Exact analytic theory of the finite-size expansions of the critical internal energy and specific heat of the two-dimensional Ising model with free boundaries

For the critical two-dimensional Ising model with free (open) boundaries, high-precision bond-propagation computations (Wu, Zheng, Izmailian and Guo, 2014) determined the complete finite-size expansions of the internal energy and specific heat on square-lattice rectangles and on five triangular-lattice shapes, and conjectured closed-form values for the edge and corner amplitudes — all four leading ones for square-lattice rectangles, e.g. $u_{\rm surf}=-2/\pi$ and $u_{\rm corn}=-\sqrt{2}/\pi$ for the internal energy and $c_{\rm corn}=3A_0^{\rm squ}/4$ for the specific heat, and most of the angle-resolved ones on the triangular lattice. These amplitudes have never been derived analytically. The problem is to derive (or refute) the conjectured values from an exact solution of the free-boundary model, to establish analytically the observed shape-dependent pattern of higher-order logarithmic corrections (present for triangular-lattice rhombus, trapezoid and hexagon shapes, absent for triangles and rectangles), and to determine which amplitudes are universal and which are lattice-specific relative to boundary conformal field theory. Since the source paper appeared, the corner free energies (Vernier–Jacobsen, proven by Baxter) and the exact free-boundary rectangle partition function (Hucht) have become available, making the analytic route concrete.

Background

The ferromagnetic Ising model on a two-dimensional lattice is the paradigmatic exactly solved model of a continuous phase transition. For a finite system with free (open) boundary conditions — spins live on a bounded domain with no periodic wrap-around — the critical thermodynamics receives, beyond the bulk contribution, distinct contributions from the edges and from the sharp corners of the boundary. For a rectangle of $M\times N$ sites with area $S=MN$ and aspect ratio $\rho=M/N$, the critical internal energy per site $u$ and specific heat per site $c$ (derivatives of the free energy with respect to temperature) expand as $u = u_\infty + u_{\rm surf}\frac{M\ln N+N\ln M}{S}+\frac{B_1}{S^{1/2}}+u_{\rm corn}\frac{\ln S}{S}+\frac{B_2}{S}+\cdots,$ $c = A_0\ln N + c_0 + c_{\rm surf}\frac{M\ln N+N\ln M}{S}+c_{\rm corn}\frac{\ln S}{S}+\cdots,$ where $u_\infty=\sqrt{2}$ is the exact critical bulk value in the sign convention of the source work, which computes $u=-\langle H\rangle/S$ (with the standard Hamiltonian $H=-J\sum\sigma_i\sigma_j$ the physical critical energy per site is $-\sqrt{2}\,J$), $A_0^{\rm squ}=\frac{2}{\pi}\ln^2(1+\sqrt{2})=0.49453858953231911786\ldots$ is the exactly known bulk specific-heat amplitude on the square lattice, and $A_0^{\rm tri}=\frac{3\sqrt{3}}{4\pi}\ln^2 3=0.49906937804646044983\ldots$ is its counterpart on the triangular lattice. The coefficients $u_{\rm surf},u_{\rm corn},c_{\rm surf},c_{\rm corn}$ are the edge and corner amplitudes in question.

The numerical side is essentially settled. The bond-propagation (BP) algorithm is a numerically exact sequence of local $Y$–$\Delta$ (star-triangle) network transformations that reduce the lattice to a single bond while preserving the partition function and the moment sums needed for $u$ and $c$; in quadruple precision it reaches accuracies of order $10^{-26}$ for system sizes up to $N=2000$. Wu, Zheng, Izmailian and Guo used it to fix the expansions above and to conjecture exact closed forms: for square rectangles (independent of $\rho$), $u_{\rm surf}=-2/\pi$, $u_{\rm corn}=-\sqrt{2}/\pi$, $c_{\rm surf}=3\sqrt{2}\,A_0^{\rm squ}/4$, $c_{\rm corn}=3A_0^{\rm squ}/4$; for the triangular lattice, $u_{\rm surf}^{\parallel}=u_{\rm surf}^{\perp}=-\sqrt{3}/\pi$ (edges parallel or perpendicular to a bond direction), $u_{\rm corn}^{(\pi/3)}=-\sqrt{3}/\pi$, $u_{\rm corn}^{(2\pi/3)}=1/(\sqrt{3}\,\pi)$, $c_{\rm surf}^{\parallel}=A_0^{\rm tri}/3$ (edges parallel to a bond direction; the fitted perpendicular-edge value $c_{\rm surf}^{\perp}=0.1054657691435187\ldots$ was not conjectured in closed form), $c_{\rm corn}^{(\pi/3)}=A_0^{\rm tri}/2$, $c_{\rm corn}^{(2\pi/3)}=-A_0^{\rm tri}/9$, where the superscript is the corner angle and the total corner term is assumed, and numerically confirmed, to be additive over the corners of the polygon (the triangular-lattice $\pi/2$-corner amplitudes $u_{\rm corn}^{(\pi/2)}=0.0369320330730530\ldots$ and $c_{\rm corn}^{(\pi/2)}=-0.0375102120204340\ldots$ were fitted but carry no closed-form conjecture). The source paper states the last triangular specific-heat value without the minus sign, but its own fitted value $-0.05545215311627(5)$ and the hexagon total $6\times(-A_0^{\rm tri}/9)=-0.3327129186981\ldots$ fix the sign. The pattern — simple multiples of the exactly known bulk quantities $u_\infty$ and $A_0$ — is what suggests that closed-form analytic solutions for these free-boundary cases exist.

On the analytic side, much is known but not this. The exact strip solution of Au-Yang and Fisher (two free edges, infinite length) gives the edge internal-energy term $-(2/\pi)\ln N/N$, which anchors $u_{\rm surf}=-2/\pi$. Boundary conformal field theory (CFT) — the continuum field theory describing critical systems, extended to handle finite systems with boundaries — predicts, via Cardy and Peschel, that at criticality each corner of angle $\gamma$ contributes $-\frac{c_{\rm CFT}}{24}(\pi/\gamma-\gamma/\pi)\ln L$ to the free energy, with $c_{\rm CFT}=1/2$ the central charge (the number counting the effective degrees of freedom of the critical continuum theory) of the Ising universality class; but the temperature derivatives of this corner structure at criticality, which are what the amplitudes $u_{\rm corn}$ and $c_{\rm corn}$ probe, are not fixed by CFT — indeed the numerics show that $c_{\rm corn}$ differs between the square lattice ($0.3709\ldots$) and the rectangle-shaped triangular lattice ($-0.150040848\ldots$), i.e. it is nonuniversal. Vernier and Jacobsen conjectured exact product forms for the corner free energies of Ising and Potts models on the square and triangular lattices; Baxter then derived the bulk, surface and corner free energies of the square-lattice Ising model exactly (confirming the conjecture), and obtained corresponding conjectural product forms for the triangular lattice by extrapolating 108-term series. Hucht derived the exact partition function of the free-boundary square-lattice rectangle at arbitrary temperature, in determinant form, and computed the exact universal finite-size-scaling free-energy functions, confirming the Cardy–Peschel and Kleban–Vassileva predictions. All of these results concern the free energy; the edge and corner amplitudes of the internal energy and specific heat at criticality have never been computed analytically, even though the corner free energy is now known to diverge logarithmically on approaching the critical temperature, a singularity that any temperature differentiation at criticality must handle.

A second, linked puzzle is the pattern of logarithmic corrections. For the triangular-lattice shapes that contain $2\pi/3$ corners (rhombus, trapezoid, hexagon), the expansions contain logarithmic corrections at all higher orders — terms $\ln N/N^4, \ln N/N^5,\ldots$ in $u$ and $\ln N/N^3,\ldots$ in $c$ — while for the triangular-lattice triangle and rectangle and for the square-lattice rectangle, no such higher-order logarithmic terms appear. No renormalization-group or boundary-CFT explanation of this shape dependence exists.

Problem Statement

For the critical two-dimensional Ising model with free boundaries, establish an exact analytic theory of the finite-size expansions of the internal energy and specific heat, replacing the existing high-precision numerical conjectures. Concretely:

1. Derive analytically — from a specified exact representation, such as the Kaufman spinor method (Onsager's exact solution of the square-lattice Ising model rewritten in terms of free-fermion, i.e. spinor/Grassmann, variables), the Kasteleyn–McCoy–Wu dimer method (the exact solution obtained by mapping the Ising model to close-packed dimer coverings), the Vernier–Jacobsen/Baxter corner free energies, or the determinant representations of the free-boundary rectangle — the edge and corner amplitudes of the critical internal energy and specific heat; in particular prove or refute the conjectured closed forms $u_{\rm surf}=-2/\pi$, $u_{\rm corn}=-\sqrt{2}/\pi$, $c_{\rm surf}=3\sqrt{2}\,A_0^{\rm squ}/4$, $c_{\rm corn}=3A_0^{\rm squ}/4$ for square-lattice rectangles (including their independence of the aspect ratio) and $u_{\rm surf}^{\parallel}=u_{\rm surf}^{\perp}=-\sqrt{3}/\pi$, $u_{\rm corn}^{(\pi/3)}=-\sqrt{3}/\pi$, $u_{\rm corn}^{(2\pi/3)}=1/(\sqrt{3}\,\pi)$, $c_{\rm surf}^{\parallel}=A_0^{\rm tri}/3$, $c_{\rm corn}^{(\pi/3)}=A_0^{\rm tri}/2$, $c_{\rm corn}^{(2\pi/3)}=-A_0^{\rm tri}/9$ for the triangular-lattice shapes (including the angle-resolved decomposition of the total corner term into per-corner contributions, an assumption underlying the numerical fits that an analytic derivation should justify). The remaining fitted amplitudes — the triangular-lattice perpendicular-edge specific-heat term $c_{\rm surf}^{\perp}$ and the triangular-lattice $\pi/2$-corner terms $u_{\rm corn}^{(\pi/2)}$ and $c_{\rm corn}^{(\pi/2)}$ — carry no closed-form conjecture and fall under the general derivation target of this component. Each enumerated closed form is an independently required sub-claim of this component: proving or refuting one of them settles only that claim, and the mandatory target of this component is the full enumerated set together with the general derivation of the un-conjectured amplitudes above.

2. Establish analytically the complete structure of these expansions — in particular the presence of higher-order logarithmic corrections ($\ln N/N^4$ and beyond in $u$, all higher orders in $c$) for the triangular-lattice rhombus, trapezoid and hexagon shapes and their absence for the triangular-lattice triangle and rectangle and the square-lattice rectangle — and explain this shape-dependent pattern within a renormalization-group or boundary conformal-field-theory framework, including the role of the $2\pi/3$ corners.

3. Determine which of the edge and corner amplitudes are universal and which are lattice-specific (nonuniversal), and reconcile the lattice values with the Cardy–Peschel-type conformal predictions for corner contributions.

These three components jointly constitute one objective — an exact analytic account of the critical finite-size thermodynamics of free-boundary two-dimensional Ising systems; each component addresses an open item explicitly left by the source work, all three are required, and completing one component does not substitute for another.

References

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