Exact finite-size-scaling (FSS) functions for the critical Casimir potential and force of the two-dimensional Ising model on an L×ML\times M rectangle with open boundaries were derived by Hucht (J. Phys. A 50, 265205 (2017)) from the exact lattice partition function, but two steps of the construction remained indirect. First, the surface–corner scaling contribution Θs,c(x)\Theta_{\mathrm{s,c}}(x) could not be obtained from a direct regularized FSS limit of the lattice surface–corner residual free energy — the corner free energy is logarithmically singular at TcT_{\mathrm c} — and was instead inferred from the exchange symmetry of the square at aspect ratio ρ=1\rho=1 together with numerics. Second, the critical partition-function series Σ∘(0,ρ)=1+14e−2πρ+1332e−4πρ+⋯\Sigma_\circ(0,\rho)=1+\frac14 e^{-2\pi\rho}+\frac{13}{32}e^{-4\pi\rho}+\cdots derived from the exact determinant was matched only empirically (agreement of the first thirty coefficients) to the conformal-field-theory modular expression e−πρ/48η−1/4(iρ)=(e−2πρ)∞−1/4e^{-\pi\rho/48}\eta^{-1/4}(\mathrm{i}\rho)=(e^{-2\pi\rho})_\infty^{-1/4} of Kleban and Vassileva; no analytic proof was given. The problem is to evaluate the regularized FSS limit of the exact lattice free-energy decomposition directly: to derive Θs,c(x)\Theta_{\mathrm{s,c}}(x) from the lattice expression and to prove the lattice–CFT identity for all aspect ratios ρ>0\rho>0, with the convergence of the resulting representations controlled uniformly in ρ\rho as a subsidiary goal. A successor paper (Hucht 2021) reformulated the partition function as a Hankel/Toeplitz determinant whose scaling-limit evaluation via Szegő's theorem was announced as forthcoming work that has not appeared. No derivation of this scaling limit appears in the subsequent literature.

Background

The square-lattice Ising model — spins σ=±1\sigma=\pm1 on an L×ML\times M rectangle with nearest-neighbor couplings and open (free) boundary conditions, meaning no periodic wrap-around and no boundary fields — is the canonical exactly solved model of a second-order phase transition: its bulk free energy has been known exactly since Onsager (1944), and the surface free energies of the open edges are also known exactly. On a finite rectangle the reduced free energy F(L,M)=−log⁡ZF(L,M)=-\log Z (in units of kBTk_{\mathrm B}T) decomposes into bulk, surface and corner parts,

F(L,M)=LM fb+L fs↔+M fs↕+fc+Fres,F(L,M)=LM\,f_{\mathrm b}+L\,f_{\mathrm s}^{\leftrightarrow}+M\,f_{\mathrm s}^{\updownarrow}+f_{\mathrm c}+F^{\mathrm{res}},

where fbf_{\mathrm b} is the infinite-volume free energy per spin, fsδf_{\mathrm s}^{\delta} are the surface free energies per boundary pair in the two lattice directions δ\delta, the corner free energy fcf_{\mathrm c} — the non-extensive remainder concentrated at the four right-angle corners — was derived exactly for the square lattice by Baxter (2017), and the residual FresF^{\mathrm{res}} is the thermodynamic Casimir potential responsible, near TcT_{\mathrm c}, for fluctuation-induced forces between the confining boundaries. Near criticality one studies the finite-size-scaling (FSS) limit, L,M→∞L,M\to\infty and T→TcT\to T_{\mathrm c} simultaneously at fixed temperature scaling variable x∝(T/Tc−1)Mx\propto(T/T_{\mathrm c}-1)M and fixed aspect ratio ρ∝L/M\rho\propto L/M; in this limit FresF^{\mathrm{res}} defines universal scaling functions, the Casimir potential scaling function Θ(x,ρ)\Theta(x,\rho) and the Casimir force scaling function ϑ(x,ρ)\vartheta(x,\rho). Conformal field theory (CFT), the continuum theory of two-dimensional critical fluctuations with central charge c=1/2c=1/2 in the Ising case, makes sharp predictions for this geometry: Cardy and Peschel (1988) showed that a corner of opening angle γ\gamma contributes −c24(π/γ−γ/π)-\frac{c}{24}(\pi/\gamma-\gamma/\pi) times a logarithm of the system size, so the four right angles together produce a −116log⁡(LM)-\frac{1}{16}\log(LM) term in the critical free energy; Kleban and Vassileva (1991) computed the full critical free energy of the open rectangle in terms of modular forms — the Dedekind eta function η(τ)=eπiτ/12∏n≥1(1−e2πinτ)\eta(\tau)=e^{\pi i\tau/12}\prod_{n\ge1}(1-e^{2\pi i n\tau}) and the qq-Pochhammer symbol (q)∞=∏n≥1(1−qn)(q)_\infty=\prod_{n\ge1}(1-q^n). Hucht's two-part work made this program exact at the lattice level. Part I (2017) expressed the finite-lattice partition function itself: the free energy splits exactly as F=L fb,s(M)+Fs,c(M)−log⁡det⁡(1+Y)F=L\,f_{\mathrm{b,s}}(M)+F_{\mathrm{s,c}}(M)-\log\det(\mathbf 1+\mathbf Y), where fb,sf_{\mathrm{b,s}} contains the bulk and the two horizontal-surface contributions, Fs,cF_{\mathrm{s,c}} collects the two vertical surfaces and the four corners (the surface–corner part), and Y\mathbf Y is an M/2×M/2M/2\times M/2 matrix built from the dominant eigenvalues of the row-to-row transfer matrix (the matrix whose repeated application propagates the system along its long direction). Part II (2017) evaluated the scaling limit of the strip-residual determinant and obtained an exponentially convergent series Σ(x,ρ)\Sigma(x,\rho) — a perturbative series whose terms are labeled by constrained subsets of the positive integers, namely sets with equal numbers of even and odd elements — giving the decomposition

Θ(x,ρ)=Θ(oo)(x)+ρ−1Θs,c(x)+Ψ(x,ρ),Ψ=−ρ−1log⁡Σ(x,ρ),\Theta(x,\rho)=\Theta^{(\mathrm{oo})}(x)+\rho^{-1}\Theta_{\mathrm{s,c}}(x)+\Psi(x,\rho),\qquad\Psi=-\rho^{-1}\log\Sigma(x,\rho),

where Θ(oo)\Theta^{(\mathrm{oo})} is the known strip-geometry (aspect ratio ρ→∞\rho\to\infty) Casimir potential, Ψ\Psi is the strip-residual contribution, and Θs,c(x)\Theta_{\mathrm{s,c}}(x) is the surface–corner contribution, which in the scaling limit corresponds to the residual of the lattice quantity Fs,cF_{\mathrm{s,c}}. At criticality (x=0x=0) the series begins Σ∘(0,ρ)=1+14e−2πρ+1332e−4πρ+55128e−6πρ+⋯\Sigma_\circ(0,\rho)=1+\frac14 e^{-2\pi\rho}+\frac{13}{32}e^{-4\pi\rho}+\frac{55}{128}e^{-6\pi\rho}+\cdots and was found to coincide, coefficient by coefficient over the first thirty terms, with the CFT expression Σ∘(0,ρ)=e−πρ/48η−1/4(iρ)=(e−2πρ)∞−1/4\Sigma_\circ(0,\rho)=e^{-\pi\rho/48}\eta^{-1/4}(\mathrm{i}\rho)=(e^{-2\pi\rho})_\infty^{-1/4} of Kleban and Vassileva. Two steps of this construction were, however, indirect, and they are the subject of this problem. (i) Θs,c(x)\Theta_{\mathrm{s,c}}(x) could not be calculated directly from the regularized FSS limit of the lattice surface–corner residual free energy Fs,cresF_{\mathrm{s,c}}^{\mathrm{res}}: the corner free energy is logarithmically singular at TcT_{\mathrm c}, so the naive limit diverges and requires an explicit regularization that the paper could not carry out; instead Θs,c(x)\Theta_{\mathrm{s,c}}(x) was reconstructed from the exact symmetry of the square under exchange of the two lattice directions, which forces ∂ρΘ∘(x∘,ρ)∣ρ=1=0\partial_\rho\Theta_\circ(x_\circ,\rho)|_{\rho=1}=0, where Θ∘(x∘,ρ)=ρ Θ(x,ρ)\Theta_\circ(x_\circ,\rho)=\rho\,\Theta(x,\rho) is the potential written in terms of the volume scaling variable x∘=xρ1/2x_\circ=x\rho^{1/2}, together with numerical integration. (ii) The identification of the lattice series with the modular CFT expression remained an empirically checked equivalence ('we were not able to proof this correspondence'). In addition, the series converges non-uniformly in the aspect ratio (suboptimally for ρ≲1\rho\lesssim1), which the paper works around using the same exchange symmetry. Part III (2021) later rewrote the full partition function as the determinant of a Hankel matrix — equivalently the Pfaffian of a Toeplitz matrix (a matrix with constant diagonals) whose symbol is a ratio of characteristic polynomials of the two transfer directions — and observed that the scaling limit should follow from Szegő's theorem, the classical asymptotic theory of large Toeplitz determinants; that evaluation was explicitly deferred to a forthcoming work that has not appeared.

Problem Statement

Evaluate directly and rigorously the regularized finite-size-scaling limit of the exact lattice free energy of the square-lattice Ising model on the L×ML\times M rectangle with open boundary conditions (the exact partition-function representation of part I, eqs. (I.87) and (I.91), or equivalently the Hankel/Toeplitz determinant representation of part III), in the limit L,M→∞L,M\to\infty, T→TcT\to T_{\mathrm c} with fixed temperature scaling variable x∝(T/Tc−1)Mx\propto(T/T_{\mathrm c}-1)M and fixed aspect ratio ρ∝L/M\rho\propto L/M, so as to:

(a) derive the surface–corner scaling contribution Θs,c(x)\Theta_{\mathrm{s,c}}(x) directly as the regularized FSS limit of the lattice surface–corner residual free energy Fs,cresF_{\mathrm{s,c}}^{\mathrm{res}} — with the logarithmic critical singularity of the corner free energy treated explicitly — without recourse to the ρ=1\rho=1 exchange-symmetry relation, reproducing the function obtained indirectly in part II (eqs. (80)–(84) there);

(b) prove analytically that the critical partition-function scaling function derived from the exact lattice determinant satisfies, for all aspect ratios ρ>0\rho>0,

Σ∘(0,ρ)=e−πρ/48 η−1/4(iρ)=(e−2πρ)∞−1/4,\Sigma_\circ(0,\rho)=e^{-\pi\rho/48}\,\eta^{-1/4}(\mathrm{i}\rho)=(e^{-2\pi\rho})_\infty^{-1/4},

the conformal-field-theory expression of Kleban and Vassileva, where η\eta is the Dedekind eta function and (q)∞(q)_\infty the qq-Pochhammer symbol — an equivalence currently supported only by agreement of the first thirty series coefficients.

A complete derivation should also characterize the convergence of the resulting representations in the aspect ratio, removing the non-uniformity for ρ≲1\rho\lesssim1 that presently requires the exchange-symmetry workaround; this convergence goal is subsidiary — the completion core of the problem is (a) and (b).

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