Classical Six-Vertex (Ice-Rule) Model on the Square Lattice

Overview

The six-vertex model is the canonical exactly-solvable lattice model of constrained statistical mechanics. Each edge of the square lattice carries an arrow; at every vertex the ice rule (Pauling 1935; Lieb 1967a) demands two incoming and two outgoing arrows, leaving exactly six allowed local configurations. Lieb's three 1967 papers solved each of the named sub-models in closed form, and Sutherland 1967 followed up with the analytic continuation across the full disordered phase.

Each of the six vertex configurations is weighted in pairs:

\[\omega_1 = \omega_2 = a, \qquad \omega_3 = \omega_4 = b, \qquad \omega_5 = \omega_6 = c.\]

All thermodynamic information at fixed temperature is encoded in the single dimensionless invariant

\[\Delta = \frac{a^2 + b^2 - c^2}{2 a b}\]

which divides parameter space into three exactly solvable phases:

The square-ice point $a = b = c = 1$ (so $\Delta = 1/2$) sits inside the disordered phase and admits the celebrated Lieb 1967a residual entropy.

Square-Ice Residual Entropy (Lieb 1967a)

Statement

At the symmetric point $a = b = c = 1$ the zero-temperature configurational entropy per vertex is the closed form

\[\frac{S}{N} = \frac{3}{2} \log\frac{4}{3} \approx 0.4315231086776713\ldots\]

Physical Context

This is the per-vertex residual entropy of two-dimensional square ice — the ground-state ensemble of the ice-rule manifold without any energy bias. The number of admissible configurations grows exponentially with $N$ (the lattice volume) at exactly this rate. Lieb's derivation uses a Bethe-ansatz solution of the transfer matrix in the special unit-weight limit; the analogous Pauling 1935 mean-field estimate $S/N = \log(3/2)$ is not tight.

API

m = QAtlas.square_ice()
QAtlas.fetch(m, ResidualEntropy(), Infinite())     # → 0.4315231086776713

The implementation returns the closed form (3/2) * log(4/3) directly.

Free-Energy Density: Ferroelectric Phase (Lieb 1967c)

Statement

For $\Delta > 1$ the ground state is the unique frozen configuration in which all arrows are parallel along the dominant axis, and

\[f_{\mathrm{FE}}(a, b, c) = -\log \max(a, b).\]

Physical Context

The partition function is dominated by a single configuration: $Z \sim \big(\max(a, b)\big)^N$, so $f = -\log\max(a, b)$. At the KDP point $a > 2$, $b = c = 1$ this reduces to the Lieb 1967c result $f = -\log a$. The phase boundary $\Delta = 1$ is the KDP critical point.

API

m = QAtlas.kdp_model(2.0)   # a = 2, b = c = 1, Δ > 1 ⇒ FE
QAtlas.fetch(m, FreeEnergy(), Infinite())          # → -log(2)
QAtlas.fetch(m, ResidualEntropy(), Infinite())     # → 0.0  (frozen)

Free-Energy Density: Disordered Phase (Lieb / Sutherland 1967)

Statement

For $|\Delta| \le 1$, parameterise $\Delta = -\cos\mu$ with $0 \le \mu \le \pi$. The free-energy density is the trigonometric integral

\[-f(a, b, c) = \log c + \frac{1}{\pi} \int_0^\infty \frac{\sinh\big((\pi - \mu) x\big)\, \tanh(\mu x)}{x \cosh(\mu x)} \, dx\]

(Lieb 1967a, Sutherland 1967; cf. Baxter 1982 §8.8). The integrand decays exponentially at large $x$ and its small-$x$ Taylor expansion is $\mu (\pi - \mu) x + \mathcal{O}(x^3)$, so the integral is well-conditioned.

Physical Context

Inside the disordered phase the system has algebraic correlations and a continuously varying critical exponent set as a function of $\mu$. The square-ice point is the high-symmetry interior point $\mu = \pi/3$ (so $\Delta = 1/2$), and at that special value the integral evaluates to the closed-form Lieb 1967a result, which we use as a machine-precision cross-check on the numerical quadrature.

API

m = SixVertex(; a=1.0, b=1.0, c=0.5)   # Δ = 0.875, disordered
QAtlas.fetch(m, FreeEnergy(), Infinite())     # finite, evaluated by QuadGK to ~1e-12

# Square-ice cross-check
m = QAtlas.square_ice()
QAtlas.fetch(m, FreeEnergy(), Infinite())     # ≈ -(3/2) log(4/3)

The implementation evaluates the integral with QuadGK.quadgk on $(0, \infty)$ with rtol = 1e-12, atol = 1e-14, using a small-$x$ Taylor expansion for $|x| < 10^{-8}$ to suppress floating-point cancellation.

Free-Energy Density: Antiferroelectric Phase (Lieb 1967b) — Deferred

For $\Delta < -1$, parameterise $\Delta = -\cosh\lambda$ with $\lambda > 0$. The closed form is the Lieb 1967b elliptic-function expression involving Jacobi theta functions; it is heavier than the disordered branch and is deferred to phase 3 of issue #163. Calls to fetch(::SixVertex, ::FreeEnergy, ::Infinite) in the AFE phase currently raise an informative ArgumentError. The phase boundary at $\Delta = -1$ (e.g. f-model with $c = 2$) is already covered by the disordered-branch limit $\mu \to \pi$.

Convenience constructors

QAtlas.square_ice()          # a = b = c = 1                       (Δ = 1/2)
QAtlas.f_model(c::Real)      # a = b = 1, c free; AFE for c > 2    (Δ = 1 − c²/2)
QAtlas.kdp_model(a::Real)    # b = c = 1, a free; FE for a > 2    (Δ = a/2)

These are thin wrappers around SixVertex(; a, b, c).

Verification

The standalone test test/standalone/test_six_vertex.jl covers:

• Square ice $S/N = (3/2) \log(4/3) pprox 0.4315231087$ at atol = 1e-14.
• Phase classification on $\Delta = \pm 1$ boundaries (KDP at $a = 2$, F-model at $c = 2$).
• FE plateau $f = -\log \max(a, b)$ at the KDP point ($a = 3$, $b = c = 1$) and at a $b$-dominated point ($a = 1$, $b = 3$, $c = 1$).
• Square-ice FreeEnergy closed form $f = -(3/2) \log(4/3)$.
• AFE deferral and generic-disordered deferral both asserted via @test_throws ArgumentError.

Run it as

julia --project=test test/standalone/test_six_vertex.jl

References

• E. H. Lieb, Residual entropy of square ice, Phys. Rev. 162, 162 (1967a). Closed form for the square-ice residual entropy.
• E. H. Lieb, Exact solution of the F model of an antiferroelectric, Phys. Rev. Lett. 18, 1046 (1967b). AFE elliptic free energy.
• E. H. Lieb, Exact solution of the two-dimensional Slater KDP model of a ferroelectric, Phys. Rev. Lett. 19, 108 (1967c). KDP / FE.
• B. Sutherland, Exact solution of a two-dimensional model for hydrogen-bonded crystals, Phys. Rev. Lett. 19, 103 (1967). Disordered-phase trigonometric integral.
• R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, 1982), ch. 8 — textbook treatment.

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