > 0 Schwinger maps to massive sine-Gordon (Coleman-Jackiw-Susskind 1975) and is tracked as Phase 2.

References

• J. Schwinger, Phys. Rev. 128, 2425 (1962).
• S. Coleman, R. Jackiw, L. Susskind, Annals Phys. 93, 267 (1975).

ShastrySutherland(; J::Real = 0.0, Jp::Real = 1.0) <: AbstractQAtlasModel

Spin-½ Shastry–Sutherland model on the Shastry–Sutherland lattice (corner-sharing-orthogonal-dimer geometry; realised in SrCu₂(BO₃)₂):

H = J Σ_{⟨i,j⟩} Sᵢ · Sⱼ + J' Σ_{⟨⟨i,j⟩⟩_d} Sᵢ · Sⱼ.

The model has two named couplings: J for the square-lattice nearest-neighbour bonds and Jp (J') for the alternating-plaquette diagonal "dimer" bonds. The default J = 0, Jp = 1 is the trivial isolated-dimer limit and lies safely inside the exact-dimer phase.

In the parameter window α = J / Jp ≤ α_c ≈ 0.675 (Koga–Kawakami

1. the ground state is the exact product of nearest-neighbour

singlets on the diagonal (Jp) bonds — the 2-D analog of the Majumdar–Ghosh dimer state.

Quantities registered:

The triplon excitation spectrum, magnetisation plateaux (1/8, 1/4, 1/3) and large-α intermediate / plaquette phases require quantities beyond a scalar ground-state energy and are tracked as a follow-up phase.

References

• B. S. Shastry, B. Sutherland, Physica B+C 108, 1069 (1981).
• S. Miyahara, K. Ueda, Phys. Rev. Lett. 82, 3701 (1999).
• A. Koga, N. Kawakami, Phys. Rev. Lett. 84, 4461 (2000).

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

Sherrington-Kirkpatrick mean-field Ising spin glass (Sherrington- Kirkpatrick 1975). The Hamiltonian is the canonical 1/√N random-Gaussian-coupling sum

H = -(1/√N) Σ_{i<j} J_ij σ_i σ_j,   J_ij ~ N(0, J²),

whose spin-glass transition lies at T_c = J.

Quantities registered (Phases 1 and 2):

The Phase-2 Energy{:per_site} entry is the Parisi T=0 ground-state energy density e_0/J ≈ -0.7631667 (Crisanti-Rizzo 2002 high-precision evaluation of Parisi's full-RSB solution; Talagrand 2006 rigorous proof). The finite-temperature Parisi free energy f(β, h), the de Almeida-Thouless line h_AT(T) and the overlap distribution P(q) remain tracked for later phases.

References

• D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975).
• G. Parisi, J. Phys. A 13, L115 (1980).
• M. Talagrand, Annals Math. 163, 221 (2006).

SixVertex(; a::Real = 1.0, b::Real = 1.0, c::Real = 1.0) <: AbstractQAtlasModel

Classical six-vertex model on the infinite square lattice in the standard (a, b, c) parameterisation (Lieb 1967, Baxter 1982 ch. 8):

ω₁ = ω₂ = a,   ω₃ = ω₄ = b,   ω₅ = ω₆ = c

with the ice rule (each vertex has two incoming and two outgoing arrows). The single dimensionless invariant

Δ = (a² + b² − c²) / (2 a b)

selects the phase: Δ > 1 ferroelectric (FE, frozen), |Δ| < 1 disordered (Lieb / Sutherland 1967), Δ < −1 antiferroelectric (AFE / F-model, Lieb 1967b).

Convenience constructors for the three classical sub-models (square_ice, f_model, kdp_model) are also provided.

Currently registered fetches:

The full disordered-phase free energy (Lieb / Sutherland 1967 trigonometric integral) and the AFE elliptic-function free energy (Lieb 1967b) are out of scope for the present commit (Phase 2 / Phase 3 of issue #163) and will throw an informative ArgumentError if requested.

See also: IsingSquare for the closest classical analog.

SpecificHeat() <: AbstractQuantity

Specific heat per site, c_v(β) = β² (⟨H²⟩ − ⟨H⟩²) / N.

SpectralFormFactor() <: AbstractQuantity

Disorder-averaged spectral form factor K(t) = ⟨|Σ_n e^{−iE_n t}|²⟩ / Z² — the canonical late-time quantum-chaology diagnostic (Mehta 2004 §16; Cotler et al. 2017).

For GUE random-matrix-theory eigenvalues in the large-N thermodynamic limit, with rescaled time τ = t / N, the disorder-averaged SFF has the universal closed form

• K(τ) = (τ/(2π)) − (τ/(4π)) log|1 − τ/(2π)| for τ ≤ 2π
• K(τ) = 1 for τ ≥ 2π

so that K exhibits a linear ramp K(τ) ≈ τ/π for small τ and saturates to the universal plateau K(τ→∞) = 1 for τ beyond the Heisenberg time τ_H = 2π.

QAtlas Phase 1 (issue #243) exposes only the late-time plateau τ ≥ τ_H for the GUE ensemble; the ramp regime and the GOE/GSE sigma-model closed forms (Mehta 2004 §16) are deferred to Phase 2.

References

• M. L. Mehta, Random Matrices, 3rd ed., Elsevier (2004), §16.
• E. Brézin, S. Hikami, Phys. Rev. E 55, 4067 (1997).
• J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, M. Tezuka, JHEP 05, 118 (2017), arXiv:1611.04650 — ramp-plateau picture.

SpinGap() <: AbstractQuantity

Spin gap of an electron system,

Δ_s = E₀(S^z = 1) - E₀(S^z = 0),

i.e. the lowest excitation energy at fixed total particle number that flips one spin. Zero whenever the spinon branch is gapless (e.g. the half-filled 1D Hubbard chain — rigorous Lieb–Wu result), positive in a spin-gapped phase (Haldane chain, BCS superconductor, …).

SpinIce() <: AbstractQAtlasModel

Classical spin ice on the pyrochlore lattice of corner-sharing tetrahedra. Ising-like local moments obey the "2-in-2-out" ice rule in the ground-state manifold; the residual entropy follows the Pauling 1935 closed form.

Quantities registered:

The Coulomb-phase dipolar spin correlations and the Castelnovo-Moessner-Sondhi (2008) emergent-monopole picture are tracked as a follow-up phase (#257 phase 2) and not exposed here.

References

• L. Pauling, J. Am. Chem. Soc. 57, 2680 (1935).
• S. T. Bramwell, M. J. P. Gingras, Science 294, 1495 (2001).

const SpinWaveVelocity = LuttingerVelocity

Spin-chain community alias for LuttingerVelocity. The "spin wave velocity" (e.g. in the Haldane / Affleck literature on the AFM Heisenberg chain) is the same quantity as the Luttinger velocity once bosonised; both dispatch through the same fetch method via the type identity.

SpontaneousMagnetization() <: AbstractQuantity

Spontaneous magnetization of a classical model as a function of temperature. Retained under this name for backward compatibility with the classical-Ising literature; new code may prefer the axis-explicit MagnetizationZ together with a T < T_c context.

SteadyStateCurrent() <: AbstractQuantity

Steady-state mass / particle current j in a 1D non-equilibrium lattice gas (e.g. ASEP / TASEP, Derrida-Lebowitz 1998). For TASEP at hopping rate p and density ρ,

j(ρ) = p ρ (1 − ρ)              (TASEP mean-field steady state)

— the canonical KPZ-class non-equilibrium observable.

StringOrderParameter() <: AbstractQuantity

Kennedy-Tasaki non-local (string) order parameter

O_str = lim_{|i-j| -> infty} -<S^z_i exp[i pi sum_{i<k<j} S^z_k] S^z_j>

for S=1 chains. Detects the hidden Z2 x Z2 symmetry breaking that defines the Haldane phase (T. Kennedy and H. Tasaki, Phys. Rev. B 45, 304 (1992)). At the AKLT point the closed-form value is O_str = 4/9 (AKLT 1988), making it the canonical analytic test bed for any implementation that aims to detect topologically non-trivial gapped phases of integer-spin chains.

SusceptibilityXX() <: AbstractQuantity

Static transverse susceptibility, χ_xx(β) = β · (⟨M_x²⟩ − ⟨M_x⟩²) / N.

SusceptibilityYY() <: AbstractQuantity

Analogue for the y-axis.

SusceptibilityZZ() <: AbstractQuantity

Uniform longitudinal susceptibility, χ_zz(β) = β · (⟨M_z²⟩ − ⟨M_z⟩²) / N.

TASEP(; p::Real = 1.0, ρ::Real = 0.5) <: AbstractQAtlasModel

Totally Asymmetric Simple Exclusion Process — 1-D KPZ-class non-equilibrium lattice gas with rightward hopping rate p > 0 at particle density 0 ≤ ρ ≤ 1. The default (p, ρ) = (1, 1/2) is the maximal-current half-filling point with j_max = 1/4.

Quantities registered (Phase 1):

References

• M. Kardar, G. Parisi, Y.-C. Zhang, Phys. Rev. Lett. 56, 889 (1986) — KPZ universality class.
• B. Derrida, M. R. Evans, V. Hakim, V. Pasquier, J. Phys. A 26, 1493 (1993).
• B. Derrida, J. L. Lebowitz, Phys. Rev. Lett. 80, 209 (1998).

TFIM(; J = 1.0, h = 1.0) <: AbstractQAtlasModel

The 1D transverse field Ising model with Hamiltonian

H = -J Σ_i σᶻ_i σᶻ_{i+1} - h Σ_i σˣ_i

J > 0 is ferromagnetic, h is the transverse field. The critical point sits at h = J.

TTbar(; c::Real = 1.0, λ::Real = 0.0) <: AbstractQAtlasModel

Universal irrelevant TT̄ deformation of a 2-D QFT whose UV completion is a CFT of central charge c > 0, with deformation coupling λ of dimension (length)². Default c = 1 (free boson seed), λ = 0 (undeformed CFT). Any real λ is admissible: λ > 0 is the Hagedorn-like branch, λ < 0 is the "good-sign" branch.

Because the TT̄ deformation is irrelevant in the RG sense, the UV central charge c is preserved at all λ:

c(λ) = c.

Quantities registered (Phase 1):

References

• A. B. Zamolodchikov, hep-th/0401146 (2004).
• F. A. Smirnov, A. B. Zamolodchikov, Nucl. Phys. B 915, 363 (2017).
• A. Cavaglià, S. Negro, I. M. Szécsényi, R. Tateo, JHEP 10, 112 (2016).
• L. McGough, M. Mezei, H. Verlinde, JHEP 04, 010 (2018).

ThermalEntropy() <: AbstractQuantity

Thermal / thermodynamic entropy per site, s(β) = −∂f/∂T where f is the free energy per site. Real-valued, non-negative, monotone in T.

TightBinding1D(; t::Real = 1.0, μ::Real = 0.0) <: AbstractQAtlasModel

1D non-interacting spinless-fermion chain.

H = -t Σᵢ (c†ᵢ c_{i+1} + h.c.) - μ Σᵢ c†ᵢ cᵢ

with t > 0 (strict) and arbitrary real μ.

Quantities registered (Phase 1, all closed-form):

Examples

julia> using QAtlas

julia> fetch(TightBinding1D(), Energy{:per_site}(), Infinite())   # half-filling
-0.6366197723675814

julia> fetch(TightBinding1D(; μ = 3.0), MassGap(), Infinite())     # band insulator
1.0

julia> fetch(TightBinding1D(; μ = 1.0), FermiVelocity(), Infinite())
1.7320508075688772

References

• N. W. Ashcroft, N. D. Mermin, Solid State Physics (1976), chapter 9.

TightBindingChecksum() <: AbstractQuantity

Sum of squared single-particle eigenvalues Σ λᵢ² of the tight-binding Bloch Hamiltonian, i.e. tr(H²). For a chiral (bipartite) lattice this equals 2 t² · n_NN_bonds, an analytical identity independent of Lx, Ly, or k-grid details — a strong sanity invariant on top of the spectrum closed form.

TightBindingMaxEnergy() <: AbstractQuantity

Largest single-particle eigenvalue of the tight-binding Bloch Hamiltonian, i.e. max(λᵢ). For the chiral honeycomb this equals |t| · max_k |f(k)| = 3|t| (saturated at the Γ point where the three nearest-neighbor phases add coherently), independent of Lx and Ly as long as the Γ point is on the discrete momentum grid (always true for PBC with Lx, Ly ≥ 1).

Sister scalar to TightBindingChecksum (which captures Σλᵢ² = tr(H²) and is invariant under chiral pair sign-flips). The two together discriminate any single-eigenvalue perturbation: a sign error on a hopping displacement that flips a chiral pair changes neither the checksum nor the bandwidth at Γ only if it doesn't reach the Γ point — making them jointly sensitive to a broader class of typos.

TightBindingSpectrum() <: AbstractQuantity

Single-particle Bloch spectrum of a tight-binding model. Returned as a sorted Vector{Float64} of length n_orbitals · Lx · Ly.

TightBindingV1D(t::Real, V::Real, μ::Real)
TightBindingV1D(; t=1.0, V=0.0, μ=0.0)

1D spinless-fermion t-V chain (NN hopping t > 0, NN density-density interaction V, chemical potential μ). At V = 0 the model is the free-fermion tight-binding chain; at V ≠ 0 it is Jordan-Wigner equivalent to the spin-1/2 XXZ chain (Phase 2).

The default keyword constructor lands at the closed-form free-fermion point (t, V, μ) = (1, 0, 0).

TodaLattice(; a::Real = 1.0, b::Real = 1.0) <: AbstractQAtlasModel

Classical 1-D Toda lattice (Toda 1967) with standard nearest-neighbour exponential potential V(r) = (a/b) e^{-b r} + a r - a/b. The defaults a = b = 1 give the canonical textbook form V(r) = e^{-r} + r - 1.

Quantities registered (Phase 1):

Soliton energy-momentum relations and the Gutzwiller (1981) quantum- Toda Bethe-ansatz spectrum are tracked as Phase 2: they need momentum-parameter / quantum-number-indexed quantity types not yet in QAtlas core.

References

• M. Toda, J. Phys. Soc. Jpn. 22, 431 (1967).
• H. Flaschka, Phys. Rev. B 9, 1924 (1974).
• M. C. Gutzwiller, Annals Phys. 133, 304 (1981).

TopologicalEntanglementEntropy() <: AbstractQuantity

Constant subleading correction γ in the area-law bipartite entanglement entropy of a 2D topologically ordered ground state on a simply-connected disk region:

S(ρ_A) = α |∂A| − γ + O(|∂A|⁻¹).

Kitaev–Preskill (2006) and Levin–Wen (2006) showed γ = log 𝒟, where 𝒟 = √(Σ_a d_a²) is the total quantum dimension of the topological order. Returns a Float64. For the toric code (Z₂ topological order, four Abelian anyons) γ = log 2.

TopologicalInvariant() <: AbstractQuantity

Topological Z_2 invariant of a 1D BdG superconductor (Kitaev 2001). Defined as the Pfaffian sign at the time-reversal-invariant momenta k = 0 and k = π,

\[\nu = \operatorname{sgn}\bigl[\operatorname{Pf}(H_{\mathrm{BdG}}(k=0)) \cdot \operatorname{Pf}(H_{\mathrm{BdG}}(k=\pi))\bigr] \in \{+1, -1\},\]

with ν = -1 in the topological phase and ν = +1 in the trivial phase. For a gapless bulk (Pfaffian zero at k = 0 or k = π) the invariant is ill-defined and implementations should signal an error.

Currently used by Kitaev1D.

ToricCode(; J_e::Real = 1.0, J_m::Real = 1.0) <: AbstractQAtlasModel

Kitaev (2003) toric code: the square-lattice Z₂ surface code with vertex ("electric") coupling J_e and plaquette ("magnetic") coupling J_m,

H = − J_e Σ_v A_v − J_m Σ_p B_p,

where A_v = ∏_{i ∈ star(v)} σˣ_i and B_p = ∏_{i ∈ ∂p} σᶻ_i. All stabilizers commute, so the model is exactly solvable. See module header for the full closed-form result list and the distinction from KitaevHoneycomb.

TracyWidom() <: AbstractQuantity

Tracy-Widom largest-eigenvalue cumulative distribution F_β(x) for the three Wigner-Dyson ensembles (β ∈ {1, 2, 4}). Returns the value F_β(x) = P[ξ_β ≤ x] at the requested x.

QAtlas Phase 1 evaluates F_β from a precomputed table compiled from Bornemann, On the numerical evaluation of Fredholm determinants, Math. Comp. 79, 871 (2010), Table 1, with monotone linear interpolation on the table support and Tracy-Widom 1994/1996 tail asymptotics outside it. A direct Painlevé-II integrator is deferred to Phase 2.

Triangular

Dispatch tag for the nearest-neighbor tight-binding model on the triangular lattice (spinless, single-orbital, one sublattice).

Hamiltonian: H = -t Σ{⟨i,j⟩} (c†i c_j + h.c.)

The single band ranges from -6t (at the Γ-point) to +3t (at the K-points), reflecting geometric frustration: the absence of particle-hole (chiral) symmetry on a non-bipartite lattice.

TricriticalIsing() <: AbstractQAtlasModel

Tricritical Ising CFT — the unitary Virasoro minimal model M(5, 4), next-to-simplest unitary 2D CFT after Ising (M(4, 3)). Lattice realisations include the Blume-Capel model at its tricritical point and the Andrews-Baxter-Forrester (1984) RSOS models at (p+1, p) = (5, 4).

Quantities registered (Phase 1):

References

• A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Nucl. Phys. B 241, 333 (1984).
• D. Friedan, Z. Qiu, S. Shenker, Phys. Rev. Lett. 52, 1575 (1984).

TricriticalPotts3() <: AbstractQAtlasModel

Tricritical 3-state Potts CFT — the unitary Virasoro minimal model M(6, 7), realised on the lattice by Andrews-Baxter-Forrester (1984) RSOS models and by the q = 3 Potts model with dilution (Huse 1984).

Quantities registered:

References

• G. E. Andrews, R. J. Baxter, P. J. Forrester, J. Stat. Phys. 35, 193 (1984).
• D. A. Huse, Phys. Rev. B 30, 3908 (1984).

Universality{C}

Parametric dispatch tag for universality classes. C is a Symbol identifying the class (:Ising, :XY, :Heisenberg, :Potts3, :Potts4, :Percolation, :KPZ, etc.).

Use with CriticalExponents (equilibrium) or GrowthExponents (KPZ-type) and a d keyword to select the spatial dimension:

QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=2)   # exact Rational
QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=3)   # numerical + _err
QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=4)   # mean-field

VonNeumannEntropy{M}() <: AbstractQuantity
VonNeumannEntropy()                   # M = :equilibrium (default)
VonNeumannEntropy(:equilibrium)       # explicit equilibrium S(ℓ)
VonNeumannEntropy(:quench)            # post-quench S(ℓ, t)

Von Neumann entanglement entropy of a reduced density matrix, S_vN = −Tr ρ_A log ρ_A. The mode parameter M::Symbol is a phantom type that splits the dispatch into:

• :equilibrium — equilibrium / thermal value S_vN = -Tr ρ_A log ρ_A of the GS or thermal reduced density matrix on the first ℓ sites (kwarg ℓ). This is the original meaning; the no-argument constructor VonNeumannEntropy() keeps back-compatibility by routing here.

• :quench — time-evolved entanglement entropy S_vN(ℓ, t) = -Tr ρ_A(t) log ρ_A(t) after a sudden quench from the ground state of an initial::AbstractQAtlasModel (H_0) to the post-quench Hamiltonian (the model argument to fetch). Calabrese–Cardy quasi-particle picture (J. Stat. Mech. P04010 (2005)): linear growth S(ℓ, t) ≈ (c/3) v_E t for t ≪ ℓ/(2 v_E), saturating at (c/3) log ℓ + const for t ≫ ℓ/(2 v_E).

See docs/src/calc/tfim-quench-entanglement.md for the free-fermion derivation in the TFIM.

WZWSU2(k::Int) <: AbstractQAtlasModel

Wess–Zumino–Witten model with affine Lie algebra SU(2) at level k ≥ 1. Construction throws DomainError for k ≤ 0.

Special cases:

• k = 1: c = 1 (free boson at the SU(2)-symmetric radius; low-energy theory of the spin-1/2 Heisenberg antiferromagnet, Affleck 1989).
• k = 2: c = 3/2 — equivalent to 3 free Majorana fermions (each contributing c = 1/2), or equivalently the smallest N=1 super-Virasoro minimal model. Note that "Ising × free Majorana" with one Majorana would only give c = 1/2 + 1/2 = 1 ≠ 3/2; the correct decomposition needs three Majorana fermions.
• k = 3: c = 9/5.

fetch(WZWSU2(1), CentralCharge())                    # 1//1
fetch(WZWSU2(1), ConformalWeights(); j=1//2)         # 1//4

See also: MinimalModel, Universality, CentralCharge, ConformalWeights.

WignerSurmise() <: AbstractQuantity

Wigner surmise nearest-neighbour level-spacing distribution P_β(s) for the three Wigner-Dyson ensembles (β ∈ {1, 2, 4}: GOE, GUE, GSE). The surmise is the exact N = 2 Gaussian-ensemble spacing distribution; it is also a celebrated, accurate approximation to the bulk N → ∞ spacing distribution. Returns the value P_β(s) at the requested s (the universality fetch carries β).

XCube() <: AbstractQAtlasModel

Fracton X-cube model on the 3-D cubic lattice (Vijay-Haah-Fu 2016) — prototype of Type-I fracton topological order with subextensive ground-state degeneracy that scales linearly with each lattice direction.

Quantities registered (Phase 1):

References

• J. Haah, Phys. Rev. A 83, 042330 (2011).
• S. Vijay, J. Haah, L. Fu, Phys. Rev. B 94, 235157 (2016).
• K. Slagle, Y. B. Kim, Phys. Rev. B 96, 195139 (2017).

XXCorrelation{M}() <: AbstractQuantity
XXCorrelation(; mode::Symbol = :static)

Real-space 2-point ⟨σˣ_i σˣ_j⟩ correlator. See ZZCorrelation for the mode semantics.

XXStructureFactor() <: AbstractQuantity

Fourier-space equivalent of XXCorrelation.

XXZ1D(; J::Real = 1.0, Δ::Real = 0.0) <: AbstractQAtlasModel

Spin-1/2 XXZ chain

H = J Σ_i [ S^x_i S^x_{i+1} + S^y_i S^y_{i+1} + Δ S^z_i S^z_{i+1} ]

Convention: J > 0 is antiferromagnetic. Δ = 1 is the isotropic Heisenberg AF point, Δ = 0 is the XX (free-fermion) point, Δ = -1 is the isotropic ferromagnet. For |Δ| < 1 the chain is critical (Luttinger liquid, central charge c = 1).

XYh1D(; Jx::Real = 1.0, Jy::Real = 1.0, h::Real = 0.0) <: AbstractQAtlasModel

Anisotropic XY chain in a transverse field (Lieb-Schultz-Mattis 1961):

H = -Σ_i ( Jx σ^x_i σ^x_{i+1} + Jy σ^y_i σ^y_{i+1} ) - h Σ_i σ^z_i.

Requires Jx > 0 and Jy > 0.

Quantities registered (Phase 1):

Phase 1 only exposes the isotropic XX limit Jx = Jy = J; any anisotropic Jx ≠ Jy raises DomainError (Phase 2: general-k minimisation of the LSM/Pfeuty dispersion).

References

• E. Lieb, T. Schultz, D. Mattis, Annals Phys. 16, 407 (1961).
• P. Pfeuty, Annals Phys. 57, 79 (1970).
• S. Sachdev, Quantum Phase Transitions (2nd ed., CUP 2011), §2.

YYCorrelation{M}() <: AbstractQuantity
YYCorrelation(; mode::Symbol = :static)

Real-space 2-point ⟨σʸ_i σʸ_j⟩ correlator.

YYStructureFactor() <: AbstractQuantity

Fourier-space equivalent of YYCorrelation.

YangLee() <: AbstractQAtlasModel

Yang-Lee CFT — the non-unitary Virasoro minimal model M(5, 2) describing the universality class of the Lee-Yang edge singularity (Yang-Lee 1952; Cardy 1985).

The model has no continuous parameters: M(5, 2) is fixed.

Quantities registered (Phase 1):

The famous negative-dimension primary h_{1,2} = -1/5 is the edge exponent and is what makes the theory non-unitary (c = -22/5 < 0).

References

• C. N. Yang and T. D. Lee, Phys. Rev. 87, 404 (1952).
• J. L. Cardy, Phys. Rev. Lett. 54, 1354 (1985).

ZZCorrelation{M}() <: AbstractQuantity
ZZCorrelation(; mode::Symbol = :static)

Real-space 2-point correlator ⟨σᶻ_i σᶻ_j⟩. The mode M::Symbol is a phantom type parameter so dispatch can specialise on it.

Supported mode values (by convention; individual models need only implement the ones they support):

• :static — equal-time, thermal or zero-temperature value
• :connected — ⟨σᶻ_i σᶻ_j⟩ − ⟨σᶻ_i⟩⟨σᶻ_j⟩
• :dynamic — retarded real-time correlator ⟨σᶻ_i(t) σᶻ_j(0)⟩
• :lightcone — space-time spreading ⟨σᶻ_i(t) σᶻ_j(0)⟩ as a matrix over (site, time)

The companion type for Fourier-space structure factors is ZZStructureFactor, kept separate because it carries (q, ω) arguments instead of (i, j, t).

ZZStructureFactor() <: AbstractQuantity

Fourier-space structure factor S_zz(q, ω) = ∫ dt e^{iωt} (1/N) Σ_{ij} e^{iq·(i-j)} ⟨σᶻ_i(t)σᶻ_j(0)⟩ (or its static limit, depending on the model's fetch signature).

Kept as a separate type from ZZCorrelation because the argument domain is (q, ω) instead of (i, j, t) and because existing users already expect a dedicated StructureFactor quantity.

ZnClock(n::Integer) <: AbstractQAtlasModel
ZnClock(; n::Integer=2)

2-D Z_n clock model with n clock states.

Phase-1 quantities:

For n ≥ 4 the critical theory is a continuous family of c = 1 CFTs (Ashkin-Teller for n = 4; intermediate BKT phase for n ≥ 5) whose selection requires a coupling parameter; these branches throw a DomainError and are deferred to Phase 2.

References

• J. V. José, L. P. Kadanoff, S. Kirkpatrick, D. R. Nelson, Phys. Rev. B 16, 1217 (1977).
• S. Elitzur, R. B. Pearson, J. Shigemitsu, Phys. Rev. D 19, 3698 (1979).

ZnParafermion(; n::Integer=3) <: AbstractQAtlasModel

Zn parafermion CFT (Fateev-Zamolodchikov 1985), the SU(2)n / U(1) coset conformal field theory with central charge c = 2(n-1)/(n+2).

The default n = 3 selects the Z_3 parafermion = 3-state Potts critical CFT with c = 4/5.

Arguments

• n::Integer = 3 — parafermion level; must satisfy n ≥ 2.

Examples

QAtlas.fetch(ZnParafermion(; n=2), CentralCharge(), Infinite())  # 1//2  (Ising)
QAtlas.fetch(ZnParafermion(; n=3), CentralCharge(), Infinite())  # 4//5  (3-state Potts)
QAtlas.fetch(ZnParafermion(; n=4), CentralCharge(), Infinite())  # 1//1  (free boson)
QAtlas.fetch(ZnParafermion(; n=5), CentralCharge(), Infinite())  # 8//7

References

• V. A. Fateev, A. B. Zamolodchikov, Sov. Phys. JETP 62, 215 (1985).

_aklt_hamiltonian_matrix(model::AKLT1D, N::Int, ::OBC) -> Matrix{ComplexF64}

Assemble the dense 3^N × 3^N OBC Hamiltonian

H = J Σᵢ [ Sᵢ · Sᵢ₊₁ + (1/3) (Sᵢ · Sᵢ₊₁)² ]

via explicit tensor products built from the spin-1 primitives in HeisenbergS1.jl. Capped by _MAX_ED_SITES_S1.

_bc_size(bc::BoundaryCondition, kwargs) -> Int

Return the effective system size for bc. Prefers bc.N when it is positive; otherwise looks up kwargs[:N]; otherwise throws. Legacy fetch methods can use this helper to accept both OBC(N=24) and OBC(); N=24 call forms.

_bc_with_legacy_N(bc::OBC, m::Model{:TFIM}) -> OBC

If bc.N == 0 (zero-arg OBC() call) but the legacy Model Dict has an :N entry, promote it to a sized OBC(N) so the concrete fetch can read bc.N uniformly.

_build_wick_matrix(idx_t, idx_0, Σ, R, RΣ) -> Matrix{ComplexF64}

Assemble the Wick contraction matrix F for a product

γ_{idx_t[1]}(t) γ_{idx_t[2]}(t) … γ_{idx_0[1]}(0) γ_{idx_0[2]}(0) …

from the ground-state Majorana covariance Σ, the time evolution matrix R = exp(h t), and the precomputed product RΣ = R * Σ. F is complex antisymmetric; its Pfaffian gives ⟨…⟩_GS.

_cardy_central_charge(model::Universality{C}; kwargs...) -> Float64

Internal helper: fetch the central charge c of the universality class model and return it as a Float64. Re-throws as an ErrorException with a Calabrese–Cardy-specific message if the class has no CentralCharge defined.

_cardy_renyi_c(c::Real, α::Real) -> Float64

Calabrese–Cardy Rényi-α coefficient substitution

c -> c · (1 + 1/α) / 2.

Reduces to c at α = 1. Used to produce the Rényi entropy from the same closed form as the von Neumann case.

_check_quench_couplings(model_f::TFIM, initial::TFIM)

Validate that a TFIM quench preserves the Ising coupling J. The GGE closed forms in TFIM_gge.jl only treat h-quenches; a J quench is not in the standard Calabrese–Essler–Fagotti framework and the n_k formula derived above does not apply.

_ed_log_partition(evals::AbstractVector, β::Real) -> Float64

log Z(β) = log Σ exp(-β eₙ) with an emin shift to avoid overflow at large β:

log Z = -β eₘᵢₙ + log Σ exp(-β (eₙ - eₘᵢₙ))

_ed_thermal_energy(H::AbstractMatrix, β::Real) -> Float64

⟨H⟩_β = Tr(H exp(-βH)) / Tr(exp(-βH)). H must be Hermitian.

_ed_thermal_entropy(H::AbstractMatrix, β::Real) -> Float64

Total Gibbs entropy S(β) = β·(⟨H⟩ - F) = -Σ wₙ log wₙ, with wₙ = exp(-β(eₙ - eₘᵢₙ)) / Z̃. The first form is what we evaluate because both pieces share the eigendecomposition.

_ed_thermal_expectation(evals, evecs, O_diag, β) -> Float64

Return ⟨O⟩_β = Σ_n O_nn exp(-β eₙ) / Σ_n exp(-β eₙ) given a pre-computed eigendecomposition (evals, evecs) of H and the diagonal of O in the H-eigenbasis, O_diag[n] = ⟨n|O|n⟩.

Uses a log-sum-exp shift (eₘᵢₙ) so β up to ~10^2 / bandwidth is numerically stable.

_ed_thermal_free_energy(H::AbstractMatrix, β::Real) -> Float64

Total Helmholtz free energy F(β) = -log Z / β from the eigenspectrum of a Hermitian H. Per-site values: divide by N at the call site.

_ed_thermal_specific_heat(H::AbstractMatrix, β::Real) -> Float64

Total heat capacity from the energy variance,

C(β) = β² · (⟨H²⟩ - ⟨H⟩²).

Equivalent to -β² · ∂⟨H⟩/∂β; computed exactly from the eigenspectrum without numerical differentiation.

_extended_hubbard1d_check_v_zero(model::ExtendedHubbard1D)

Throw DomainError if model.V is not zero. Phase 1 only exposes the V = 0 limit (delegated Lieb–Wu) — the V ≠ 0 phase diagram is deferred to Phase 2.

_free_energy_fe(a, b, c) -> Float64

Closed-form free-energy density of the six-vertex model in the ferroelectric phase Δ > 1 (Lieb 1967c, Baxter 1982 §8.10):

f = −log max(a, b)

The ground state is the unique frozen configuration with all arrows parallel along the favoured axis, so the partition function is dominated by (max(a,b))^N and the free energy density is −log of the larger weight. At the KDP point (a > 2, b = c = 1) this reduces to −log a.

_heisenberg_szz_muller(J, q, ω) -> Float64

Müller-ansatz approximation to the longitudinal dynamic structure factor S^{zz}(q, ω) of the spin-½ XXX AFM chain. The closed-form expression, valid inside the two-spinon continuum, is

S^{zz}(q, ω) ≈  Θ[ω - ε_L(q)] · Θ[ε_U(q) - ω]
                ----------------------------------
                    2 √(ω² - ε_L(q)²)

with ε_L(q) = (π J / 2) |sin q| and ε_U(q) = π J |sin(q/2)|. Outside the closed continuum (i.e. ω ≤ ε_L or ω ≥ ε_U) the expression returns exactly 0.0.

The square-root singularity at the lower edge is integrable, so we return the raw analytical value (no regulator); callers performing ω-integrals are expected to use a routine that handles the integrable singularity (e.g. Gauss–Chebyshev or substitution ω² = ε_L² + s).

This is Phase 1 — the Müller ansatz captures the lower-edge behaviour and the support correctly but mis-estimates the spectral weight near the upper edge. The Phase 2 exact result is given by the Caux–Hagemans (2006) algebraic Bethe ansatz form factor sum; implementing it is tracked as a TODO in docs/src/calc/heisenberg-spinons.md.

_hubbard1d_charge_gap(t, U) -> Float64

Lieb–Wu (1968) Mott (charge) gap at half filling:

Δ_c = (16 t² / U) ∫_1^∞ dω  √(ω² - 1) / sinh(2π t ω / U).

The integrand vanishes at ω = 1 like √(ω-1) and decays exponentially at large ω; quadgk resolves both endpoints with the default Gauss–Kronrod adaptive subdivision.

_hubbard1d_check_half_filling(model::Hubbard1D)

Throw a DomainError if model.μ is not the half-filling chemical potential U/2. Phase 1 only exposes the Lieb–Wu closed forms at the particle-hole-symmetric point; general filling requires the coupled integral equations and is tracked as Phase 2 follow-up.

_hubbard1d_e0(t, U) -> Float64

Lieb–Wu (1968) ground-state energy per site at half filling:

E₀/N = -4 t ∫_0^∞ dω  J₀(ω) J₁(ω) / [ ω (1 + exp(ω U / 2t)) ].

_hubbard1d_e0_integrand(ω, t, U)

Integrand of the Lieb–Wu ground-state-energy integral at half filling,

f(ω) = J₀(ω) J₁(ω) / [ ω (1 + exp(ω U / 2t)) ].

The integrand is regular at ω = 0 (J₁(ω)/ω → 1/2 as ω → 0); we evaluate the limit explicitly to avoid 0/0 at the lower endpoint that some quadgk adaptive subdivisions produce.

_ising_sq_log_z(m, Lx, Ly, β, J) -> Real

log tr(T^Lx) for the Lx × Ly torus — the finite-N partition function in log form, generic in β and J so ForwardDiff Duals propagate through the transfer matrix.

_ising_sq_transfer_matrix(Ly, β, J) -> Matrix

Return the 2^Ly × 2^Ly symmetric transfer matrix for a row of Ly Ising spins with PBC along the row direction.

The symmetric (Hermitian) form is defined by:

T_{σ,σ'} = exp(βJ/2 · Eₕ(σ)) · exp(βJ · Eᵥ(σ,σ')) · exp(βJ/2 · Eₕ(σ'))

where Eₕ(σ) = Σⱼ₌₁^Ly σⱼ σ_{(j mod Ly)+1} (horizontal bonds within row, PBC) Eᵥ(σ,σ') = Σⱼ₌₁^Ly σⱼ σ'ⱼ (vertical bonds between rows)

Note: for Ly = 2, each horizontal bond is counted twice by the PBC sum (σ₁σ₂ + σ₂σ₁ = 2σ₁σ₂). The same applies along the transfer direction: tr(T^Lx) double-counts the vertical bonds when Lx = 2, because the cyclic product T[σ,σ'] T[σ',σ] weights the single pair of rows with the vertical Boltzmann factor twice. This doubling is an intrinsic property of the PBC transfer-matrix formula at L = 2 and is used consistently by the brute-force cross-check in test/util/classical_partition.jl (its bond list is constructed to match this convention by enumerating (i, j) ↔ (i, (j mod Ly) + 1) and (i, j) ↔ ((i mod Lx) + 1, j) for every site).

The function is generic in β and J so that automatic-differentiation dual numbers (e.g. ForwardDiff.Dual) propagate through the matrix elements. The element type of the returned matrix is inferred from typeof(exp(β * J)).

_kitaev1d_bdg_spectrum(N, μ, t, Δ) -> Vector{Float64}

Return the N non-negative BdG quasiparticle energies of the OBC Kitaev1D chain with N sites and parameters (μ, t, Δ).

The BdG matrix in Nambu basis is H_BdG = [[A, B]; [-B, -A]] (the same real, antisymmetric 2N × 2N convention used by _tfim_bdg_spectrum), with

A_{ii}   = -μ
A_{i,i+1} = A_{i+1,i} = -t
B_{i,i+1} = +Δ,  B_{i+1,i} = -Δ.

Eigenvalues come in ± pairs; the function returns the non-negative half, sorted ascending. The BdG zero modes of the topological phase (lifted by exponentially small N⁻¹ corrections) appear as the smallest entry.

This is a strict generalisation of _tfim_bdg_spectrum: at (μ, t, Δ) = (-2h, J, J) the matrix coincides with the TFIM BdG matrix and the returned spectrum matches _tfim_bdg_spectrum(N, J, h).

_kitaev1d_majorana_bloch(μ, t, k::Symbol) -> Matrix{Float64}

Return the 2 × 2 real antisymmetric Majorana Bloch matrix A(k) of the infinite Kitaev1D chain at the time-reversal-invariant momenta k ∈ {:zero, :pi}. At those momenta the pairing amplitude 2Δ sin k vanishes, so the matrix depends only on μ and t:

A(k=0)  = [ 0  μ + 2t; -(μ + 2t)  0 ]   ⇒   Pf A(k=0)  = μ + 2t
A(k=π)  = [ 0  μ - 2t; -(μ - 2t)  0 ]   ⇒   Pf A(k=π)  = μ - 2t

(Convention: the diagonal Majorana Bloch matrix at TR-invariant momenta has off-diagonal entry equal to the on-site mass term μ + 2t cos k.) This is the matrix on which the Kitaev (2001) Z₂ Pfaffian invariant is evaluated.

_kitaev_bz_integral(integrand, model; rtol) -> Float64

Adaptive nested Gauss-Kronrod quadrature of integrand(θ₁, θ₂) over the 2D Brillouin zone (θ₁, θ₂) ∈ [0, 2π)². Mirrors the inner / outer tolerance choice already used by the T = 0 Energy integration in KitaevHoneycomb.jl. Returns the integral value (the contribution of the prefactor 1/((2π)²) is left to the caller).

fetch(model::KitaevHoneycomb, ::Energy{:per_site}, ::Infinite;
      beta=nothing, rtol=1e-8, kwargs...) -> Float64

Per-site matter-sector energy of the infinite Kitaev honeycomb.

• beta = nothing (default) routes to the existing T = 0 ground-state formula ε_gs = -(1/(8π²)) ∫ |f| d²θ — preserved bit-for-bit (the method dispatched is the original one in KitaevHoneycomb.jl).
• beta::Real returns the finite-temperature value ε(β) = -(1/(2·(2π)²)) ∫ (λ/2) tanh(βλ/2) d²θ in the matter-sector free-fermion approximation.

Approximation regime: valid for T ≪ Δ_v (flux gap; Δ_v ≈ 0.07|K| at the isotropic gapless point). Above the flux gap the full finite-temperature thermodynamics needs flux-sector Monte Carlo (Nasu et al. 2014).

fetch(model::KitaevHoneycomb, ::Energy{:per_site}, ::OBC;
      Lx, Ly, beta=nothing) -> Float64

Per-site matter-sector energy on a Lx × Ly OBC honeycomb strip.

• beta = nothing (default) returns the T = 0 ground-state energy -Σ σ_k / (2·Lx·Ly) already provided by KitaevHoneycomb.jl — the original method is preserved unchanged.
• beta::Real returns the matter-sector thermal value ε(β) = -(1/(2·Lx·Ly)) Σ_k (σ_k/2) tanh(βσ_k/2).

Same matter-sector approximation regime as the Infinite variant.

_kitaev_fk_abs²(m, θ₁, θ₂) -> Float64

|f(k)|² = Kₓ² + Kᵧ² + K_z² + 2Kₓ Kᵧ cos θ₁ + 2Kᵧ K_z cos θ₂ + 2K_z Kₓ cos(θ₁ − θ₂).

_kitaev_logcosh2(x) -> typeof(x)

Numerically stable evaluation of log(2 cosh(x)) = |x| + log1p(exp(-2|x|)). The naive log(2 cosh(x)) overflows for |x| ≳ 700; this rewriting is exact and stays bounded (→ |x|) at large arguments.

Named with a _kitaev_ prefix to avoid shadowing the analogous helper in TFIM_thermal.jl once both files are included in the same module.

_kitaev_pbc_sector_energy(m, Lx, Ly, νx, νy) -> Float64

Per-site ground state energy on the Lx × Ly torus within a fixed topological flux sector (νx, νy) ∈ {0, 1/2}². νx = 1/2 flips every bond that crosses the m1 = Lx → m1 = 1 seam (anti-periodic fermion boundary condition along the x-direction); νy = 1/2 does the same in the y-direction. The four sectors correspond to the four inequivalent choices (W_x, W_y) ∈ {±1}² of the two non-contractible Wilson-loop operators.

Built as a singular-value decomposition of the Majorana hopping matrix M with entries ±K_γ, where the sign flips whenever a bond of type γ crosses one of the selected seams. Per-site energy = −Σ σ_k / (2·Lx·Ly).

_kitaev_thermo_infinite(quantity, model, β; rtol) -> Float64

Per-site matter-sector thermodynamic potential of the infinite Kitaev honeycomb at inverse temperature β. quantity is one of (:free_energy, :entropy, :specific_heat, :energy).

The integrals are evaluated via adaptive Gauss-Kronrod quadrature over the BZ (see _kitaev_bz_integral). The leading factor 1/(2·(2π)²) accounts for the per-site (two sublattice atoms) and the 2D BZ volume.

_kitaev_thermo_obc(quantity, model, Lx, Ly, β) -> Float64

Per-site matter-sector thermodynamic potential of the Lx × Ly OBC honeycomb strip at inverse temperature β, computed by summing each BdG positive Majorana mode's contribution.

The bipartite hopping matrix M = _obc_hopping_matrix(model, Lx, Ly) has Lx·Ly positive singular values σ_k; the lattice carries N_sites = 2·Lx·Ly sites, so per-site potentials scale as (1/(2·Lx·Ly)) Σ_k g(σ_k) with g the free-fermion-mode kernel.

_majorana_covariance_gs(h::AbstractMatrix) -> Matrix{Float64}

Ground-state Majorana covariance Σ of a quadratic Hamiltonian whose Majorana matrix is h. Σ is real antisymmetric, with ⟨γ_a γ_b⟩_GS = δ_{ab} + i Σ_{ab}. The formula Σ = -i · sign(i h) is used, evaluated through an eigendecomposition of the Hermitian matrix i h.

_majorana_evolution(h::AbstractMatrix, t::Real) -> Matrix{Float64}

exp(h * t) for a real antisymmetric h, returned as a real orthogonal matrix. Tiny imaginary noise from the matrix exponential is projected away.

_majorana_ham(N, J, h) -> Matrix{Float64}

Return the 2N × 2N real antisymmetric Majorana Hamiltonian matrix h of the OBC TFIM such that H = (i/4) Σ h_{ab} γ_a γ_b.

_majorana_thermal_covariance(h::AbstractMatrix, β::Real) -> Matrix{Float64}

Thermal-equilibrium Majorana covariance at inverse temperature β. The T = 0 ground-state formula Σ_GS = -i sign(i h) generalises to

Σ(β) = -i tanh((β/2) · i h)

so that ⟨γ_a γ_b⟩_β = δ_{ab} + i Σ(β)_{ab}. In the canonical 2×2 BdG block this gives the off-diagonal entry tanh(βΛ/2), recovering the Fermi–Dirac population of each quasiparticle and reducing to ±1 as β → ∞.

β = Inf falls back to _majorana_covariance_gs.

_obc_hopping_matrix(model, Lx, Ly) -> Matrix{Float64}

Lx·Ly × Lx·Ly bipartite hopping matrix for the flux-free-sector Majorana problem with open boundary conditions in both lattice directions. M[a, b] = K_γ whenever the A-sublattice site of unit cell a connects to the B-sublattice site of unit cell b via a bond of type γ (matching Lattice2D's Honeycomb topology conventions: type1 ↔ Kz within the same cell, type2 ↔ Kx crossing (+a₁, −a₂), type3 ↔ Ky crossing (−a₂)). Bonds that would leave the Lx × Ly strip are dropped.

Row indexing: a = (m1 − 1) Ly + m2 for (m1, m2) ∈ 1..Lx × 1..Ly.

_onsager_log_z_per_site(K) -> Float64

Onsager (1944) thermodynamic-limit log Z / N = -βf per site of the classical 2D Ising model on the square lattice, parameterised by the reduced coupling K = βJ. The bond-counting convention matches _ising_sq_transfer_matrix(Lx, Ly, β, J) (each bond enumerated once for Lx, Ly ≥ 3):

-β f(K) = log(2) + (1/(2π)) ∫₀^π dφ · log[(A(φ) + √(A(φ)² − B²))/2]
A(φ)    = cosh²(2K) − sinh(2K) · cos(φ),
B       = sinh(2K).

The integrand is finite for all K ≥ 0 (A ≥ B strictly except at the critical point K_c = ln(1+√2)/2, where A − B → 0 at φ = 0). QuadGK's adaptive quadrature handles the integrable square-root edge without intervention.

Reference: L. Onsager, Phys. Rev. 65, 117 (1944), Eq. (105).

_pauli_string(N::Int, site_ops::Pair{Int,Matrix{ComplexF64}}...) -> Matrix{ComplexF64}

Return the 2^N × 2^N tensor product that places each σ_k at its listed site and the identity elsewhere. site_ops is an arbitrary sequence of (site => matrix) pairs; missing sites get _σ0.

# σˣ at site 3 in an N=5 chain:
_pauli_string(5, 3 => _σx)

# σˣᵢ σᶻⱼ for (i,j) = (2,4):
_pauli_string(5, 2 => _σx, 4 => _σz)

_peschel_mode_entropy(ν::Float64) -> Float64

Per-mode von Neumann entropy

s(ν) = -[(1-ν)/2 · ln((1-ν)/2) + (1+ν)/2 · ln((1+ν)/2)]

as a function of the Majorana covariance singular value ν ∈ [0, 1]. Numerically stable at ν → 0 (s → ln 2) and ν → 1 (s → 0).

_peschel_mode_renyi(ν::Real, α::Real) -> Float64

Per-mode Rényi entropy of order α (α ≠ 1) for a Gaussian fermionic mode with covariance singular value ν ∈ [0, 1]:

s_α(ν) = (1 / (1 - α)) · log[ ((1+ν)/2)^α + ((1-ν)/2)^α ].

Numerically stable at the corner cases:

• ν → 1 (pure mode): the vanishing branch is dropped rather than clamped to eps(Float64) — clamping injects an α · log(eps) error that becomes visible at small α < 1 (the dominant branch is α log 1 ≈ 0, so a clamped contribution of exp(α log eps) no longer underflows). See test/models/test_TFIM_renyi.jl "Compare against full ED".
• ν → 0 (maximally mixed): both branches are (1/2)^α, giving s_α = log(2^{1-α}) / (1 − α) = log 2.
• α → ∞: s_α → -log p_max = -log((1+ν)/2) (min-entropy limit).
• α → 0⁺: s_α → log 2 (Hartley entropy for a 2-level Bernoulli).

_s1_heisenberg_hamiltonian_matrix(model::S1Heisenberg1D, N::Int) -> Matrix{ComplexF64}

Assemble the 3^N × 3^N OBC Hamiltonian

H = J Σᵢ [Sˣᵢ Sˣᵢ₊₁ + Sʸᵢ Sʸᵢ₊₁ + Sᶻᵢ Sᶻᵢ₊₁]

via explicit tensor products built from the spin-1 primitives. Capped by _MAX_ED_SITES_S1.

_s1_partial_trace_A(ρ::AbstractMatrix, ℓ::Int, N::Int) -> Matrix{ComplexF64}

Trace out sites ℓ+1 .. N of the 3^N × 3^N density matrix ρ and return the 3^ℓ × 3^ℓ reduced density matrix on sites 1 .. ℓ (local dimension d = 3 for spin-1).

The convention matches _spin1_string: site 1 is the leftmost (slowest- running) tensor factor, so the row/column index decomposes as (i_A, i_B) with i_A = 0:dA-1, i_B = 0:dB-1 where dA = d^ℓ, dB = d^(N-ℓ). Reshaping ρ as (dA, dB, dA, dB) and contracting the B indices then gives ρ_A.

_s1_thermal_expectation(kernel, O) -> Float64

Trace Tr(O ρ) from a precomputed _s1_thermal_kernel and a dense operator O of matching size. Returns the real part (Hermitian operators give real expectation values; small imaginary residue is numerical noise).

_s1_thermal_kernel(model, N, beta) -> NamedTuple

Compute the eigendecomposition of the OBC spin-1 Heisenberg Hamiltonian on N sites and return everything downstream observables need to evaluate thermal expectation values:

• H — the dense 3^N × 3^N Hermitian Hamiltonian matrix
• evals — sorted real eigenvalues (F.values)
• evecs — eigenvector matrix (F.vectors)
• weights — Boltzmann weights wₙ = exp(-β(eₙ - emin))/Z̃, normalised so sum(weights) == 1
• ρ — evecs · diagm(weights) · evecs', the thermal density matrix in the computational (product) basis

For beta = Inf we collapse onto the (possibly degenerate) ground-state manifold by zeroing all weights above evals[1] + 1e-12.

Constructed once per fetch call (so a routine that needs both an energy and a magnetisation does not pay for two eigen calls). No caching across calls — that would require a per-model mutable cache and hasn't been profiled to matter yet.

_six_vertex_delta(a, b, c) -> Float64

Lieb / Baxter phase parameter Δ = (a² + b² − c²) / (2 a b). All exact six-vertex thermodynamics is a function of Δ together with the overall energy scale.

_six_vertex_phase(a, b, c) -> Symbol

Return one of :ferroelectric, :disordered, :antiferroelectric according to whether Δ > 1, |Δ| ≤ 1, or Δ < −1. Phase boundaries |Δ| = 1 are treated as members of the disordered phase (KDP / F-model critical points).

_spin1_string(N::Int, site_ops::Pair{Int,Matrix{ComplexF64}}...) -> Matrix{ComplexF64}

Tensor-product analogue of _pauli_string for spin-1 chains. Returns the 3^N × 3^N operator that places each Sᵅ at its listed site and the 3×3 identity elsewhere.

_sx_expect(Σ, i) -> Float64

⟨σˣ_i⟩_GS = Σ[2i-1, 2i], since σˣ_i = -i γ_{2i-1} γ_{2i}.

_sx_sx_corr(N, J, h, i, j, t; β = Inf) -> ComplexF64

⟨σˣ_i(t) σˣ_j(0)⟩_β for the OBC TFIM. Reduces to a 4×4 Pfaffian since σˣ_k = -i γ_{2k-1} γ_{2k}. β = Inf ⇒ ground state.

_sx_sx_static_thermal(N, J, h, β) -> Matrix{Float64}

C[i, j] = ⟨σˣ_i σˣ_j⟩_β at OBC, evaluated as the t = 0 slice of the free-fermion Pfaffian / Wick formula. Symmetric, diagonal = 1 exactly (σˣ² = I). One Majorana diagonalisation amortised over N(N-1)/2 small (4×4) Pfaffians.

_sy_sy_corr(N, J, h, i, j, t; β = Inf) -> ComplexF64

⟨σʸ_i(t) σʸ_j(0)⟩_β for the OBC TFIM at inverse temperature β. Wraps _sy_sy_corr_from_cached with a fresh Majorana diagonalisation and time-evolution matrix.

_sy_sy_corr_from_cached(Σ, R, i, j) -> ComplexF64

⟨σʸ_i(t) σʸ_j(0)⟩_β from a precomputed thermal Majorana covariance Σ and time-evolution matrix R = exp(h·t). Same structure as _sz_sz_corr_from_cached (in TFIM_dynamics.jl) — only the index lists differ.

The overall phase is (-i)^{i+j-2}: each σʸ_k contributes -(-i)^{k-1}; the two minus signs cancel, leaving (-i)^{i+j-2}.

_sy_sy_static_thermal(N, J, h, β) -> Matrix{Float64}

C[i, j] = ⟨σʸ_i σʸ_j⟩_β at OBC. Same structure as _sx_sx_static_thermal but uses the σʸ Majorana index list defined in TFIM_yy.jl.

_sz_sz_corr(N, J, h, i, j, t; β = Inf) -> ComplexF64

⟨σᶻ_i(t) σᶻ_j(0)⟩_β for the OBC TFIM at inverse temperature β (default Inf ⇒ ground state).

Implementation: Wick / Pfaffian over the (2i-1) + (2j-1) = 2(i+j)-2 Majorana operators constituting the product. The overall phase is (-i)^{i+j-2}. The thermal generalisation enters only through the Majorana 2-point function

⟨γ_a γ_b⟩_β = δ_{ab} + i Σ(β)_{ab},   Σ(β) = -i tanh((β/2) i h),

so the body of the Pfaffian assembly is unchanged.

_sz_sz_spreading(N, J, h, center, times) -> Matrix{ComplexF64}

Return C[it, ix] = ⟨σᶻ_ix(t_it) σᶻ_center(0)⟩_GS for ix ∈ 1:N, t_it ∈ 1:length(times). Reuses the Majorana Hamiltonian, covariance, and (per time-step) evolution matrix, so the cost is O(length(times) · N · M³) with M = 2(center + N) - 2.

_sz_sz_static_thermal(N, J, h, β; i = nothing, j = nothing) -> Matrix{Float64}

Static ⟨σᶻ_i σᶻ_j⟩_β for the OBC TFIM at inverse temperature β. If both i and j are given, returns a single value (wrapped in a 1×1 matrix); otherwise returns the full N×N matrix of equal-time correlators.

_tfim_bdg_diagonalise(N, J, h) -> (Λ::Vector{Float64}, ϕ::Matrix{Float64}, ψ::Matrix{Float64})

Diagonalise the OBC BdG matrix of the TFIM with N sites, returning the N positive quasiparticle energies Λn and the corresponding particle / hole amplitudes (ϕn, ψ_n) ∈ ℝ^N (Lieb-Schultz-Mattis convention).

The Bogoliubov quasiparticles are

η_n = Σ_i [ (ϕ_n,i + ψ_n,i)/2 c_i + (ϕ_n,i − ψ_n,i)/2 c_i^† ]

so the orthogonal (N + N) × (2N) Bogoliubov transformation has rows (g_n; h_n) = ((ϕ_n + ψ_n)/2, (ϕ_n − ψ_n)/2).

_tfim_bdg_full(N, J, h) -> (Λ::Vector{Float64}, U::Matrix{Float64}, V::Matrix{Float64})

Full diagonalisation of the 2N × 2N TFIM OBC BdG Hamiltonian. Returns the N positive quasiparticle energies Λ[n] and the Bogoliubov amplitudes U[i, n], V[i, n] (site-index i, mode-index n) defining

η_n = Σᵢ U[i, n] cᵢ + V[i, n] cᵢ†,

normalised by Σᵢ (U[i, n]² + V[i, n]²) = 1.

The 2N × 2N BdG matrix is H_BdG = [A B; -B -A] with A symmetric (transverse field + hopping) and B antisymmetric (pairing); its spectrum is symmetric about zero by particle-hole symmetry. We pick the N eigenvectors with positive eigenvalues; for each such eigenvector (x, y) ∈ ℝ²ᴺ, we identify U[:, n] = x, V[:, n] = y.

Diagonalising the full 2N × 2N matrix (rather than the squared N × N system (A−B)(A+B) φ = Λ² φ) is robust to the near-zero edge mode in the ordered phase (h < J), which becomes Λ ≈ 10⁻¹⁵ and renders the ψ = (A+B)φ / Λ reconstruction ill-conditioned at certain N.

_tfim_bdg_spectrum(N, J, h) -> Vector{Float64}

Return the N positive BdG quasiparticle energies Λₙ > 0 for the OBC TFIM with N sites, Ising coupling J, and transverse field h.

The 2N×2N BdG matrix is: H_BdG = [[A, B]; [-B, -A]] where A (tridiagonal, symmetric) encodes hopping + onsite energy, and B (antisymmetric) encodes the pairing terms from JW transformation.

A_{ii}   = 2h
A_{i,i±1} = -J
B_{i,i+1} = +J,  B_{i+1,i} = -J

_tfim_cc_entanglement(J, h, ℓ, β; α=1.0) -> Float64

Calabrese-Cardy closed form for the Rényi-α entanglement entropy of a contiguous block of length ℓ in the infinite TFIM at inverse temperature β (use β = Inf for the ground state).

The Ising central charge c = 1/2 is hard-coded; the prefactor

P_α = (c / 6) · (1 + 1/α)

reduces to c/3 at α = 1 (von Neumann) and to (c/6)(1 + 1/α) otherwise. At criticality (h ≈ J) the gapped form is replaced by its ξ → ∞ limit log(2 ℓ) (T = 0) or log[(β/π) sinh(π ℓ / β)] (T > 0).

Returns the leading-log term only — the non-universal additive constant S_0 is dropped, so downstream fits should include an offset.

Off-critical at finite β requires composing the gapped CC mass with the thermal CFT scaling and is not implemented here.

_tfim_chi_F_infinite(J, h; rtol=1e-10) -> Float64

Per-site fidelity susceptibility χ_F / L for the infinite TFIM with respect to the transverse field h, computed by Gauss–Kronrod quadrature of the closed-form Bogoliubov-vacuum overlap integral

χ_F / L = (1 / 8π) ∫₀^π (J sin k)² / ε_k(h)⁴ dk,
ε_k(h)  = √(J² + h² − 2 J h cos k) = Λ_k(h) / 2.

Equivalent to (2/π) ∫₀^π (J sin k)² / Λ_k(h)⁴ dk if expressed in terms of the doubled BdG dispersion Λ_k = 2 ε_k.

The closed form χ_F / L = J² / (8 (J² - h²)²) (h ≠ J) follows by residue integration; the routine still does numerical quadrature so that small numerical errors propagate consistently with the OBC counterpart and so that the implementation is robust to future generalisations (e.g. χ_F with respect to J).

Throws DomainError at the critical point h = J (logarithmic divergence of the integrand at k = 0).

_tfim_chi_F_obc(N, J, h) -> Float64

Total fidelity susceptibility (NOT per site) of the OBC TFIM with N sites, with respect to the transverse field h.

Formula (2nd-order perturbation theory on the BdG diagonalisation):

χ_F = Σ_{p < q} 4 X_{pq}² / (Λ_p + Λ_q)²,
X_{pq} = Σⱼ [U_{q,j} V_{p,j} − U_{p,j} V_{q,j}],

where U[j, n], V[j, n] are the Bogoliubov particle/hole amplitudes returned by _tfim_bdg_full.

Cost: O(N³) eigendecomposition + O(N³) for the X matrix + O(N²) summation.

_tfim_chi_imag_zz_dynamic_proxy(J, h, β, q, ω, N_proxy, t_max, dt) -> Float64

OBC large-N proxy implementation of the imaginary part of the dynamic longitudinal susceptibility χ''_zz(q, ω; β). Used by the Infinite() fetch router; the heavy lifting lives here so the public method stays a thin wrapper.

The Kubo formula (Kubo 1957; Mahan, Many-Particle Physics, ch. 3) gives the spectral function as the time-Fourier transform of the expectation value of the commutator,

χ''_zz(q, ω; β) = (1/2) ∫dt e^{iωt} · (1/N_b) Σ_{i,j ∈ bulk}
                      e^{-iq(i-j)} ⟨[σᶻ_i(t), σᶻ_j(0)]⟩_β.

For Hermitian operators in a thermal state Hermiticity gives

⟨[σᶻ_i(t), σᶻ_j(0)]⟩_β = 2i Im ⟨σᶻ_i(t) σᶻ_j(0)⟩_β,

so each (i, j, t) point reuses the cached Pfaffian correlator _sz_sz_corr_from_cached and only the imaginary part participates. The result is real (the explicit i from the commutator combines with the (1/2) and the imaginary part to give a real spectral function); we return real(...) defensively against round-off.

Discretisation: t ∈ [-t_max, t_max] with spacing dt, (i, j) restricted to the central bulk window [N/4, 3N/4] of an N_proxy-site OBC chain. The Majorana Hamiltonian and thermal covariance are computed once; the evolution matrix R(t) is recomputed at every time step (single 2N × 2N expm) — same machinery as _tfim_zz_structure_factor_dynamic_proxy.

_tfim_dispersion(k, J, h) -> Float64

Single-particle BdG quasiparticle energy at momentum k for the TFIM with couplings J (Z-Z coupling) and h (transverse field):

Λ(k) = 2 √(J² + h² - 2 J h cos k).

_tfim_from_legacy_model(m::Model{:TFIM}) -> TFIM

Extract J, h from the legacy Model{:TFIM}(params) Dict and build the concrete struct.

_tfim_gge_energy_density(J, h0, hf) -> Float64

Per-site GGE energy density of the post-quench TFIM (J, hf) starting from the ground state of (J, h0):

ε_GGE = -(1/π) ∫₀^π dk · (Λ_k(hf) / 2) · (1 − 2 n_k(h0, hf))

with Λ_k(h) = 2 √(J² + h² − 2 J h cos k) and n_k = sin²(θ_k(h0) − θ_k(hf)).

Sanity:

• h0 == hf ⇒ n_k ≡ 0 ⇒ ε_GGE = -(1/π) ∫ Λ/2 dk = ε_GS(hf).

The integral is evaluated via adaptive Gauss-Kronrod quadrature with rtol = 1e-10.

_tfim_gge_magnetization_x(J, h0, hf) -> Float64

Per-site GGE transverse magnetisation ⟨σˣ⟩_GGE of the post-quench TFIM (J, hf) starting from the ground state of (J, h0):

⟨σˣ⟩_GGE = (2/π) ∫₀^π dk · (hf − J cos k)/Λ_k(hf) · (1 − 2 n_k)

The factor (hf − J cos k)/Λ_k(hf) equals cos(2 θ_k(hf)) / 2, so equivalently

⟨σˣ⟩_GGE = (1/π) ∫₀^π dk · cos(2 θ_k(hf)) · (1 − 2 n_k).

We use the explicit (hf - J cos k)/Λ_k form to share the integrand structure with the equilibrium MagnetizationX routine in TFIM_thermal.jl.

Sanity:

• h0 == hf ⇒ n_k ≡ 0 ⇒ recovers the T = 0 / β = ∞ equilibrium value (2/π) ∫ (h − J cos k)/Λ_k dk.

_tfim_gge_occupation(k, J, h0, hf) -> Float64

Conserved mode occupation n_k = sin²((θ_k(h0) − θ_k(hf))) generated by a sudden quench from initial transverse field h0 to final transverse field hf at momentum k.

Numerical detail: Δ(2θ) = 2θ_k(h0) − 2θ_k(hf) is computed via the two-argument atan (atan2) for branch safety; we then divide by 2 inside sin to get θ_k(h0) − θ_k(hf).

_tfim_gge_two_theta(k, J, h) -> Float64

The Bogoliubov double-angle 2 θ_k(h) of the TFIM at momentum k,

2 θ_k(h) = atan2(J sin k, h − J cos k) ∈ (-π, π].

The atan2 form keeps the angle on the principal branch through the critical point h = J and through k = 0, π, which a naive atan(J sin k / (h - J cos k)) misses (the denominator changes sign).

_tfim_loschmidt_obc_log_echo(N, J, h0, hf, t) -> Float64

Compute log L(t) for the OBC TFIM quench h0 → hf at finite N.

Implementation: diagonalise BdG of H0 and Hf, build the rotation matrix between their Bogoliubov bases, and evaluate the per-mode Loschmidt product.

Concretely, the Bogoliubov transformations relate fermion operators to quasiparticles as

η_n = Σ_i ( g_n,i c_i + h_n,i c_i^† ),

with the two row matrices G = (ϕ + ψ)/2, H = (ϕ − ψ)/2. The vacuum overlap of H0 and Hf Bogoliubov vacua decomposes into a per-mode product whose factors are

cos² θ_n + sin² θ_n e^{-2 i Λ_n^{(f)} t}

with cos² θn = Σm |P^{(+)}{n,m}|², sin² θn = Σm |P^{(-)}{n,m}|², where

P^{(+)} = G^{(f)} G^{(0)†} + H^{(f)} H^{(0)†}
P^{(-)} = G^{(f)} H^{(0)†} + H^{(f)} G^{(0)†}.

In the diagonal-pair limit (translationally invariant), this reduces to the issue's (cos²Δθ_k + sin²Δθ_k e^{-2iΛ_k t}) factor; for OBC the row-norm form folds residual mode mixing into per-mode angles consistent with unitarity (cos² + sin² = 1 is enforced by row normalisation).

Returns log L(t), suitable for direct λ = −log L / N conversion.

_tfim_pbc_log_Z(N, J, h, β) -> Float64

log Z (eq. (1)) for the N-site PBC TFIM at inverse temperature β.

_tfim_pbc_mass_gap(N, J, h) -> Float64

Lowest single excitation energy of the N-site PBC TFIM.

The Hilbert space splits by Z₂ parity P = ∏ᵢ σˣᵢ:

• even parity (NS): GS energy -½ Σ_k Λ_k, lowest excited state is a pair of quasiparticles → minimum two-particle energy is Λ_{k₁} + Λ_{k₂} over the two smallest NS Λ's.
• odd parity (R): GS energy -½ Σ_k Λ_k, lowest excited state is a single quasiparticle → minimum one-particle energy is min_k Λ_k^R.

The mass gap is min(E_excited) − min(E_GS) across both sectors.

_tfim_pbc_momenta(N, sector::Symbol) -> Vector{Float64}

Return the N momenta for the requested fermion sector, sector ∈ (:NS, :R).

• :NS (anti-periodic / even parity): k = (2j-1)π/N, j = 1..N
• :R (periodic / odd parity): k = 2jπ/N, j = 0..N-1 (includes k = 0; k = π only if N is even.)

_tfim_pbc_thermo(quantity, N, J, h, β) -> Float64

Compute one of the per-site thermodynamic potentials of the PBC TFIM. quantity is a Symbol:

• :free_energy → f = -T log Z / N
• :energy_per_site → ε = -∂_β log Z / N
• :entropy → s = β(ε - f)
• :specific_heat → cv = β² ∂²β log Z / N
• :transverse_magnetization → mx = ∂h log Z / (Nβ)
• :transverse_susceptibility→ χxx = ∂²h log Z / (Nβ)

_tfim_pfeuty_mz(J, h) -> Float64

Pfeuty (1970) closed-form spontaneous longitudinal magnetisation per site for the infinite TFIM at T = 0 (Pauli convention, eigenvalues ±1):

m_z = (1 - (h/J)²)^{1/8}    (h < J, ordered phase)
    = 0                     (h ≥ J, disordered phase)

The exponent β = 1/8 is the Onsager–Yang exponent of the 2D Ising universality class.

Returns the positive branch of the spontaneously-broken Z₂ doublet (the negative branch is -m_z). m_z = 0 exactly at the critical point h = J.

_tfim_sigma_x_quench_infinite(J, h_0, h_f, t) -> Float64

Closed-form integral

⟨σˣ⟩(t) = (1/π) ∫₀^π dk [
             cos(2 θ_k^f) · cos(2 Δθ_k)
           + sin(2 θ_k^f) · sin(2 Δθ_k) · cos(2 Λ_k^f t) ]

with Δθk = θk(hf) − θk(h0) for the infinite TFIM after a sudden quench h0 → h_f at fixed J. Adaptive Gauss–Kronrod quadrature with rtol = 1e-12 (the integrand is bounded, smooth on (0, π) for any non-critical h, and has a finite limit at the endpoints).

_tfim_sigma_x_quench_obc(N, J, h_0, h_f, i, t) -> Float64

Exact ⟨σˣ_i⟩(t) for the OBC TFIM after a sudden quench h_0 → h_f at fixed J, computed from the Majorana covariance of the initial ground state propagated by the post-quench Hamiltonian:

Σ_0   = _majorana_covariance_gs(_majorana_ham(N, J, h_0))
R(t)  = _majorana_evolution(_majorana_ham(N, J, h_f), t)
Σ(t)  = R(t) Σ_0 R(t)^T
⟨σˣ_i⟩(t) = Σ(t)[2i-1, 2i]   (real).

A single (i, t) point costs one 2N × 2N eigendecomposition for Σ0 plus one matrix exponential for R(t). Sweeps in i should reuse the returned matrix; sweeps in t should be hoisted to a custom loop that caches `Σ0and recomputes onlyR(t)`.

_tfim_thermo_infinite(quantity, J, h, β) -> Float64

Compute one of the per-site thermodynamic potentials of the infinite TFIM at inverse temperature β. quantity is a Symbol from (:free_energy, :entropy, :specific_heat, :transverse_magnetization, :transverse_susceptibility).

The integrals are evaluated via adaptive Gauss-Kronrod quadrature.

_tfim_thermo_obc(quantity, N, J, h, β) -> Float64

Per-site thermodynamic potential for the OBC finite-N TFIM, computed by summing the contribution of each BdG quasiparticle mode.

The transverse magnetisation and its susceptibility require the full single-particle Bogoliubov coefficients, not just the spectrum, so this routine diagonalises the BdG matrix internally to obtain them.

_tfim_transverse_obc(quantity, N, J, h, β) -> Float64

Compute m_x or χ_xx per site for OBC finite N by direct site-resolved BdG expectation. Uses the Majorana covariance formula

Σ(β) = -i tanh(β/2 · i h_BdG)

(see TFIM_dynamics.jl) and identifies ⟨σˣ_i⟩ = Σ[2i-1, 2i].

The transverse susceptibility is computed via _xx_uniform_susceptibility (exact Wick contraction, no numerical differentiation).

_tfim_zz_structure_factor_dynamic_proxy(J, h, β, q, ω, N_proxy, t_max, dt) -> Float64

OBC large-N proxy implementation of the dynamic longitudinal structure factor S_zz(q, ω; β). Used by the Infinite() fetch router; the heavy lifting is performed here so the public method stays a thin wrapper.

The Fourier integral is replaced by a finite Riemann sum on t ∈ [-t_max, t_max] with spacing dt, and i, j run over the central bulk window [N/4, 3N/4] of an N_proxy-site OBC chain to suppress boundary contamination. The Majorana Hamiltonian and thermal covariance are computed once; the evolution matrix R(t) is recomputed at every time step (single 2N × 2N expm).

_xx_static_structure_factor(N, J, h, β, q) -> Float64

S_xx(q, β) = (1/N) Σ_{i,j} e^{-iq(i-j)} ⟨σˣ_i σˣ_j⟩_β for OBC. Direct double sum over the N×N matrix returned by _sx_sx_static_thermal.

_xx_uniform_susceptibility(N, J, h, β) -> Float64

Exact transverse susceptibility per site for the OBC TFIM,

χ_xx(β) = (β/N) Var(Σᵢ σˣᵢ)
        = (β/N) Σᵢⱼ [ ⟨σˣᵢ σˣⱼ⟩_β − ⟨σˣᵢ⟩_β ⟨σˣⱼ⟩_β ]

Uses the Majorana covariance matrix Σ[a,b] = ⟨γₐγᵦ⟩ − δₐᵦ. With σˣᵢ = -i γ_{2i-1} γ_{2i} the connected correlators follow from Wick's theorem:

Diagonal (i = j): ⟨(σˣᵢ)²⟩c = 1 − Σ[2i-1, 2i]² Off-diagonal (i ≠ j): ⟨σˣᵢ σˣⱼ⟩c = −Σ[2i-1,2j-1]·Σ[2i,2j] + Σ[2i-1,2j]·Σ[2i,2j-1]

No numerical differentiation; no Pfaffian library calls.

_xxz1d_energy_yang_yang(J, Δ; rtol = 1e-12, atol = 1e-14) -> Float64

Ground-state energy per site of the spin-½ XXZ chain in the thermodynamic limit, evaluated by the Yang–Yang single-integral formula

e₀(Δ) = (J cos γ)/4 − J sin² γ · ∫_{-∞}^{∞} dλ /
        [2 cosh(πλ) · (cosh(2γλ) − cos γ)],

with Δ = cos γ, γ ∈ (0, π) (critical / gapless regime). The three canonical points Δ ∈ {-1, 0, 1} are dispatched in the caller for exactness; this helper is invoked only for general -1 < Δ < 1 with Δ ∉ {0}. Tolerance defaults give relative error ≤ 1e-12 against the closed-form values at Δ = 0, ±1 (verified in test_XXZ1D.jl).

Cost: a few QuadGK panel splits, ≈ 50 µs per call on a recent CPU.

_xxz1d_from_legacy_model(m::Model{:XXZ1D}) -> XXZ1D

_xxz1d_hamiltonian_matrix(model::XXZ1D, N::Int) -> Matrix{ComplexF64}

Assemble the 2^N × 2^N OBC Hamiltonian

H = J Σᵢ [ Sˣ_i Sˣ_{i+1} + Sʸ_i Sʸ_{i+1} + Δ Sᶻ_i Sᶻ_{i+1} ]
  = (J/4) Σᵢ [ σˣ σˣ + σʸ σʸ + Δ σᶻ σᶻ ]

via explicit tensor products built from the Pauli primitives in src/core/dense_ed.jl. Capped by _MAX_ED_SITES.

_xxz1d_partial_trace_B(ρ, ℓ, N) -> Matrix{ComplexF64}

Trace out the right N - ℓ sites from a 2^N × 2^N density matrix ρ, leaving the reduced state ρ_A on sites 1..ℓ (a 2^ℓ × 2^ℓ matrix). Implementation: reshape ρ to a 4-tensor (d_A, d_B, d_A, d_B) and contract the two d_B legs.

_xxz1d_reduced_density_matrix(model, N, ℓ, β) -> Matrix{ComplexF64}

Reduced density matrix of the first ℓ sites at inverse temperature β. For β = Inf we use the ground-state pure state |0⟩ (the lowest eigenvector of H); for finite β we build the full thermal density matrix and partial-trace.

The full thermal path costs O(D²) memory (D = 2^N); at the _MAX_ED_SITES = 12 ceiling that's a 4096×4096 complex matrix (~256 MB), still cheap.

_xxz1d_thermal_density_matrix(F::NamedTuple) -> Matrix{ComplexF64}

Build the density matrix ρ = exp(-βH)/Z = U diag(w) U† from a kernel result. Allocates a 2^N × 2^N matrix; only call for entanglement quantities that actually need ρ explicitly.

_xxz1d_thermal_expectation_op(F::NamedTuple, A::AbstractMatrix) -> Float64

Compute ⟨A⟩_β = Σₙ wₙ ⟨n|A|n⟩ for a Hermitian operator A, using a _xxz1d_thermal_kernel result F. Returns the real part (assumes the imaginary part is round-off only).

_xxz1d_thermal_kernel(model, N, beta) -> NamedTuple

Diagonalise the OBC XXZ Hamiltonian once and return everything that downstream observables need:

• evals :: Vector{Float64} — eigenvalues
• evecs :: Matrix{ComplexF64} — eigenvectors (columns)
• weights :: Vector{Float64} — Boltzmann weights wₙ, normalized so Σ wₙ = 1 (uses an emin shift)
• H :: Matrix{ComplexF64} — the Hamiltonian (kept for callers that need it, e.g. EnergyLocal)

Callers that only need the spectrum can ignore evecs.

_xxz1d_yang_yang_integrand(λ, γ)

Integrand of the Yang–Yang single-integral form for the XXZ ground-state energy density at anisotropy Δ = cos γ, namely

f(λ) = 1 / [ 2 cosh(π λ) · (cosh(2 γ λ) − cos γ) ].

Smooth and exponentially decaying for 0 < γ < π. Used by _xxz1d_energy_yang_yang.

_yy_static_structure_factor(N, J, h, β, q) -> Float64

S_yy(q, β) = (1/N) Σ_{i,j} e^{-iq(i-j)} ⟨σʸ_i σʸ_j⟩_β for OBC.

_zz_static_structure_factor(N, J, h, β, q) -> Float64

S_zz(q, β) = (1/N) Σ_{i,j} e^{-i q (i-j)} ⟨σᶻ_i σᶻ_j⟩_β evaluated by direct double sum from the thermal Pfaffian correlator. For OBC the lattice lacks translation invariance so this is the boundary-aware definition; in the bulk of a long enough chain it converges to the translation-invariant value.

_zz_uniform_susceptibility(N, J, h, β) -> Float64

Uniform (q = 0) longitudinal susceptibility per site,

χ_zz(β) = (β/N) Σ_{i,j} ⟨σᶻ_i σᶻ_j⟩_β,

obtained from a finite-temperature direct double sum (assumes ⟨σᶻ⟩β = 0 in the OBC ground manifold of the TFIM, which holds for any h ≠ 0 finite N). This is the static isothermal susceptibility — the fluctuation-dissipation form `χ = β · ⟨M²⟩c / Nfor the magnetisationM = Σᵢ σᶻᵢ`.

f_model(c::Real) -> SixVertex

F-model sub-family a = b = 1, free c (Lieb 1967b). Phase boundary at c = 2:

• c < 2 — disordered (Δ = 1 − c²/2 ∈ (-1, 1)).
• c = 2 — critical (Δ = −1).
• c > 2 — antiferroelectric / F-model phase (Δ < −1).

References

• E. H. Lieb, Phys. Rev. Lett. 18, 1046 (1967b).

fetch(model::AKLT1D, ::CorrelationLength, ::Infinite) -> Float64

Closed-form bulk correlation length of the AKLT chain,

ξ = 1 / log 3 ≈ 0.91024

(AKLT 1988). Connected ⟨S^z_0 S^z_r⟩ decays as (−1)^r (4/3) · 3^{−|r|} in the VBS state, giving ξ = 1/log 3 independent of J.

fetch(model::AKLT1D, ::Energy{:per_site}, ::Infinite) -> Float64

Per-site ground-state energy e₀ = −2J/3 of the infinite AKLT chain. Numerically identical to fetch(::AKLT1D, ::GroundStateEnergyDensity, ::Infinite); provided so the BC-explicit Energy(:per_site) API resolves through the same analytic path.

fetch(model::AKLT1D, ::ExactSpectrum, ::OBC; N::Int) -> Vector{Float64}

Sorted full eigenvalue spectrum of the OBC AKLT chain on N sites, computed by dense ED on the 3^N-dimensional Hilbert space. Capped by _MAX_ED_SITES_S1 (so N ≤ 8, 3^8 = 6561).

Under OBC the AKLT ground state is 4-fold degenerate (S=½ edge modes at each end, total spin S_tot ∈ {0, 1} from singlet ⊕ triplet of the two edge ½-spins), and dense ED on N ≤ 8 already exhibits this degeneracy with Δ_low_4 = E₃ − E₀ of order 10^{-13} (only floating-point noise).

fetch(model::AKLT1D, ::GroundStateEnergyDensity, ::Infinite) -> Float64

Closed-form ground-state energy density of the AKLT chain in the thermodynamic limit:

e₀ = −(2/3) J

Derived analytically from the projector form H = 2J Σ P₂(i,i+1) − (2J/3) (N − 1) (AKLT 1988): the VBS state is the exact null space of every bond P₂ projector, so per-bond energy is −2J/3 and per-site energy density (one bond per site in the bulk) is −2J/3.

fetch(model::AKLT1D, ::MassGap, ::Infinite) -> Float64

Haldane gap of the AKLT chain in the thermodynamic limit,

Δ ≈ 0.350 J

(numerical-exact DMRG value; A. García-Saez, V. Murg, and F. Verstraete, Phys. Rev. B 88, 245118 (2013), arXiv:1308.3631). No closed form is known; reliability=:medium in the registry.

fetch(model::AKLT1D, ::StringOrderParameter, ::Infinite) -> Float64

Kennedy–Tasaki non-local (string) order parameter of the AKLT chain,

O_str = 4/9

(closed form; AKLT 1988, Kennedy–Tasaki 1992). This is the infinite-distance limit of

O_str(r) = −⟨ S^z_i exp[iπ Σ_{i<k<j} S^z_k] S^z_j ⟩

evaluated in the VBS ground state, and detects the hidden Z₂ × Z₂ symmetry breaking that defines the Haldane phase. Independent of J.

fetch(model::AKLT1D, ::ZZCorrelation{:static}, ::Infinite; r::Integer) -> Float64

Exact equal-time spin-z two-point function of the AKLT VBS ground state at separation r:

⟨Sᶻ₀ Sᶻ_r⟩ = (−1)^r · (4/3) · 3^{−|r|}   (r ≠ 0),
⟨(Sᶻ)²⟩    = 2/3                          (r = 0).

Closed form (AKLT 1988); J-independent (the VBS ground state does not depend on J > 0). Exponential decay with rate log 3, consistent with fetch(::AKLT1D, ::CorrelationLength, ::Infinite) ξ = 1/log 3. Since ⟨Sᶻ⟩ = 0 in the VBS this equal-time value is already the connected correlation.

fetch(model::AKLT1D, ::ZZStructureFactor, ::Infinite; q::Real) -> Float64

Exact static (equal-time) spin-z structure factor of the AKLT chain,

S_zz(q) = Σ_r e^{iqr} ⟨Sᶻ₀ Sᶻ_r⟩ = 2 (1 − cos q) / (5 + 3 cos q)

(Arovas–Auerbach–Haldane 1988). S_zz(0) = 0 by total-Sᶻ conservation; antiferromagnetic peak S_zz(π) = 2. J-independent. This is the lattice-Fourier sum of the closed-form ZZCorrelation (−1)^r (4/3) 3^{−|r|}.

fetch(::AKLT2D, ::Energy{:per_site}, ::Infinite; J=m.J) -> Float64

Ground-state energy density of the 2D AKLT model on the honeycomb lattice (Affleck-Kennedy-Lieb-Tasaki 1988):

e_0 / N = 0

The Hamiltonian is a sum of non-negative spin-3 projectors, and the valence-bond-solid (VBS) state is annihilated by every term → frustration-free with exact zero ground-state energy density. The VBS state is the prototypical 2D AKLT-class SPT and a paradigmatic non-trivial PEPS (bond dimension 2).

References

• I. Affleck, T. Kennedy, E. H. Lieb, H. Tasaki, Commun. Math. Phys. 115, 477 (1988).
• F. Verstraete, J. I. Cirac, cond-mat/0407066 (2004) — PEPS realisation.

fetch(::BCFT, ::ResidualEntropy, ::Infinite; state::Symbol=:fixed) -> Float64

Return the Affleck-Ludwig boundary entropy log g_a for an Ising-CFT Cardy boundary state a.

The :free and :sigma symbols are synonyms: the physical free-spin boundary of the critical Ising model coincides with the |σ⟩ Cardy state, whose g-factor is g_σ = S_{σ,1}/√S_{1,1} = 1. The fixed-spin boundaries |±⟩ ∝ |1⟩ ± |ε⟩ each have g = 1/√2, as do the bare primary Cardy states |1⟩ and |ε⟩.

Throws a DomainError for any other state symbol (Phase 1 only supports Ising Cardy boundary states; other CFTs are Phase 2).

fetch(::ChernSimons3D, ::CentralCharge, ::Infinite; N=m.N, k=m.k)
    -> Rational{Int}

Sugawara central charge of the boundary WZW theory ŝu(N)_k dual to 3-D SU(N)_k Chern-Simons:

c = k (N² − 1) / (k + N),

returned as an exact Rational{Int}.

References

• E. Witten, Comm. Math. Phys. 121, 351 (1989).
• V. G. Knizhnik, A. B. Zamolodchikov, Nucl. Phys. B 247, 83 (1984).

fetch(::ChernSimons3D, ::PartitionFunction, ::Infinite; N=m.N, k=m.k) -> Float64

Closed-form partition function of 3-D SU(N)_k Chern-Simons on the three-sphere S³ (Witten 1989), equal to the S_{0,0} entry of the SU(N)_k modular S-matrix (Verlinde formula):

Z(S³; SU(N)_k) = N^{-1/2} (k + N)^{-(N-1)/2}
                 ∏_{1 ≤ j < l ≤ N} 2 sin( π (l − j) / (k + N) ).

For SU(2)_k this simplifies to Z = √(2 / (k + 2)) · sin(π / (k + 2)).

Boundary condition

Infinite() — S³ is a closed compact 3-manifold without boundary; no transfer-matrix BC label is meaningful, so the BC slot is the catch-all Infinite tag also used for thermodynamic-limit quantities elsewhere in QAtlas.

Verified values

• SU(2)_1: Z = 1/√2 ≈ 0.7071067811865476
• SU(2)_2: Z = 1/2 = 0.5
• SU(2)_3: Z = √(2/5) · sin(π/5) ≈ 0.3717480344601845
• SU(3)_1: Z = 1/√3 ≈ 0.5773502691896258

References

• E. Witten, Comm. Math. Phys. 121, 351 (1989).
• E. P. Verlinde, Nucl. Phys. B 300, 360 (1988).

fetch(::Cluster1D, ::Energy{:per_site}, ::Infinite; J=m.J) -> Float64

Ground-state energy density E_0 / N = -J. Every stabiliser K_i = σ^z_{i-1} σ^x_i σ^z_{i+1} contributes -J in the cluster-state ground state.

fetch(::Cluster1D, ::MassGap, ::Infinite; J=m.J) -> Float64

Spectral gap Δ = 2J. Flipping a single stabiliser eigenvalue K_i: +1 → -1 raises the energy by 2J; all other states differ by integer multiples of 2J.

fetch(model::Compass1D, ::MassGap, ::Infinite) -> Float64

Bulk mass gap of the 1D alternating-bond compass chain:

Δ = 2 |J_x − J_y|.

Closed-form result obtained from the Jordan–Wigner dual dimerised Kitaev / p-wave wire (Brzezicki–Dziarmaga–Oleś 2007). Δ vanishes at the symmetric point J_x = J_y, which is a first-order quantum phase transition between the X-bond-dimerised and Y-bond-dimerised ground states.

References

• Brzezicki–Dziarmaga–Oleś, PRB 75, 134415 (2007).

fetch(::CurieWeissIsing, ::CriticalExponents, ::Infinite; kwargs...) -> NamedTuple

Mean-field (Landau-Ginzburg) critical exponents α = 0, β = 1/2, γ = 1, δ = 3, ν = 1/2, η = 0 delegated to MeanField. Independent of h (universality is a zero-field concept).

References

• L. D. Landau, Phys. Z. Sowjet. 11, 26 (1937).
• H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena (1971).

fetch(::CurieWeissIsing, ::CriticalTemperature, ::Infinite; J=m.J) -> Float64

Mean-field critical temperature T_c = J (with k_B = 1) for the zero-field Curie-Weiss Ising model. For J ≤ 0 no ferromagnetic order at any positive temperature; returns 0.

At h ≠ 0 there is no sharp transition — this dispatch still returns J as the natural reference scale (the temperature at which χ(h=0) diverges), independent of the model's h field.

References

• L. D. Landau, E. M. Lifshitz, Statistical Physics §149.

fetch(::CurieWeissIsing, ::Energy{:per_site}, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Internal energy per site

u(β, J, h) = -J m*² / 2 - h m*.

At J ≤ 0: single-spin, u = -h tanh(βh). At J > 0, h = 0, T > T_c: u = 0 (paramagnet). At J > 0, T → 0, h ≥ 0: u → -J/2 - h (saturated).

References

• L. D. Landau, E. M. Lifshitz, Statistical Physics §149.

fetch(::CurieWeissIsing, ::FreeEnergy, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Saddle-point free energy per site

f(β, J, h) = J m*²/2 - β⁻¹ log[2 cosh(β(J m* + h))],

with m*(β, J, h) the unique stable SCE root. Limits:

• J ≤ 0: non-interacting single spin in field, f = -β⁻¹ log(2 cosh(βh)).
• J > 0, h = 0, T > T_c: paramagnet, f = -log(2)/β.
• J > 0, T → 0: saturated, f → -J/2 - |h|.

References

• L. D. Landau, E. M. Lifshitz, Statistical Physics §149.

fetch(::CurieWeissIsing, ::SpecificHeat, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Specific heat per site

c_v(β, J, h) = β² (J m* + h)² (1 - m*²) / (1 - β J (1 - m*²)).

Derived from u(β) = -J m*²/2 - h m* and dm*/dβ via implicit differentiation of the SCE. At J > 0, h = 0: zero for T > T_c, jumps to 3/2 at T_c⁻ (mean-field Landau jump). At J ≤ 0: non-interacting, c_v = (βh)² sech²(βh).

References

• L. D. Landau, E. M. Lifshitz, Statistical Physics §149.

fetch(::CurieWeissIsing, ::SpontaneousMagnetization, ::Infinite;
      beta::Real, J=m.J) -> Float64

Spontaneous magnetisation m*(β) = lim_{h → 0⁺} m(β, J, h) of the Curie-Weiss Ising model: the ℤ₂-positive nontrivial root of m = tanh(βJm).

By definition this dispatch always uses the zero-field SCE branch (the model's h field is ignored), since "spontaneous" magnetisation is the symmetry-broken value that survives in the field-free limit. For T ≥ T_c = J only the trivial m = 0 root exists and 0.0 is returned; otherwise bisection on g(m) = m - tanh(βJm) over [m_min, 1) converges in ~52 steps (no rate degeneracy at criticality).

References

• L. D. Landau, E. M. Lifshitz, Statistical Physics §149.

fetch(::CurieWeissIsing, ::SusceptibilityZZ, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Longitudinal isothermal susceptibility per site

χ(β, J, h) = β (1 - m*²) / (1 - β J (1 - m*²)).

Derived by implicit differentiation of m* = tanh(β(Jm* + h)) w.r.t. h. Reduces to the Curie-Weiss law χ = β / (1 - βJ) above T_c (where m* = 0). Diverges at T_c⁻ from the m* = 0 side. At J ≤ 0: non-interacting, χ = β sech²(βh).

References

• L. D. Landau, E. M. Lifshitz, Statistical Physics §149.

fetch(::CurieWeissIsing, ::ThermalEntropy, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Per-site entropy via the Gibbs identity s = β(u - f):

s(β, J, h) = log[2 cosh(β(J m* + h))]  -  β(J m* + h) m*.

Bounded between 0 (T → 0, saturated) and log 2 (T → ∞, h finite).

References

• L. D. Landau, E. M. Lifshitz, Statistical Physics §149.

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).

fetch(::E8, ::E8Spectrum, ::Infinite) -> Vector{Float64}

Returns the analytical E8 mass spectrum [m₁, m₂, …, m₈] normalised by m₁ = 1. m₂/m₁ = 2 cos(π/5) = ϕ (golden ratio) is the hallmark of the E8 symmetry.

fetch(model::ExtendedHubbard1D, ::ChargeGap, ::Infinite) -> Float64

Charge (Mott) gap of the t-U-V Hubbard chain at half filling. Phase 1 only supports the V = 0 limit, where the gap reduces to the Lieb–Wu (1968) integral and is delegated to Hubbard1D at μ = U/2. Any V ≠ 0 raises DomainError.

References

• Lieb, Wu, Phys. Rev. Lett. 20, 1445 (1968).
• Voit, Rep. Prog. Phys. 58, 977 (1995).
• Nakamura, Phys. Rev. B 61, 16377 (2000).

fetch(::FibonacciAnyons, ::TopologicalEntanglementEntropy, ::Infinite; kwargs...) -> Float64

Topological entanglement entropy of the Fibonacci-anyon non-Abelian topological order:

γ = log 𝒟 = (1/2) log(2 + φ)  ≈  0.6429653906

where 𝒟 = √(d1² + dτ²) = √(1 + φ²) = √(φ + 2) is the total quantum dimension and φ = (1 + √5)/2 is the golden ratio (= d_τ, the quantum dimension of the non-trivial τ anyon).

Fibonacci anyons are universal for topological quantum computation (Freedman-Kitaev-Larsen-Wang 2003) and arise as edge excitations of the Read-Rezayi Z_3 parafermion state (Read-Rezayi 1999).

References

• M. Freedman, A. Kitaev, M. Larsen, Z. Wang, Bull. Amer. Math. Soc. 40, 31 (2003).
• N. Read, E. Rezayi, Phys. Rev. B 59, 8084 (1999).
• A. Kitaev, J. Preskill, Phys. Rev. Lett. 96, 110404 (2006).
• M. Levin, X.-G. Wen, Phys. Rev. Lett. 96, 110405 (2006).

fetch(::GrossNeveu, ::CentralCharge, ::Infinite; N=m.N, g=m.g) -> Int

Free-fermion UV central charge of the Gross-Neveu model:

c(g = 0) = N

(N Dirac fermions in 1+1-D, each c = 1). At any g ≠ 0 the theory is massive in the IR (dynamical chiral symmetry breaking; Gross-Neveu 1974) and the IR central charge is zero; this Phase 1 exposes only the UV g = 0 value and raises DomainError for non-zero coupling, deferring the RG-flow case to Phase 2.

References

• D. J. Gross, A. Neveu, Phys. Rev. D 10, 3235 (1974).

fetch(::GrossNeveu, ::MassGap, ::Infinite; Λ::Real, N=m.N, g=m.g) -> Float64

Dynamically generated fermion mass m_F in the large-N Gross-Neveu model (Gross-Neveu 1974):

m_F = Λ · exp(−π / (N · g²)).

Λ is the renormalisation scheme's UV cutoff / dimensional-transmutation scale and is a required keyword argument — it is intentionally not stored on the GrossNeveu struct because the choice of scheme is up to the caller. The exponential suppression at weak coupling is the textbook signature of asymptotic-freedom-driven mass-gap formation.

Λ > 0, g > 0, N ≥ 1 required (DomainError otherwise).

Examples

• fetch(GrossNeveu(; N=1, g=1.0), MassGap(), Infinite(); Λ=1.0) → exp(-π) ≈ 0.0432139
• fetch(GrossNeveu(; N=2, g=1.0), MassGap(), Infinite(); Λ=1.0) → exp(-π/2) ≈ 0.2078796

References

• D. J. Gross, A. Neveu, Phys. Rev. D 10, 3235 (1974).
• N. Andrei, J. H. Lowenstein, Phys. Rev. Lett. 43, 1698 (1979).

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(model::Heisenberg1D, ::ZZStructureFactor, ::Infinite;
      q::Real, ω::Real, method::Symbol = :muller, J::Real = 1.0,
      kwargs...) -> Float64

Longitudinal dynamic structure factor S^{zz}(q, ω) of the infinite-volume spin-½ XXX antiferromagnetic Heisenberg chain in the thermodynamic limit.

The default method = :muller evaluates the Müller-ansatz closed form inside the two-spinon continuum (see _heisenberg_szz_muller):

S^{zz}(q, ω) ≈  Θ[ω − ε_L(q)] · Θ[ε_U(q) − ω]
                ----------------------------------
                    2 √(ω² − ε_L(q)²)

with edges defined by heisenberg_two_spinon_lower_edge and heisenberg_two_spinon_upper_edge. Outside the continuum the routine returns 0.0. No regulator is added at the lower edge: the singularity is integrable (∝ 1/√(ω − εL) for ω → εL⁺), and any quadrature scheme used downstream should treat it analytically rather than relying on a numerical cap.

Arguments

• q::Real: total momentum (radians).
• ω::Real: frequency (energy units consistent with J).
• method::Symbol = :muller: ansatz selector; only :muller is implemented in Phase 1. Reserved future values: :caux_hagemans for the exact form-factor sum (Phase 2).
• J::Real = 1.0: Heisenberg coupling.

References

G. Müller, H. Thomas, H. Beck, J. C. Bonner, "Quantum spin
  dynamics of the antiferromagnetic linear chain in zero and
  nonzero magnetic field", Phys. Rev. B 24, 1429 (1981).
J.-S. Caux, R. Hagemans, "The four-spinon dynamical structure
  factor of the Heisenberg chain", J. Stat. Mech. P12013 (2006)
  (Phase 2 reference, not yet implemented).

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, ::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(model::Hubbard1D, ::ChargeGap, ::Infinite) -> Float64

Lieb–Wu Mott (charge) gap at half filling:

Δ_c = (16 t² / U) ∫_1^∞ dω  √(ω² - 1) / sinh(2π t ω / U).

Strictly positive for any U > 0 (no Mott transition in 1D — the chain is insulating at half filling for arbitrarily small U, the celebrated Lieb–Wu result).

Asymptotic limits

• U → 0: Δ_c → 0 (exponentially small, ∝ exp(-2π t / U)).
• U → ∞: Δ_c → U - 4t + 8 t² log 2 / U.

References

• Lieb–Wu, PRL 20, 1445 (1968).

fetch(model::Hubbard1D, ::GroundStateEnergyDensity, ::Infinite) -> Float64

Lieb–Wu ground-state energy density E₀/N at half filling. Currently only implemented for μ = U/2; off-half-filling raises a DomainError.

Asymptotic limits

• U/t → 0: E₀/N → -4t/π (free 1D fermion).
• U/t → ∞: E₀/N → -4 t² log 2 / U (Heisenberg AFM reduction).

Both are exercised in test/standalone/test_hubbard1d.jl.

References

• Lieb–Wu, PRL 20, 1445 (1968).
• Essler et al., The One-Dimensional Hubbard Model (Cambridge, 2005).

fetch(::Hubbard1D, ::LuttingerParameter, ::Infinite; t=m.t, U=m.U, μ=m.μ) -> Float64

Luttinger-liquid parameter of the 1D Hubbard model in the free-fermion limit U = 0:

K = 1               (non-interacting spinful fermions; Voit 1995)

For U > 0, the Lieb–Wu Bethe-ansatz solution gives a non-closed-form expression for both Kρ (charge) and Kσ (spin); deferred to Phase 2 (or Phase 3). This entry exposes only the U = 0 free-fermion limit and throws DomainError for any U ≠ 0.

References

• E. H. Lieb, F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1968).
• J. Voit, Rep. Prog. Phys. 58, 977 (1995) — TLL review for Hubbard.

fetch(model::Hubbard1D, ::SpinGap, ::Infinite) -> Float64

Spin gap of the half-filled 1D Hubbard chain. Returns 0.0 exactly: the spinon branch is gapless for any U > 0 (rigorous Lieb–Wu result; spin-charge separation in the low-energy effective theory).

References

• Lieb–Wu, PRL 20, 1445 (1968).
• Essler et al., The One-Dimensional Hubbard Model (Cambridge, 2005).

fetch(::Ising2D, ::CriticalExponents) -> NamedTuple

Backward-compatible alias for fetch(Universality(:Ising), CriticalExponents(); d=2).

fetch(::IsingChain1D, ::CorrelationLength, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Spin-spin correlation length ξ(β, h) = 1 / log(λ_+ / λ_-) of the 1-D Ising chain, with λ_± the two transfer-matrix eigenvalues. At h = 0 this reduces to the Ising 1925 form

ξ(β, 0) = 1 / log(coth(β J)).

The correlation length is finite for every T > 0 (no finite-temperature phase transition) and diverges only in the zero-temperature ferromagnetic limit T → 0⁺ at J > 0.

References

• E. Ising, Z. Phys. 31, 253 (1925).

fetch(::IsingChain1D, ::CriticalTemperature, ::Infinite; kwargs...) -> Float64

Critical temperature of the 1-D Ising chain. Ising (1925) proved that no finite-temperature phase transition occurs in 1-D; the only singular point is T = 0. Returns 0.0.

References

• E. Ising, Z. Phys. 31, 253 (1925).

fetch(::IsingChain1D, ::Energy{:per_site}, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Internal energy per site at zero field (Ising 1925):

u(β, h=0) = ⟨H⟩/N = -J tanh(β J).

References

• E. Ising, Z. Phys. 31, 253 (1925).

fetch(::IsingChain1D, ::FreeEnergy, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Helmholtz free energy per site f(β, h) = -β⁻¹ log λ_+(β, J, h) of the 1-D Ising chain in the thermodynamic limit, with λ_+ the larger transfer-matrix eigenvalue. At h = 0 this reduces to the textbook form f = -β⁻¹ log[2 cosh(β J)].

References

• E. Ising, Z. Phys. 31, 253 (1925).

fetch(::IsingChain1D, ::SpecificHeat, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Specific heat per site at zero field (Ising 1925):

c_v(β, h=0) = (β J)² sech²(β J).

References

• E. Ising, Z. Phys. 31, 253 (1925).

fetch(::IsingChain1D, ::SpontaneousMagnetization, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Spontaneous magnetization per site of the 1D Ising chain — 0 for all T > 0 and any J ≠ 0 (Ising 1925: no spontaneous symmetry breaking in 1D at finite temperature).

References

• E. Ising, Z. Phys. 31, 253 (1925).

fetch(::IsingChain1D, ::SusceptibilityZZ, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Zero-field longitudinal susceptibility per site (Brush 1967):

χ(β, h=0) = β e^{2 β J}.

References

• E. Ising, Z. Phys. 31, 253 (1925).
• S. G. Brush, "History of the Lenz-Ising model", Rev. Mod. Phys. 39, 883 (1967), Eq. (4.18).

fetch(::IsingChain1D, ::ThermalEntropy, ::Infinite;
      beta::Real, J=m.J, h=m.h) -> Float64

Entropy per site at zero field (Ising 1925), via the Gibbs identity s = β(u − f) applied to the 1D Ising closed forms:

s(β, h=0) = log(2 cosh(β J)) − β J tanh(β J).

Bounded between 0 (T → 0) and log 2 (T → ∞).

References

• E. Ising, Z. Phys. 31, 253 (1925).

fetch(::IsingSquare, ::CriticalExponents, ::Infinite; kwargs...) -> NamedTuple

Onsager 1944 critical exponents of the 2D square-lattice Ising model at T_c (= 2J / log(1 + √2)):

α = 0,  β = 1/8,  γ = 7/4,  δ = 15,  ν = 1,  η = 1/4.

Delegated to the existing Universality(:Ising) infrastructure at d = 2. Rushbrooke (α + 2β + γ = 2), Widom (γ = β(δ − 1)), and Fisher (η = 2 − γ/ν) hyperscaling relations all check out.

The returned NamedTuple also carries the central charge c = 1//2 inherited from the CFT minimal model M(3,4) — same payload as fetch(Universality(:Ising), CriticalExponents(); d=2).

References

• L. Onsager, Phys. Rev. 65, 117 (1944) — exact 2D Ising solution.

fetch(::IsingSquare, ::CriticalTemperature; J=1.0) -> Float64

Exact critical temperature of the 2D Ising model on the square lattice:

T_c = 2J / ln(1 + √2) ≈ 2.269 J

Equivalently, the critical reduced coupling is Kc = J/Tc = ln(1+√2)/2, or sinh(2K_c) = 1.

References

L. Onsager, "Crystal Statistics. I.", Phys. Rev. 65, 117 (1944).

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

Per-site thermal energy ε(β) = -∂(log Z / N)/∂β from the Onsager closed form via ForwardDiff. At low T (β → ∞) ε → -2J — the ferromagnetic ground state has every spin aligned, contributing -J per bond and 2 bonds per site.

fetch(m::IsingSquare, ::Energy{:per_site}, ::PBC; beta, Lx, Ly, J) -> Float64

Per-site thermal energy ε(β) = -∂(log Z)/∂β / (Lx · Ly) for the finite torus, via ForwardDiff over the transfer-matrix log Z.

fetch(m::IsingSquare, ::FreeEnergy, ::Infinite; beta, J=m.J) -> Float64

Per-site Helmholtz free energy f(β) = -β⁻¹ · log Z / N of the classical 2D Ising model on the infinite square lattice (Onsager 1944).

fetch(m::IsingSquare, ::FreeEnergy, ::PBC; beta, Lx=m.Lx, Ly=m.Ly, J=m.J) -> Float64

Per-site Helmholtz free energy f(β) = -log Z / (β · Lx · Ly) for the Lx × Ly torus. Builds the 2^Ly × 2^Ly transfer matrix and trace-cubes; cost is O(2^{3 Ly}), so practical Ly ≤ 10–12.

fetch(::IsingSquare, ::PartitionFunction; Lx, Ly, β, J=1.0) -> Real

Exact partition function Z = Tr(T^Lx) for the classical 2D Ising model on an Lx × Ly square lattice with periodic boundary conditions in both directions.

The transfer matrix T acts along the Lx direction (row-to-row transfer), with each row containing Ly spins and PBC along the Ly direction.

Bond-counting convention

The transfer-matrix sum double-counts bonds along any dimension of length 2 (PBC wraparound of a length-2 ring produces the factor σ_1 σ_2 + σ_2 σ_1 = 2 σ_1 σ_2). The brute-force enumeration in test/util/classical_partition.jl is built under the same PBC convention so that Z_transfer-matrix == Z_bruteforce exactly for every (Lx, Ly, β, J). For Lx ≥ 3 and Ly ≥ 3 each physical bond is enumerated exactly once and the result coincides with the standard physical Z of a PBC lattice.

Special values

• β = 0 (any Lx, Ly, J): Z = 2^(Lx·Ly) — all configurations equally weighted
• J = 0 (any β, Lx, Ly): Z = 2^(Lx·Ly) — no interactions, same as β = 0

Automatic differentiation

fetch is generic in β and J so that ForwardDiff.Dual numbers propagate through it. This allows macroscopic thermodynamic quantities to be recovered from Z by differentiation — e.g.

⟨E⟩ = -∂(log Z)/∂β
C_v = β² · ∂²(log Z)/∂β²

See test/verification/test_ising_ad_thermodynamics.jl for a cross-check against direct ensemble averages.

Arguments

• Lx::Int: number of rows (transfer direction)
• Ly::Int: number of columns (row length, PBC)
• β::Real: inverse temperature (β = 1/(k_B T))
• J::Real: Ising coupling constant (default 1.0; J > 0 ferromagnetic)

References

L. Onsager, Phys. Rev. 65, 117 (1944).
B. M. McCoy and T. T. Wu, "The Two-Dimensional Ising Model" (1973).

fetch(m::IsingSquare, ::SpecificHeat, ::Infinite; beta, J=m.J) -> Float64

Per-site specific heat c_v(β) = β² · ∂²(log Z / N)/∂β² via ForwardDiff (twice). Diverges logarithmically at the Onsager critical point K_c = ln(1+√2)/2 (i.e. T_c = 2J/ln(1+√2)). Caller is responsible for staying off that slice; finite values everywhere else.

fetch(m::IsingSquare, ::SpecificHeat, ::PBC; beta, Lx, Ly, J) -> Float64

Per-site specific heat c_v(β) = β² · Var(H) / (Lx · Ly) for the finite torus, via ForwardDiff (twice) on log Z.

fetch(::IsingSquare, ::SpontaneousMagnetization; β, J=1.0) -> Float64

Exact spontaneous magnetization of the 2D Ising model on the infinite square lattice:

M(T) = (1 − sinh⁻⁴(2βJ))^{1/8}    for T < T_c  (i.e. sinh(2βJ) > 1)
M(T) = 0                            for T ≥ T_c

The critical exponent β = 1/8 is visible in the approach M → 0 as T → T_c⁻.

Special values:

• T = 0 (β → ∞): M = 1 (fully ordered)
• T = T_c: M = 0 (onset of disorder)

Arguments

• β::Real: inverse temperature (β = 1/(k_B T))
• J::Real: Ising coupling constant (default 1.0; J > 0 ferromagnetic)

References

C. N. Yang, "The spontaneous magnetization of a two-dimensional Ising
model", Phys. Rev. 85, 808 (1952).

fetch(m::IsingSquare, ::ThermalEntropy, ::Infinite; beta, J=m.J) -> Float64

Per-site Gibbs entropy s(β) = β · (ε − f) from the Onsager free-energy and energy paths.

fetch(m::IsingSquare, ::ThermalEntropy, ::PBC; beta, Lx, Ly, J) -> Float64

Per-site Gibbs entropy s(β) = β · (ε − f) for the finite torus.

fetch(::IsingTriangular, ::CriticalExponents, ::Infinite; kwargs...) -> NamedTuple

2D Ising universality critical exponents (Onsager 1944), shared by the square and triangular lattices via universality:

α = 0,  β = 1/8,  γ = 7/4,  δ = 15,  ν = 1,  η = 1/4.

Delegated to Universality(:Ising) at d = 2. The triangular and square 2D Ising lattices have different microscopic T_c (Onsager's $2/log(1+sqrt(2))$ for the square; Houtappel's $4/log 3$ for the FM triangular) but identical universal exponents — the canonical textbook example of universality.

References

• L. Onsager, Phys. Rev. 65, 117 (1944).
• R. M. F. Houtappel, Physica 16, 425 (1950) — exact triangular-lattice Ising solution.

fetch(::IsingTriangular, ::CriticalTemperature, ::Infinite; J=m.J) -> Float64

Exact critical temperature of the classical 2D Ising model on the triangular lattice in the Wannier 1950 sign convention $H = +J Σ σ_i σ_j$:

• J > 0 (AFM, frustrated) — T_c = 0. No long-range order at any positive temperature (Wannier 1950).
• J < 0 (FM, Houtappel) — T_c = 4 |J| / ln 3 ≈ 3.6409 |J| (Houtappel 1950).
• J = 0 — T_c = 0 (no interaction; degenerate value, kept finite).

References

• G. H. Wannier, Phys. Rev. 79, 357 (1950).
• R. M. F. Houtappel, Physica 16, 425 (1950).

fetch(::IsingTriangular, ::ResidualEntropy, ::Infinite; J=m.J) -> Float64

Zero-temperature residual entropy per site of the classical Ising model on the triangular lattice in the Wannier 1950 sign convention $H = +J Σ σ_i σ_j$.

• J > 0 (frustrated AFM) — Wannier (1950) closed form

  S/(N k_B) = (2/π) ∫₀^{π/3} ln(2 cos θ) dθ ≈ 0.32306594722…

  The integral is evaluated by QuadGK.quadgk to ~1e-12 precision.

• J ≤ 0 (FM or non-interacting Ising on a triangular lattice) — there is a unique pair of degenerate FM ground states related by the global ℤ₂ flip, so S_residual = 0.

References

• G. H. Wannier, "Antiferromagnetism. The triangular Ising net", Phys. Rev. 79, 357 (1950).
• R. M. F. Houtappel, Physica 16, 425 (1950) — independent derivation in the same period.