Critical behavior near $T_c$ of the boundary-row spin-spin correlation function of the finite-size two-dimensional Ising model with periodic-free boundary conditions

For the two-dimensional rectangular Ising model on an $L\times N$ lattice that is periodic in the horizontal direction and free in the vertical direction, in zero magnetic field, the correlation $\langle\sigma_{1,1}\sigma_{1+P,1}\rangle$ between two spins of the boundary row is known exactly for every separation $P=1,\dots,L-1$ and all finite $L,N$ (Mei, 2026), as an explicit spectral sum that decomposes as $\Lambda_P+Y_P$. Only the special-root term $\Lambda_P$ has been analyzed near the critical temperature $T_c$, giving a correlation length $[\rho(1-T/T_c)]^{-1}$ and exponent $\nu=1$; the source explicitly states that these are not the correlation length and critical exponent of the full correlation, because the remainder $Y_P$ has not been discussed near $T_c$. The open problem is to complete this near-critical analysis: determine the behavior of $Y_P$ near $T_c$; the correlation length, critical exponents, amplitudes and scaling function of the full correlation $\Lambda_P+Y_P$; the finite-size scaling that interweaves $L$ and $N$ in the critical region, including the crossover between the two non-commuting thermodynamic limits; and the relation of the exact finite-size expression to the form-factor, Painlevé and conformal-field-theory descriptions of Ising correlations. The thermodynamic-limit surface exponents are classical, so the genuinely new content is the derivation from the exact expression, the exact amplitudes and scaling functions, and the two-parameter finite-size crossover.

Background

The two-dimensional Ising model is the archetypal exactly solved statistical system. In its rectangular (anisotropic) form, spins $\sigma_{j,k}=\pm 1$ occupy an $L\times N$ lattice with nearest-neighbor couplings $J$ along rows and $J'$ between rows, in zero magnetic field. This problem concerns the periodic-free geometry: the horizontal direction is closed into a ring of circumference $L$ (periodic boundary conditions), while the vertical direction of $N$ rows terminates in free (open) boundaries; the boundary row is the outermost row of spins. It is convenient to use $K=J/kT$ together with the dual coupling $K'$ of the transverse coupling, defined by $e^{-2K'}=\tanh(J'/kT)$; the critical temperature satisfies $\sinh(2J/kT_c)\cosh(2J'/kT_c)=\cosh(2J/kT_c)$, equivalently $K=K'$ at $T_c$, with $K'<K$ for $T<T_c$ and $K'>K$ for $T>T_c$.

Exact results for this model rest on Kaufman's spinor analysis of the transfer matrix, in which the transfer matrix is represented on a Clifford algebra so that its eigenvalues are obtained in closed form. The monograph of McCoy and Wu developed the boundary physics of the half-plane: the boundary spontaneous magnetization $M_1$ (the limiting expectation value of a boundary spin as a vanishing boundary field is removed; it is nonzero below $T_c$, vanishing as $M_1\sim(T_c-T)^{1/2}$) and exact integral representations of the boundary-row correlation $S_{11}(n,0)=\langle\sigma_{1,0}\sigma_{1,n}\rangle$ in the thermodynamic limit, with and without a boundary magnetic field (Chapter VII, 'Boundary Spin-Spin Correlation Functions').

The source (Mei, 2026) extends these classical results to finite size. For the $L\times N$ periodic-free lattice it derives, by spinor analysis, the exact formula

$\langle\sigma_{1,1}\sigma_{1+P,1}\rangle=2\sum_{n=1}^{N}\Omega_{n-1}^{2}\,\frac{\cosh\!\left[\left(\tfrac{L}{2}-P\right)\gamma_{n-1}\right]}{\cosh\!\left(\tfrac{L}{2}\gamma_{n-1}\right)},\qquad P=1,\dots,L-1,$

valid for all temperatures and all finite $L,N$. Here $\gamma_{n-1}>0$ is fixed by $\cosh\gamma_{n-1}=\cosh 2K'\cosh 2K-x_{n-1}\sinh 2K'\sinh 2K$, where $x_{n-1}$ ($n=1,\dots,N$) are the roots of the $N$-th order polynomial equation $G_N(x)=0$ with $G_k(x)=g_k(x)-2g_{k-1}(x)\coth 2K'\tanh K+g_{k-2}(x)\tanh 2K$ and $g_k$ the Chebyshev polynomial of the second kind, and $\Omega_{n-1}$ are explicit coefficients of the diagonalizing transformation (Eqs. (A11), (A12), (A25) of the source). For $T<T_c$ the sum decomposes as $\langle\sigma_{1,1}\sigma_{1+P,1}\rangle=\Lambda_P+Y_P$: the $n=1$ term $\Lambda_P$ is governed by a special root $x_0$ that lies in the interval $(1,\tanh 2K/\tanh 2K')$, outside the band $[-1,1]$ of the remaining roots, and carries the long-range order, $\Lambda_P\to M_1(0^+)^2=(\cosh 2K-\cosh 2K')/(2\sinh^2 K)$ as $N\to\infty$; the remainder $Y_P$ ($n\geq 2$) tends to a function $F_P$ independent of the order in which the $L$ and $N$ limits are taken. A striking exact property is the non-commutativity of the thermodynamic limits: for $T<T_c$ and long-range $P$, $\lim_{L\to\infty}\lim_{N\to\infty}\neq\lim_{N\to\infty}\lim_{L\to\infty}$, demonstrated numerically in the source.

What is not known is the critical behavior of this exact expression. Near $T_c$ from below, the source finds $\gamma_0\approx\rho(1-T/T_c)$ and $\Omega_0^2\approx\tfrac{1}{4}\coth(2J/kT_c)(1-T/T_c)$, so that $\Lambda_P$ alone has correlation length $\xi=[\rho(1-T/T_c)]^{-1}$ and critical exponent $\nu=1$ ($\rho$ a constant given by Eq. (73) of the source); but it explicitly warns that these are not the correlation length and critical exponent of the full correlation, since $Y_P$ has not been discussed near $T_c$. By correlation length one means the inverse exponential decay rate of the connected correlation as a function of the separation; the critical exponent $\nu$ describes its divergence as $T\to T_c$. For orientation, surface critical behavior of the ordinary transition (the surface analogue of bulk criticality, applicable to a free boundary of the Ising model) is understood at the level of scaling and conformal field theory (Cardy, 1984): at $T_c$ the boundary two-point function decays as $P^{-2x_b}$ with boundary scaling dimension $x_b=1/2$, i.e. as $P^{-1}$, and the boundary correlation length diverges as $|T-T_c|^{-1}$; below $T_c$ the correlation saturates at $M_1^2$. It is not known how the exact finite-size expression reproduces this behavior: what the full scaling function combining $\Lambda_P$ and $Y_P$ is, what its amplitudes are, and how the two finite dimensions $L$ and $N$ interleave in the critical region. Here finite-size scaling means the scaling form of observables with the ratios of the system dimensions to the correlation length, together with the associated shift of the apparent critical point in finite systems. Separately, Ising correlations possess well-developed alternative descriptions: form factor expansions (spectral-sum representations of scaling-limit correlation functions), Painlevé transcendents (special functions satisfying certain nonlinear ordinary differential equations, which for the full-plane two-point function give the celebrated Wu-McCoy-Tracy-Barouch scaling function), conformal field theory at criticality, and finite-size correction formulas. The relation of the new exact expression to these frameworks is, per the source's own summary, yet to be systematically studied.

Problem Statement

Consider the two-dimensional rectangular Ising model on an $L\times N$ lattice with periodic boundary conditions in the horizontal direction (period $L$) and free boundary conditions in the vertical direction ($N$ rows), in zero magnetic field, with row coupling $J$ and transverse coupling $J'$. Write $K=J/kT$ and let $K'$ be the dual of $J'$, defined by $e^{-2K'}=\tanh(J'/kT)$; the critical temperature satisfies $\sinh(2J/kT_c)\cosh(2J'/kT_c)=\cosh(2J/kT_c)$, equivalently $K=K'$ at $T_c$.

The exact boundary-row correlation function is (Mei, 2026, Eq. (44); notation as in the Background):

$\langle\sigma_{1,1}\sigma_{1+P,1}\rangle=2\sum_{n=1}^{N}\Omega_{n-1}^{2}\,\frac{\cosh\!\left[\left(\tfrac{L}{2}-P\right)\gamma_{n-1}\right]}{\cosh\!\left(\tfrac{L}{2}\gamma_{n-1}\right)},\qquad P=1,\dots,L-1,$

with $\cosh\gamma_{n-1}=\cosh 2K'\cosh 2K-x_{n-1}\sinh 2K'\sinh 2K$, the $x_{n-1}$ the roots of $G_N(x)=0$ where $G_k(x)=g_k(x)-2g_{k-1}(x)\coth 2K'\tanh K+g_{k-2}(x)\tanh 2K$ and $g_k$ is the Chebyshev polynomial of the second kind, and $\Omega_{n-1}$ the coefficients of Eq. (A25) of the source. For $T<T_c$ write $\langle\sigma_{1,1}\sigma_{1+P,1}\rangle=\Lambda_P+Y_P$, where $\Lambda_P$ is the $n=1$ special-root term and $Y_P$ the $n\geq 2$ remainder.

Determine the critical behavior of this correlation function near $T_c$. Parts (i)-(iii) below and the framework relation in the closing paragraph are all required for a complete resolution:

(i) the asymptotic behavior of $Y_P$ as $T\to T_c$ from below (the decomposition $\Lambda_P+Y_P$ being defined for $T<T_c$), together with the asymptotic behavior of the full correlation $\langle\sigma_{1,1}\sigma_{1+P,1}\rangle$ as $T\to T_c$ from above, in the thermodynamic limit, for both short-range and long-range separations $P$; this part is not a separate obligation but a derived consistency check on (ii): since the source already fixes $\Lambda_P$ near $T_c$, determining the full correlation in (ii) also determines $Y_P=\langle\sigma_{1,1}\sigma_{1+P,1}\rangle-\Lambda_P$ below $T_c$ and the full correlation above $T_c$, so a complete answer to (ii) discharges (i);

(ii) the correlation length, critical exponents, amplitudes and scaling function of the full correlation $\Lambda_P+Y_P$ near $T_c$; in particular, whether the $\Lambda_P$-only results $\xi=[\rho(1-T/T_c)]^{-1}$ and $\nu=1$ persist for the full function, and how the full scaling function interpolates between the critical decay $\propto P^{-1}$ at $T_c$, the exponential decay above $T_c$, and the saturation at $M_1(0^+)^2=(\cosh 2K-\cosh 2K')/(2\sinh^2 K)$ below $T_c$;

(iii) the finite-size scaling of $\langle\sigma_{1,1}\sigma_{1+P,1}\rangle$ in the critical region at finite $L$ and $N$: the scaling form describing the interplay of the two dimensions, the finite-size shift of the apparent critical temperature, and the crossover between the two non-commuting limit orders $\lim_{L\to\infty}\lim_{N\to\infty}$ and $\lim_{N\to\infty}\lim_{L\to\infty}$;

and relate the results to the existing frameworks for Ising correlations (form factor expansions, Painlevé representations, conformal field theory, finite-size corrections), making explicit how the exact finite-size expression reduces to, or refines, those descriptions in their respective regimes. This final part is discharged by exhibiting, for each framework, the limit or regime in which the exact expression matches that framework's known result — the scaling limit $T\to T_c$, $P\to\infty$ for the form-factor and Painlevé descriptions, the critical point $T=T_c$ for conformal field theory, and the strip and cylinder limits for the finite-size corrections — rather than by a qualitative analogy.

The extension to a nonzero boundary magnetic field at finite size and the extension of the spinor method to other models or boundary conditions are explicitly outside the scope of this problem.

References

1. T. Mei, 'Exact expressions of correlation functions between two spins in the boundary row of the two-dimensional rectangular Ising model with periodic-free boundary conditions and finite size', arXiv:2608.19274 (2026). DOI: 10.48550/arxiv.2608.19274. URL: https://arxiv.org/abs/2608.19274.
2. B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (Harvard University Press, 1973); Chapter VII, 'Boundary Spin-Spin Correlation Functions', pp. 142-176. DOI: 10.4159/harvard.9780674180758.c6. URL: https://doi.org/10.4159/harvard.9780674180758.c6.
3. B. Kaufman, 'Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis', Physical Review 76, 1232-1243 (1949). DOI: 10.1103/PhysRev.76.1232. URL: https://doi.org/10.1103/PhysRev.76.1232.
4. T. Mei, 'An Exact Closed Formula of a Spin-spin Correlation Function of the Two-Dimensional Ising Model with Finite Size', International Journal of Theoretical Physics 54, 3462-3489 (2015). DOI: 10.1007/s10773-015-2587-1. URL: https://doi.org/10.1007/s10773-015-2587-1.
5. T. Mei, 'Exact Expressions of Spin-Spin Correlation Functions of the Two-Dimensional Rectangular Ising Model on a Finite Lattice', Entropy 20(4), 277 (2018). DOI: 10.3390/e20040277. URL: https://doi.org/10.3390/e20040277.
6. J. L. Cardy, 'Conformal invariance and surface critical behavior', Nuclear Physics B 240, 514-532 (1984). DOI: 10.1016/0550-3213(84)90241-4. URL: https://doi.org/10.1016/0550-3213(84)90241-4.
7. T. W. Burkhardt and I. Guim, 'Universal scaling form of the correlation length in Ising strips with periodic, free, fixed, and mixed boundary conditions', Physical Review B 35, 1799-1806 (1987). DOI: 10.1103/PhysRevB.35.1799. URL: https://doi.org/10.1103/PhysRevB.35.1799.
8. M. Suzuki, H. Suzuki, and S.-C. Chang, 'Study on boundary-boundary correlation functions of the two-dimensional Ising model using topological interaction method', Journal of Mathematical Physics 46, 033301 (2005). DOI: 10.1063/1.1845954. URL: https://doi.org/10.1063/1.1845954.
9. T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch, 'Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region', Physical Review B 13, 316-374 (1976). DOI: 10.1103/PhysRevB.13.316. URL: https://doi.org/10.1103/PhysRevB.13.316.
10. V. V. Mangazeev and A. J. Guttmann, 'Form factor expansions in the 2D Ising model and Painlevé VI', Nuclear Physics B 838, 391-412 (2010). DOI: 10.1016/j.nuclphysb.2010.05.021. URL: https://doi.org/10.1016/j.nuclphysb.2010.05.021.
11. M. A. Rajabpour, 'Finite size corrections to scaling of the formation probabilities and the Casimir effect in the conformal field theories', Journal of Statistical Mechanics: Theory and Experiment (2016) 123101. DOI: 10.1088/1742-5468/2016/12/123101. URL: https://doi.org/10.1088/1742-5468/2016/12/123101.
12. E. Granet, M. Fagotti, and F. H. L. Essler, 'Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions', SciPost Physics 9, 033 (2020). DOI: 10.21468/SciPostPhys.9.3.033. URL: https://doi.org/10.21468/SciPostPhys.9.3.033.
13. M. Assis and B. M. McCoy, 'The energy density of an Ising half-plane lattice', Journal of Physics A: Mathematical and Theoretical 44, 095003 (2011). DOI: 10.1088/1751-8113/44/9/095003. URL: https://doi.org/10.1088/1751-8113/44/9/095003. Preprint: arXiv:1011.5463, https://arxiv.org/abs/1011.5463.
14. O. Miroshnichenko, 'Differential equation for local magnetization in the boundary Ising model', Nuclear Physics B 811, 385-394 (2009). DOI: 10.1016/j.nuclphysb.2008.11.005. URL: https://doi.org/10.1016/j.nuclphysb.2008.11.005. Preprint: arXiv:0808.3808, https://arxiv.org/abs/0808.3808.