[Paper Review] The Ising model on a cylinder: universal finite size corrections and diagonalized action

Rafael L. Greenblatt|arXiv (Cornell University)|Sep 5, 2014

Theoretical and Computational Physics8 references3 citations

TL;DR

This paper derives universal finite-size corrections to the free energy of the 2D Ising model on a cylindrical lattice using an exact solution expressed via the determinant of a block-diagonalized action matrix. By diagonalizing the action through Fourier transforms and solving for eigenvalues via a characteristic polynomial, the authors derive an explicit expression for the logarithm of the partition function, confirming conformal field theory predictions with a universal correction term proportional to $\kappa(M/N)$, where $\kappa$ is given in terms of Jacobi theta functions and matches $\pi/24 \cdot N/M$ in the $N \gg M$ limit.

ABSTRACT

Finite size corrections to the pressure (free energy) of the Ising model on a 2 dimensional cylinder are calculated and shown to be consistent with the predictions of conformal field theory. The exact solution of the model is expressed in terms of the determinant of a block-diagonal matrix.

Motivation & Objective

• To derive exact finite-size corrections to the free energy of the 2D Ising model on a cylindrical geometry with periodic boundary conditions in the transverse direction.
• To provide a complete analytical treatment of the partition function by diagonalizing the action matrix, which is essential for verifying conformal field theory predictions in non-trivial boundary conditions.
• To resolve the absence in the literature of exact finite-size corrections for the Ising model on a cylinder, despite known solutions for other topologies like the Möbius strip or Klein bottle.
• To establish that the universal correction term $\kappa(M/N)$ in the logarithm of the partition function matches the conformal field theory prediction $\pi/24 \cdot N/M$ in the $N \gg M$ limit.

Proposed method

• The partition function is expressed as $Z = \frac{1}{2} (2\cosh\beta J)^{MN} \sqrt{|S|}$, where $S$ is a large antisymmetric matrix representing the action, with entries defined via nearest-neighbor couplings and periodic boundary conditions in the $y$-direction.
• A Fourier transform in the $y$-direction block-diagonalizes the action matrix into $N$ blocks of size $4M \times 4M$, which are further decomposed into two $2M \times 2M$ blocks using a transformation based on eigenvalues of a tridiagonal matrix.
• The eigenvalues of the resulting matrix are determined by solving a characteristic polynomial equation (Eq. 23), and their properties are analyzed in Lemma 1, particularly near $k=0$ and $k=2\pi$, which govern the asymptotic behavior.
• The logarithm of the partition function is expanded using the Euler-Maclaurin formula and asymptotic analysis of the product over $k$-modes, with the dominant contribution arising from the term $\prod_k (1 + z_+(k)^{-2M})$.
• This product is expressed in terms of Jacobi theta functions via the identity $\prod_{r=0}^\infty (1 + e^{-(2r-1)\pi \zeta})^2 = \theta_3(e^{-2\pi\zeta}) / \theta_0(e^{-2\pi\zeta})$, where $\zeta = M/N$.
• The final expression for the universal correction term is derived as $\kappa(\zeta) = \frac{1}{6} \log\left( \frac{\theta_3^2(e^{-2\pi\zeta})}{2\theta_2(e^{-2\pi\zeta})\theta_4(e^{-2\pi\zeta})} \right)$, which is verified asymptotically to match CFT expectations.

Experimental results

Research questions

• RQ1 Does the exact solution of the 2D Ising model on a cylinder yield a universal finite-size correction term in the free energy that matches the conformal field theory prediction?
• RQ2 Can the action matrix of the Ising model on a cylinder be fully diagonalized to allow exact computation of the partition function and its logarithmic expansion?
• RQ3 What is the explicit functional form of the universal correction term $\kappa(M/N)$ for the Ising model on a cylinder, and how does it compare to known results for other boundary conditions?
• RQ4 How do the asymptotic behaviors of $\kappa(\zeta)$ for $\zeta \to 0$ and $\zeta \to \infty$ match the CFT prediction $\pi/24 \cdot N/M$ and $\pi/6 \cdot N/M$ respectively?

Key findings

• The exact partition function of the 2D Ising model on a cylinder is derived using the determinant of a block-diagonalized action matrix, with the diagonalization achieved via Fourier transform and solution of a characteristic polynomial.
• The universal finite-size correction term in the logarithm of the partition function is found to be $\kappa(\zeta) = \frac{1}{6} \log\left( \frac{\theta_3^2(e^{-2\pi\zeta})}{2\theta_2(e^{-2\pi\zeta})\theta_4(e^{-2\pi\zeta})} \right)$, where $\zeta = M/N$.
• In the limit $N \gg M$ ($\zeta \to 0$), the correction term asymptotically approaches $\frac{\pi}{48} \zeta^{-1}$, which matches the CFT prediction $\frac{\pi}{24} \cdot \frac{N}{M}$ for open boundary conditions.
• In the limit $M \gg N$ ($\zeta \to \infty$), the correction term behaves as $\frac{\pi}{12} \zeta = \frac{\pi}{12} \cdot \frac{M}{N}$, consistent with the CFT prediction $\frac{\pi}{6} \cdot \frac{N}{M}$ for periodic boundary conditions.
• The derivation confirms that the Ising model on a cylinder exhibits universal finite-size corrections predicted by conformal field theory, even though the correction function $\kappa$ differs from those on other topologies such as the Möbius strip or Klein bottle.