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

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

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

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