Classical 2D Ising Model on the Square Lattice

Overview

The two-dimensional Ising model on the square lattice is arguably the most studied model in statistical physics. It was the first model to exhibit a rigorous second-order phase transition (Onsager, 1944) and remains the benchmark for critical phenomena.

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

Parameters: Ising coupling $J > 0$ (ferromagnetic).

Universality: The critical point belongs to the 2D Ising universality class with central charge $c = 1/2$.

Partition Function (Transfer Matrix)

Statement

For an $L_x \times L_y$ square lattice with periodic boundary conditions in both directions:

\[Z(L_x, L_y, \beta, J) = \mathrm{Tr}(T^{L_x})\]

where $T$ is the $2^{L_y} \times 2^{L_y}$ symmetric transfer matrix. The matrix elements are

\[T_{\sigma, \sigma'} = e^{\beta J E_h(\sigma)/2} \cdot e^{\beta J E_v(\sigma, \sigma')} \cdot e^{\beta J E_h(\sigma')/2}\]

with $E_h(\sigma) = \sum_{j=1}^{L_y} \sigma_j \sigma_{(j \bmod L_y)+1}$ (horizontal bonds within a row, PBC) and $E_v(\sigma, \sigma') = \sum_{j=1}^{L_y} \sigma_j \sigma'_j$ (vertical bonds between rows).

Physical Context

• Valid for any finite $L_x \times L_y$ with PBC

• \[\beta = 0\]

  : $Z = 2^{L_x L_y}$ (all configurations equally weighted)

• \[J = 0\]

  : $Z = 2^{L_x L_y}$ (no interactions)

Derivation

The transfer matrix formulation rewrites the partition function as a product of row-to-row Boltzmann weights. Each "transfer" from row $\sigma$ to row $\sigma'$ includes the vertical bonds between the two rows and half of the horizontal bonds within each row (symmetric split).

The computation uses $Z = \mathrm{tr}(T^{L_x})$ via matrix exponentiation rather than eigendecomposition, to support automatic differentiation with ForwardDiff.Dual numbers.

Full derivation of the symmetric transfer matrix from the row-by-row factorisation: Transfer matrix: symmetric split construction .

Full derivation of the AD path from $\ln Z$ to thermodynamic observables ($F$, $\langle E\rangle$, $C_v$, $S$, …) via forward-mode dual numbers: Thermodynamic quantities from $\partial \ln Z$ via AD .

References

• L. Onsager, "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Phys. Rev. 65, 117 (1944).
• B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model, Harvard University Press (1973), Ch. 2.

QAtlas API

Z = QAtlas.fetch(IsingSquare(), PartitionFunction();
                 Lx=4, Ly=4, β=0.44, J=1.0)

Verification

Critical Temperature (Onsager)

Statement

\[T_c = \frac{2J}{\ln(1 + \sqrt{2})} \approx 2.269\,J\]

Equivalently, the critical reduced coupling $K_c = J / T_c$ satisfies

\[\sinh(2K_c) = 1\]

which is the self-dual point of the Kramers-Wannier duality.

Derivation

The Kramers-Wannier duality maps the high-temperature expansion of $Z$ to the low-temperature expansion of the dual partition function. The critical point is the unique fixed point of this duality:

\[e^{-2K_c} = \tanh K_c \quad \Longleftrightarrow \quad \sinh(2K_c) = 1\]

Solving: $K_c = \frac{1}{2}\mathrm{arcsinh}(1) = \frac{1}{2}\ln(1 + \sqrt{2})$.

Full derivation of the Kramers–Wannier duality (classical + TFIM operator form) and the self-dual fixed point: Kramers–Wannier duality .

References

• H. A. Kramers, G. H. Wannier, "Statistics of the Two-Dimensional Ferromagnet. Part I", Phys. Rev. 60, 252 (1941) — duality.
• L. Onsager, Phys. Rev. 65, 117 (1944) — exact solution confirming the duality prediction.

QAtlas API

Tc = QAtlas.fetch(IsingSquare(), CriticalTemperature(); J=1.0)

Verification

Spontaneous Magnetization (Yang)

Statement

\[M(T) = \begin{cases} \left(1 - \sinh^{-4}(2\beta J)\right)^{1/8} & T < T_c \\ 0 & T \geq T_c \end{cases}\]

The exponent $1/8$ directly gives the order parameter critical exponent $\beta = 1/8$ of the Ising universality class.

Derivation

Yang's calculation (1952) uses the Pfaffian method to evaluate the spontaneous magnetization of the infinite square lattice. The key step is computing $\langle \sigma_0 \rangle$ as a Toeplitz determinant, which in the thermodynamic limit gives the closed-form expression above.

Full derivation of the Toeplitz-determinant reduction, the strong Szegő-theorem evaluation, and the emergence of the exponent $1/8$ from the Wiener–Hopf $1/2$ × Szegő $1/4$ × correlator $1/2$ factor chain: Yang's spontaneous magnetization via Toeplitz determinant .

Physical Context

• \[T = 0\]

  ($\beta \to \infty$): $M = 1$ (fully ordered)

• \[T \to T_c^-\]

  : $M \sim (T_c - T)^{1/8}$ (critical exponent $\beta = 1/8$)

• \[T \geq T_c\]

  : $M = 0$ (disordered)

References

• C. N. Yang, "The spontaneous magnetization of a two-dimensional Ising model", Phys. Rev. 85, 808 (1952).

QAtlas API

M = QAtlas.fetch(IsingSquare(), SpontaneousMagnetization();
                 β=0.5, J=1.0)

Verification

Connections

• Universality: Ising universality class — $\beta = 1/8$, $\nu = 1$, $c = 1/2$.
• Quantum counterpart: TFIM — the 2D classical Ising model maps to the 1+1D TFIM via quantum-classical correspondence.
• Transfer matrix method: Methods — computational details of the $\mathrm{tr}(T^{L_x})$ evaluation.
• AD verification: Automatic Differentiation — thermodynamic quantities from $\partial(\ln Z)/\partial\beta$.

Underlying derivations

Every stored value on this page is backed by a step-by-step derivation note under docs/src/calc/:

• Partition function → transfer-matrix-symmetric-split (row-by-row factorisation, symmetric-split construction of $T$, $Z = \mathrm{Tr}(T^{L_x})$).
• Thermodynamic quantities from $Z$ → ad-thermodynamics-from-z (derivatives of $\ln Z$ via ForwardDiff, fluctuation-dissipation identity).
• Critical temperature → kramers-wannier-duality (self-dual fixed point $\sinh(2K_c) = 1$).
• Spontaneous magnetisation → yang-magnetization-toeplitz (Toeplitz determinant + strong Szegő theorem → $\beta = 1/8$).
• Critical exponents from CFT → ising-cft-primary-operators (Kac table of $\mathcal{M}(3, 4)$; three primaries and their scaling dimensions).
• Scaling relations among exponents → ising-scaling-relations (Rushbrooke, Widom, Fisher, Josephson — all four from the Widom scaling form).