Direct derivation of the surface–corner scaling function $\Theta_{s,c}(x)$ and rigorous proof of the series–CFT identity for the critical Ising rectangle

In the exact solution of the square-lattice Ising model on an open $L\times M$ rectangle near criticality, the universal Casimir-potential scaling function decomposes as $\Theta(x,\rho)=\Theta^{(oo)}(x)+\rho^{-1}\Theta_{s,c}(x)+\Psi(x,\rho)$, where $\rho$ is the aspect ratio and $x$ the temperature scaling variable. Hucht's 2017 finite-size-scaling analysis left two analytic gaps open. First, the surface–corner contribution $\Theta_{s,c}(x)$ — the regularized finite-size-scaling limit of the residual free energy $F^{res}_{s,c}$ that carries the contributions of the two vertical surfaces and the four corners — has never been calculated directly; it was inferred indirectly from the symmetry of the square at $\rho=1$ combined with a scaling relation. Second, the exact critical-point series for the residual strip partition function, $\Sigma_\circ(0,\rho)=1+\frac14 e^{-2\pi\rho}+\frac{13}{32}e^{-4\pi\rho}+\cdots$, is known to agree with the conformal-field-theory expression $(e^{-2\pi\rho})_\infty^{-1/4}$ of Kleban and Vassileva only through the first 30 series coefficients, without proof of equivalence. Solving both problems — for instance along the Szegő-theorem route announced for the Hankel/Toeplitz determinant representation of the partition function — would complete the analytic finite-size-scaling solution of one of the very few exactly solved geometries with corners and turn the conformal predictions of Cardy–Peschel and Kleban–Vassileva for corner-dominated Casimir potentials into rigorously established results.

Background

The Ising model on a finite lattice exhibits finite-size scaling (FSS) near its critical point: as the linear dimensions diverge with the correlation length held fixed in units of the lattice spacing, the singular part of the free energy is governed by universal functions of the aspect ratio $\rho=L/M$ and the temperature scaling variable $x\propto(T/T_c-1)M^{1/\nu}$. Subtracting from the total free energy its thermodynamic-limit value leaves the residual free energy, also called the Casimir potential, which generates finite-size forces analogous to the critical Casimir effect studied in confined near-critical fluids.

For a rectangle with open (free) boundary conditions on all four sides, the thermodynamic-limit free energy decomposes into bulk, surface, and corner parts, $F_\infty=LMf_b+Lf_s^{\leftrightarrow}+Mf_s^{\updownarrow}+f_c$. While the bulk and surface free energies of the square-lattice Ising model have been known since the classical exact solution, the corner free energy $f_c$ was obtained only recently: Vernier and Jacobsen conjectured a product formula, which Baxter derived. Near criticality the corner free energy diverges logarithmically, $f_c\simeq\frac18\log|1-T/T_c|$, which considerably complicates the FSS analysis.

In two companion papers, Hucht solved the anisotropic Ising model on the open rectangle exactly: part I gave the partition function in terms of a $2M\times2M$ transfer matrix and a residual determinant, and part II took the FSS limit. In part II the Casimir-potential scaling function decomposes as

$$\Theta(x,\rho)=\Theta^{(oo)}(x)+\rho^{-1}\Theta_{s,c}(x)+\Psi(x,\rho).$$

Here $\Theta^{(oo)}(x)$ is the known scaling function of an infinitely long open strip (obtainable from older exact solutions), $\rho\Psi(x,\rho)=-\log\Sigma(x,\rho)$ where $\Sigma$ is computed exactly from the residual determinant $\det(1+\mathbf{Y})$ of the transfer-matrix solution, and $\Theta_{s,c}(x)\simeq F^{res}_{s,c}(L_{\updownarrow})$ is the surface–corner contribution: the regularized FSS limit of the part of the exact free energy containing the contributions of the two vertical surfaces together with the four corners. At the critical point $x=0$ the analysis confirmed two conformal-field-theory (CFT) predictions: the Cardy–Peschel result that corners induce a logarithmic divergence of the critical free energy with amplitude $1/16$, and the Kleban–Vassileva formula for the shape dependence of the Casimir amplitude. A striking consequence of the corners is that the Casimir amplitude (finite) differs from the Casimir potential at criticality (divergent logarithmically) — a behavior not known from other geometries and boundary conditions.

Two steps of this derivation, however, were left open in the source. (i) The surface–corner contribution $\Theta_{s,c}(x)$ could not be computed directly from the regularized FSS limit of $F^{res}_{s,c}$, because the logarithmically divergent corner free energy makes the naive limit nonexistent. Instead it was obtained indirectly: the symmetry of the square ($\rho=1$) under exchange of the two lattice directions implies $\partial_\rho\Theta_\circ(x_\circ,\rho)|_{\rho=1}=0$ (a result of anisotropic FSS theory; here $\Theta_\circ(x_\circ,\rho)$ and $\vartheta_\circ(x_\circ,\rho)$ are the Casimir-potential and Casimir-force scaling functions of ref. [1] written in the volume scaling variable $x_\circ=x\rho^{1/2}$), which yields a scaling relation between $\Theta_\circ$ and the force function $\vartheta_\circ$, and the consistency relation $\Theta_{s,c}(x)=-\rho\Theta^{(oo)}(x)+\log\Sigma(x,\rho)+\Theta_\circ(x\rho^{1/2},\rho)$ then fixes $\Theta_{s,c}(x)$ for all $x$. (ii) At $x=0$ the exact series

$$\Sigma_\circ(0,\rho)=1+\tfrac14 e^{-2\pi\rho}+\tfrac{13}{32}e^{-4\pi\rho}+\tfrac{55}{128}e^{-6\pi\rho}+\tfrac{1235}{2048}e^{-8\pi\rho}+\cdots$$

agrees with the CFT expression of Kleban and Vassileva, $\Sigma_\circ(0,\rho)=e^{-\pi\rho/48}\eta^{-1/4}(\mathrm{i}\rho)=(e^{-2\pi\rho})_\infty^{-1/4}$, where $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\ge1}(1-e^{2\pi i n\tau})$ is the Dedekind eta function and $(q)_\infty=\prod_{n\ge1}(1-q^n)$ the infinite $q$-Pochhammer product; but the agreement was verified only numerically, through the first 30 series coefficients, and no proof of the identity is known.

A natural route to both gaps has since been identified. Hucht's part III recast the partition function of the open rectangle as the determinant of an $M/2\times M/2$ Hankel matrix (a matrix whose $(j,k)$ entry depends only on $j+k$), equivalently the Pfaffian of a skew-symmetric $M\times M$ Toeplitz matrix (the Pfaffian is the signed square root of the determinant of a skew-symmetric matrix, and a Toeplitz matrix has constant entries along its diagonals), whose elements are Fourier coefficients of a symbol function given by the ratio of the two directional characteristic polynomials. The author stated there that deriving the scaling limit from this representation by means of Szegő's theorem — the classical result giving the large-dimension asymptotics of Toeplitz determinants from the Fourier coefficients and winding data of their symbol — is 'rather straightforward' and 'will be addressed in a forthcoming work'; that work has not appeared. Precursor numerical work (bond-propagation computations — a high-precision numerical method that reduces the finite lattice to an equivalent smaller effective network — of Wu, Izmailian, and Guo on the critical open rectangle) had previously extracted bulk, surface, and corner parts numerically, confirming the framework but not providing analytic scaling functions.

Problem Statement

Complete the analytic finite-size-scaling solution of the critical square-lattice Ising model on the open $L\times M$ rectangle by closing the two gaps left open in Hucht's 2017 derivation. The common objective is a self-contained analytic derivation of the universal Casimir-potential scaling function $\Theta(x,\rho)=\Theta^{(oo)}(x)+\rho^{-1}\Theta_{s,c}(x)+\Psi(x,\rho)$ in which every contribution is obtained directly from the exact solution, and in which the conformal-field-theory correspondence is a theorem rather than a numerical coincidence. Specifically:

(1) Directly calculate the surface–corner contribution $\Theta_{s,c}(x)$, defined as the regularized finite-size-scaling limit of the residual surface–corner free energy $F^{res}_{s,c}(L_{\updownarrow})$ — the part of the exact free energy of the open rectangle that contains the contributions of the two vertical surfaces together with the four corners, with $L_{\updownarrow}$ denoting the rectangle length in the vertical ($\updownarrow$) lattice direction, in which the finite-size-scaling limit is taken. The derivation must start from the published exact finite-size expressions (the products over the zeroes of the directional characteristic polynomial, or the equivalent Hankel determinant / Toeplitz Pfaffian representation), specify the regularization that handles the logarithmically divergent corner free energy $f_c\simeq\frac18\log|1-T/T_c|$ (any regularization is admissible whose finite-size-scaling limit reproduces the surface–corner contribution $\Theta_{s,c}(x)$ fixed in ref. [1]), and produce a closed-form or systematically convergent expression for $\Theta_{s,c}(x)$ as a function of the temperature scaling variable $x$. It must not rely on the indirect route used originally: the symmetry of the square at aspect ratio $\rho=1$ under exchange of the two lattice directions ($\partial_\rho\Theta_\circ(x_\circ,\rho)|_{\rho=1}=0$), the resulting scaling relation, and the consistency relation $\Theta_{s,c}(x)=-\rho\Theta^{(oo)}(x)+\log\Sigma(x,\rho)+\Theta_\circ(x\rho^{1/2},\rho)$ — although agreement with the values fixed by that route is a required check.

(2) Prove rigorously that the exact critical-point series for the residual strip partition function,

$$\Sigma_\circ(0,\rho)=1+\tfrac14 e^{-2\pi\rho}+\tfrac{13}{32}e^{-4\pi\rho}+\tfrac{55}{128}e^{-6\pi\rho}+\tfrac{1235}{2048}e^{-8\pi\rho}+\tfrac{4615}{8192}e^{-10\pi\rho}+\cdots,$$

derived from the determinant representation of the exact solution, is identically equal to the conformal-field-theory expression

$$\Sigma_\circ(0,\rho)=e^{-\pi\rho/48}\eta^{-1/4}(\mathrm{i}\rho)=(e^{-2\pi\rho})_\infty^{-1/4}$$

for all aspect ratios $\rho>0$, where $\eta$ is the Dedekind eta function with product representation $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\ge1}(1-e^{2\pi i n\tau})$ and $(q)_\infty=\prod_{n\ge1}(1-q^n)$. The currently available evidence is numerical agreement of the series coefficients through the first 30 terms; a proof (for example via Szegő-type asymptotics of the Toeplitz/Hankel representation of the partition function, or via elliptic determinant evaluations) is required.

A solution may address the two deliverables together or separately; each is independently checkable against the published exact framework.

References

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