Classical 2D Ising Model on the Triangular Lattice

Overview

The classical Ising model on the triangular lattice is the canonical example of frustrated statistical mechanics. With antiferromagnetic coupling each elementary triangle cannot simultaneously satisfy all three bonds, leading to a macroscopically degenerate ground-state manifold and a non-zero zero-temperature entropy per site (Wannier 1950). With ferromagnetic coupling the lattice supports a standard order-disorder transition with $T_c = 4|J|/\ln 3$ (Houtappel 1950).

\[H = +J \sum_{\langle i,j \rangle} \sigma_i \sigma_j, \qquad \sigma_i \in \{-1, +1\}\]

(Wannier 1950 sign convention; each site has six nearest neighbours.)

Parameters: Ising coupling $J$. $J > 0$ — antiferromagnetic (frustrated). $J < 0$ — ferromagnetic.

Critical Temperature

Statement

Wannier 1950 / Houtappel 1950 closed forms:

\[T_c = \begin{cases} 0, & J > 0 \quad \text{(AFM, frustrated)} \\ \dfrac{4 |J|}{\ln 3} \approx 3.6410\,|J|, & J < 0 \quad \text{(FM)} \end{cases}\]

Physical context

AFM ($J > 0$): The triangular plaquette is the prototypical frustrated unit. The ground-state manifold has extensive entropy (see Residual Entropy below) and no spontaneous symmetry breaking occurs at any $T > 0$. Spin correlations decay algebraically (Stephenson 1964) — the system is effectively critical for all $T > 0$.
FM ($J < 0$): Standard 2D Ising universality class with exponents $\beta = 1/8$, $\nu = 1$, $\eta = 1/4$ (same as IsingSquare).

References

G. H. Wannier, "Antiferromagnetism. The triangular Ising net", Phys. Rev. 79, 357 (1950).
R. M. F. Houtappel, "Order-disorder in hexagonal lattices", Physica 16, 425 (1950).

QAtlas API

# AFM (frustrated): T_c = 0
Tc_afm = QAtlas.fetch(IsingTriangular(; J=1.0), CriticalTemperature(), Infinite())

# FM (Houtappel): T_c = 4 |J| / log 3
Tc_fm = QAtlas.fetch(IsingTriangular(; J=-1.0), CriticalTemperature(), Infinite())

Verification

Test file	What is checked
`test_ising_triangular.jl`	AFM `T_c = 0`; FM `T_c = 4

Residual Entropy (Wannier 1950)

Statement

For the antiferromagnetic case ($J > 0$), Wannier (1950) showed by exact transfer-matrix evaluation that the zero-temperature entropy per site of the triangular Ising net equals

\[\frac{S}{N k_B} = \frac{2}{\pi} \int_0^{\pi/3} \ln(2 \cos\theta)\, d\theta \approx 0.32306594722\ldots\]

This is strictly between $0$ and $\ln 2 \approx 0.693$ — frustration admits exponentially many ground states, but not all $2^N$ configurations.

For the ferromagnetic case ($J < 0$) the ground-state manifold consists only of the two ferromagnetically polarised states related by the global $\mathbb{Z}_2$ spin flip, so $S_\text{residual} = 0$ in the thermodynamic limit.

Physical context

The Wannier integral evaluates to $S/N \approx 0.32306594722$, consistent with a residual ground-state degeneracy of $\Omega \approx (e^{0.3231})^N \approx 1.381^N$ — a finite fraction $\approx 0.4663$ of $\log 2$.
The integrand $\ln(2 \cos\theta)$ is smooth on $[0, \pi/3]$ with $2 \cos(\pi/3) = 1$, so the upper endpoint is regular and QuadGK reaches machine-precision quadrature.

References

G. H. Wannier, Phys. Rev. 79, 357 (1950).
R. M. F. Houtappel, Physica 16, 425 (1950) — independent derivation in the same period; the kagome-lattice closed form is obtained by the same method.

QAtlas API

S = QAtlas.fetch(IsingTriangular(; J=1.0), ResidualEntropy(), Infinite())
# 0.32306594722...

fetch evaluates the Wannier integral via QuadGK.quadgk with rtol = atol = 1e-14, so the returned value is accurate to roughly $10^{-12}$.

Verification

Test file	What is checked
`test_ising_triangular.jl`	$S/N$ matches $0.32306594722$ at 1e-9 and the QuadGK recomputation at 1e-12
`test_ising_triangular.jl`	$J$-independence of $S/N$ in the AFM branch
`test_ising_triangular.jl`	$0 < S/N < \ln 2$

Future work

Two-point correlations $\langle \sigma_0 \sigma_R \rangle$: Stephenson (J. Math. Phys. 5, 1009, 1964) gave the exact asymptotic forms (algebraic decay along the symmetry axes for the AFM, exponential for the FM). Tracked as a follow-up issue.
Free energy density at finite $T$: Wannier 1950 / Houtappel 1950 give the exact integral representation; not yet wired into a FreeEnergy fetch method.
Kagome-lattice analogue: same Houtappel 1950 method gives the closed form for kagome-Ising; tracked separately.

Connections

IsingSquare — the non-frustrated square-lattice counterpart (Onsager 1944, Yang 1952).
Ising universality class — relevant for the FM branch of IsingTriangular. The AFM branch is effectively critical for all $T > 0$ and does not sit at a single RG fixed point with these exponents.

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Verified hubs

In the Verified Atlas, this model registers 10 hubs (quantity / BC pair). The badge column shows the R1 assurance level; click a hub link to see the exact verify(...) calls, references, and corroboration mechanism.

Quantity	BC	Assurance	Cards
`CriticalExponents`	`Infinite`	🟠 uncorroborated-but-feasible	0
`CriticalTemperature`	`Infinite`	🟢 corroborated-at-p	6
`Energy`	`Infinite`	🟠 uncorroborated-but-feasible	0
`FreeEnergy`	`Infinite`	🟠 uncorroborated-but-feasible	0
`ResidualEntropy`	`Infinite`	🟢 corroborated-at-p	2
`SpecificHeat`	`Infinite`	🟠 uncorroborated-but-feasible	0
`SpontaneousMagnetization`	`Infinite`	🟠 uncorroborated-but-feasible	0
`ThermalEntropy`	`Infinite`	🟠 uncorroborated-but-feasible	0
`UniversalityClass`	`Infinite`	🟠 uncorroborated-but-feasible	0
`ZZCorrelation`	`Infinite`	🟠 uncorroborated-but-feasible	0

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API

Every fetch(::Model, …) method registered for this model — together with the model struct(s) and exported helpers — generated directly from the source (in lock-step with @register):

QAtlas.IsingTriangular — Type

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:

Quantity	BC	Method
`CriticalTemperature`	`Infinite`	analytic
`ResidualEntropy`	`Infinite`	analytic (QuadGK)

Sign convention differs from `IsingSquare`

IsingTriangular uses the Wannier 1950 convention H = +J Σ σσ, so J > 0 means antiferromagnetic (frustrated). By contrast, IsingSquare uses the modern H = -J Σ σσ convention where J > 0 means ferromagnetic. Passing the same numerical value of J to the two models therefore selects opposite physics. To keep the user-facing critical-temperature comparison meaningful: IsingSquare(J=1.0) (FM) ↔ IsingTriangular(J=-1.0) (FM).

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

source

QAtlas.fetch — Method

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.

source

QAtlas.fetch — Method

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).

source

QAtlas.fetch — Method

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.

source

QAtlas.fetch — Method

fetch(::IsingTriangular, ::ZZCorrelation{:static}, ::Infinite; r=1, J=m.J) -> Float64

Zero-temperature nearest-neighbour spin–spin correlation ⟨σ_i σ_j⟩ of the classical triangular Ising model (Wannier 1950 convention H = +J Σ σσ).

For the frustrated antiferromagnet (J > 0), r = 1, the exact Wannier (1950) value

⟨σ_i σ_{i+1}⟩_{T=0} = -1/3,

a direct consequence of the ground-state rule that every elementary triangle carries exactly one unsatisfied bond (Σ σσ = -1 per triangle ⇒ -1/3 per bond). General separations r > 1 (Stephenson 1964 closed form) and the ferromagnetic branch are not implemented here.

References

G. H. Wannier, Phys. Rev. 79, 357 (1950).

source

QAtlas.fetch — Method

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

Per-site thermal energy ε(β) = -∂(log Z/N)/∂β from the Houtappel closed form via a central difference. At low T (β → ∞) ε → 3J = -3|J| — the ferromagnetic ground state aligns every spin, contributing J per bond and three bonds per site.

source

QAtlas.fetch — Method

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

Per-site Helmholtz free energy f(β) = -β⁻¹ log Z/N of the ferromagnetic (J < 0) triangular-lattice Ising model in the thermodynamic limit (Houtappel 1950). The frustrated AFM branch (J > 0) has no closed form and raises DomainError.

source

QAtlas.fetch — Method

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

Per-site specific heat c_v(β) = β² ∂²(log Z/N)/∂β² via a second central difference. Singular at the critical point T_c = 4|J|/ln 3 (2D-Ising universality); callers stay off that slice.

source

QAtlas.fetch — Method

fetch(m::IsingTriangular, ::SpontaneousMagnetization; β, J=m.J) -> Float64

Spontaneous magnetisation of the triangular-lattice Ising model.

For the ferromagnet (J < 0) the Potts–Domb closed form below T_c:

M(T) = [1 - 16 x³ / ((1-x)³ (1+3x))]^{1/8},   x = e^{-4β|J|},   T < T_c,
M(T) = 0,                                                        T ≥ T_c,

with critical exponent β = 1/8. The bracket vanishes exactly at x = 1/3, i.e. T_c = 4|J|/ln 3 (the registered Houtappel value), and M → 1 as T → 0.

For the antiferromagnet (J > 0) the triangular lattice is frustrated with no uniform long-range order at any temperature, so M = 0.

References

R. M. F. Houtappel, Physica 16, 425 (1950).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics (1982), Ch 11.

source

QAtlas.fetch — Method

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

Per-site Gibbs entropy s(β) = β(ε - f) from the Houtappel free-energy and energy paths. Bounded between 0 (T → 0) and ln 2 (T → ∞).

source

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