Finite Gluing of the Two-Dimensional Ising Free Energy

Abstract

We test the Gluing Method on the square-lattice two-dimensional Ising model by reconstructing global free-energy generators from complementary local relations. The emphasis is kept explicit throughout: the inputs used as \emph{gluing material} are separated from quantities obtained \emph{after gluing} and from data reserved for validation. At zero magnetic field, Kramers--Wannier duality is absorbed into the coordinate $\tau=\log\sinh 2K$, so that the reduced free energy decomposes as $\Phi(\tau)=\tau/2+G(|\tau|)$. The gluing materials are then only a critical logarithmic local expansion and a low-order high-temperature outer expansion; the low-temperature sector is generated by duality rather than fitted independently. Positive flat weights preserve the prescribed critical asymptotics to every finite algebraic order. With one crossover parameter fixed from three scalar anchors, the resulting bulk free energy agrees with the exact Onsager benchmark at a maximum relative error of $1.915\times10^{-5}$ on the tested interval. Internal energy and specific heat are derived from the same glued generator. Zero-field finite-size corrections are organized separately as one-variable scaling functions multiplying $L^{-2},L^{-4},\ldots$. At nonzero field in the thermodynamic limit, the gluing materials are the established high-temperature, critical, and low-temperature local expansions of the universal Ising magnetic scaling free energy $\Phi_{\rm mag}(\eta)$, $\eta=m/|h|^{8/15}$. The low-temperature input contains the $|h|$ cusp associated with spontaneous symmetry breaking. Mapping the glued scaling free energy back to the square lattice through nonlinear scaling fields yields a single generator for the magnetic free-energy increment, magnetization, and susceptibility. The construction reproduces high-temperature linear response, the critical isotherm $M\propto |H|^{1/15}$, and the two one-sided limits $M(H\to0^\pm)=\pm M_0$ without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables $x_t=Lm$ and $x_h=L^{15/8}h$. The two protected one-dimensional axes, a central mixed-response expansion, a low-temperature spectral-rounding relation, and a massive outer description are glued into a two-variable scaling surface $\mathcal Y_{\rm glue}(x_t,x_h)$. The resulting lattice free energy closes in the compact form $\Phi_L^{glue}(K,H)=\Phi_{lat}^{glue}(K,H)+\frac{\mathcal{Y}_{glue}(Lm,L^{15/8}h)- Y_{bulk}(Lm,L^{15/8}h)}{L^2}+O(L^{-4})$ where $m=m(r,H)$ and $h=h(r,H)$ are the nonlinear scaling fields and $Y_{\rm bulk}$ is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess $\mathcal Y_{\rm glue}-Y_{\rm bulk}$, while the thermodynamic lattice free energy remains the previously constructed $\Phi_{\rm lat}^{\rm glue}$. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error $3.90\times10^{-4}$ and maximum absolute error $9.55\times10^{-4}$ in $\mathcal Y$. Differentiation of the same free energy produces finite-size magnetization and susceptibility; magnetization is especially stable on the low-temperature side, while the second derivative is more sensitive in crossover regions.