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Finite Gluing of the Three-Dimensional Ising Free Energy

Yoshiki Ueoka

Published Sep 2, 2026• 0 citations
Publisher

Abstract

We construct a single reduced free-energy generator for the simple-cubic three-dimensional Ising model by finite gluing of complementary local relations. With the convention $\Phi=+\log Z/V$, the final object is $\Phi_3(K,H,L)=\Phi_0(K)+\Delta\Phi_H(K,H)+\Phi_{\rm FSS}(K,H,L),$ from which $M=\partial_H\Phi_3$, $\chi=\partial_H^2\Phi_3$, and $U_K=\partial_K\Phi_3$ are generated. The construction covers the high- and low-temperature regimes, the critical region, finite magnetic field, strong-field saturation, periodic-film finite size, and simultaneous finite $H$ and finite $L$. The materials and operations are separated explicitly. A diagnostic dimensional stage uses $d=1,2,4$ anchors in $\mu_c=-\log\tanh K_c$ to transport critical coordinates and positive film amplitudes toward $d=3$. The practical 3D generator then protects established 3D values of $K_c$, critical exponents, selected amplitude ratios, and scalar zero-field critical data. The zero-field term joins truncated simple-cubic high- and low-temperature series through a singular critical patch with $C^2$ matching. The inner magnetic increment joins an $O(K^2)$ high-temperature field expansion, a minimal 3D parametric critical free energy fixed only by the basic 3D anchors, and a low-temperature/cluster continuation using fixed quintic switches in $K$. A recovered $z^4$ strong-field expansion is integrated to an absolute free energy. The final inner-to-strong crossover is performed by constructing a monotone beta-CDF magnetization whose endpoint values and integral reproduce the two endpoint free energies, and then integrating that derivative; thus convexity is enforced at the free-energy level rather than by clipping observables. The periodic-film excess is built from dimensionally weighted 2D and leading 4D film surfaces in common correlation-length coordinates, with an $H$-independent residual stabilizer that preserves the $H=0$ axis. A previously reconstructed full 3D parametric equation-of-state benchmark and the full 3D finite-size scaling function are withheld from construction. The pure dimensional stage gives $K_c(3)=0.22301549$ and sub-percent estimates of $\beta$, $\delta$, and $\nu$. After 3D calibration, the final implementation passes 29/29 stated structural and numerical checks. In the withheld low-field window $|K-K_c|\le0.01$, $H\le0.1$, the magnetization has RMS relative error $1.21\%$ and maximum error $4.29\%$. The result is not an exact lattice solution; it is a finite, auditable reconstruction in which the input anchors, switch intervals, matching conditions, numerical tolerances, and validation data are kept distinct.

DOI
10.5281/zenodo.22261618

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Finite Gluing of the Two-Dimensional Ising Free Energy

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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.

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arXiv:1905.06183•May 8, 2019•0 citations
We construct a single reduced free-energy generator for the simple-cubic three-dimensional Ising model by finite gluing of complementary local relations. With the convention Φ=+log⁡Z/V\Phi=+\log Z/VΦ=+logZ/V, the final object is Φ3(K,H,L)=Φ0(K)+ΔΦH(K,H)+ΦFSS(K,H,L),\Phi_3(K,H,L)=\Phi_0(K)+\Delta\Phi_H(K,H)+\Phi_{\rm FSS}(K,H,L),Φ3​(K,H,L)=Φ0​(K)+ΔΦH​(K,H)+ΦFSS​(K,H,L), from which M=∂HΦ3M=\partial_H\Phi_3M=∂H​Φ3​, χ=∂H2Φ3\chi=\partial_H^2\Phi_3χ=∂H2​Φ3​, and UK=∂KΦ3U_K=\partial_K\Phi_3UK​=∂K​Φ3​ are generated. The construction covers the high- and low-temperature regimes, the critical region, finite magnetic field, strong-field saturation, periodic-film finite size, and simultaneous finite HHH and finite LLL. The materials and operations are separated explicitly. A diagnostic dimensional stage uses d=1,2,4d=1,2,4d=1,2,4 anchors in μc=−log⁡tanh⁡Kc\mu_c=-\log\tanh K_cμc​=−logtanhKc​ to transport critical coordinates and positive film amplitudes toward d=3d=3d=3. The practical 3D generator then protects established 3D values of KcK_cKc​, critical exponents, selected amplitude ratios, and scalar zero-field critical data. The zero-field term joins truncated simple-cubic high- and low-temperature series through a singular critical patch with C2C^2C2 matching. The inner magnetic increment joins an O(K2)O(K^2)O(K2) high-temperature field expansion, a minimal 3D parametric critical free energy fixed only by the basic 3D anchors, and a low-temperature/cluster continuation using fixed quintic switches in KKK. A recovered z4z^4z4 strong-field expansion is integrated to an absolute free energy. The final inner-to-strong crossover is performed by constructing a monotone beta-CDF magnetization whose endpoint values and integral reproduce the two endpoint free energies, and then integrating that derivative; thus convexity is enforced at the free-energy level rather than by clipping observables. The periodic-film excess is built from dimensionally weighted 2D and leading 4D film surfaces in common correlation-length coordinates, with an HHH-independent residual stabilizer that preserves the H=0H=0H=0 axis. A previously reconstructed full 3D parametric equation-of-state benchmark and the full 3D finite-size scaling function are withheld from construction. The pure dimensional stage gives Kc(3)=0.22301549K_c(3)=0.22301549Kc​(3)=0.22301549 and sub-percent estimates of β\betaβ, δ\deltaδ, and ν\nuν. After 3D calibration, the final implementation passes 29/29 stated structural and numerical checks. In the withheld low-field window ∣K−Kc∣≤0.01|K-K_c|\le0.01∣K−Kc​∣≤0.01, H≤0.1H\le0.1H≤0.1, the magnetization has RMS relative error 1.21%1.21\%1.21% and maximum error 4.29%4.29\%4.29%. The result is not an exact lattice solution; it is a finite, auditable reconstruction in which the input anchors, switch intervals, matching conditions, numerical tolerances, and validation data are kept distinct.
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 τ=log⁡sinh⁡2K\tau=\log\sinh 2Kτ=logsinh2K, so that the reduced free energy decomposes as Φ(τ)=τ/2+G(∣τ∣)\Phi(\tau)=\tau/2+G(|\tau|)Φ(τ)=τ/2+G(∣τ∣). 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 1.915×10−51.915\times10^{-5}1.915×10−5 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,…L^{-2},L^{-4},\ldotsL−2,L−4,…. 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 Φmag(η)\Phi_{\rm mag}(\eta)Φmag​(η), η=m/∣h∣8/15\eta=m/|h|^{8/15}η=m/∣h∣8/15. The low-temperature input contains the ∣h∣|h|∣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∝∣H∣1/15M\propto |H|^{1/15}M∝∣H∣1/15, and the two one-sided limits M(H→0±)=±M0M(H\to0^\pm)=\pm M_0M(H→0±)=±M0​ without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables xt=Lmx_t=Lmxt​=Lm and xh=L15/8hx_h=L^{15/8}hxh​=L15/8h. 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 Yglue(xt,xh)\mathcal Y_{\rm glue}(x_t,x_h)Yglue​(xt​,xh​). The resulting lattice free energy closes in the compact form ΦLglue(K,H)=Φlatglue(K,H)+Yglue(Lm,L15/8h)−Ybulk(Lm,L15/8h)L2+O(L−4)\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})ΦLglue​(K,H)=Φlatglue​(K,H)+L2Yglue​(Lm,L15/8h)−Ybulk​(Lm,L15/8h)​+O(L−4) where m=m(r,H)m=m(r,H)m=m(r,H) and h=h(r,H)h=h(r,H)h=h(r,H) are the nonlinear scaling fields and YbulkY_{\rm bulk}Ybulk​ is the thermodynamic singular scaling surface expressed in the same variables. Thus all finite-size/finite-field information is carried by the excess Yglue−Ybulk\mathcal Y_{\rm glue}-Y_{\rm bulk}Yglue​−Ybulk​, while the thermodynamic lattice free energy remains the previously constructed Φlatglue\Phi_{\rm lat}^{\rm glue}Φlatglue​. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error 3.90×10−43.90\times10^{-4}3.90×10−4 and maximum absolute error 9.55×10−49.55\times10^{-4}9.55×10−4 in Y\mathcal YY. 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.
We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin s=1,3/2,…s=1,3/2,\dots{}s=1,3/2,… and for the lattice Euclidean scalar-field theory with quartic self-interaction, on the simple-cubic and the body-centered-cubic lattices. Our bivariate high-temperature expansions, which extend through 24th order, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely, all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.
We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field hhh, for which no exact solution is known. We first conjecture a closed-form expression built from the one-dimensional Ising chain and Onsager's hypergeometric solution. Comparing with high-precision tensor renormalization group (TRG) data, we find that it deviates by up to 2.6%2.6\%2.6%, has no spontaneous magnetization, and gives an effective critical-isotherm exponent δ→1δ\to 1δ→1. However, using the Lee--Yang circle theorem, we show that the exact free energy is a superposition of one-dimensional Ising chains in the same field, weighted by a temperature-dependent (normalized, but not always positive) distribution of squared Lee--Yang gaps, of which our first expression is a single-chain approximation. Exact results (Onsager, Yang, the strong-field expansion and the Lee--Yang solution at βh=iπ/2βh = iπ/2βh=iπ/2) become sum rules for this distribution. We propose a simple form for it, fixed by Onsager's free energy and the exact strong-field limit without any fit to numerical data. This \emph{improved} conjecture agrees with TRG to 0.15%0.15\%0.15% for all βββ and hhh, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm M∝h1/δM\propto h^{1/δ}M∝h1/δ with δ=15.2δ= 15.2δ=15.2 and amplitude 1.0551.0551.055, compared with the exact δ=15δ=15δ=15 and 1.0591.0591.059. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, δ=15δ=15δ=15) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to 4×10−54\times 10^{-5}4×10−5.
This paper is a Response to Professor J.H.H. Perk's recent Comment (arXiv:1209.0731v1). We point out that the singularities of the reduced free energy βf, the free energy per site f and the free energy F of the 3D Ising model differ at β = 0. The rigorous proof presented in the Perk's Comment is only for the analyticity of the reduced free energy βf, which loses its definition at β = 0. Therefore, all of his objections lose the mathematical basis, which are thoroughly disproved. This means that the series expansions cannot serve as a standard for judging the correctness of the exact solution of the 3D Ising model. Furthermore, we note that there have been no comments on the topology-based approach developed by Zhang for the exact solution of the 3D Ising model. A Rejoinder to Professor J.H.H. Perk's open letter in arXiv:1209.0731v2 is added. We show that the analyticity in β of both the hard-core and Ising models is proved for β > 0, not for β = 0. Furthermore, the free energy per site f and the reduced free energy βf lose their definitions at β = 0, and thus either of them could have two different forms for high-temperature series expansions at/below infinite temperature. Three independent Virasoro algebras for 3D conformal field theory can be written within the 3+1 dimensional space (i.e., 3-sphere) with weight factors.
We enlighten some critical aspects of the three-dimensional (d = 3) random-field Ising model from simulations performed at zero temperature.We consider two different, in terms of the field distribution, versions of model, namely a Gaussian random-field Ising model and an equal-weight trimodal random-field Ising model.By implementing a computational approach that maps the ground-state of the system to the maximum-flow optimization problem of a network, we employ the most up-to-date version of the push-relabel algorithm and simulate large ensembles of disorder realizations of both models for a broad range of random-field values and systems sizes V = L × L × L, where L denotes linear lattice size and L max = 156.Using as finite-size measures the sampleto-sample fluctuations of various quantities of physical and technical origin, and the primitive operations of the push-relabel algorithm, we propose, for both types of distributions, estimates of the critical field h c and the critical exponent ν of the correlation length, the latter clearly suggesting that both models share the same universality class.Additional simulations of the Gaussian random-field Ising model at the best-known value of the critical field provide the magnetic exponent ratio β/ν with high accuracy and clear out the controversial issue of the critical exponent α of the specific heat.Finally, we discuss the infinite-limit size extrapolation of energyand order-parameter-based noise to signal ratios related to the self-averaging properties of the model, as well as the critical slowing down aspects of the algorithm.
There is no an accepted exact partition function (PF) for the three dimensional (3D) Ising model to our knowledge. Mainly based on the connection between the lattice Green function (LGF) for the simple cubic lattice and that for the honeycomb lattice, we infer an empirical partition function (EPF) for the simple cubic Ising model in the absence of an external magnetic field. This EPF3D=12π3∫0π∫0π∫0πlog⁡[2(2cosh⁡32z+3sinh⁡22z+2)12−2αsinh⁡2z (cos⁡ω1+cos⁡ω2+cos⁡ω3)]dω1dω2dω3,α∈[2,3]{\rm EPF}_{_{\rm 3D}}=\frac{1}{2{π^3}}\int_0^π\int_0^π\int_0^π\log [2(2{{\cosh }^3}2z + 3{{\sinh }^2}2z + 2)^{\frac{1}{2}} -2α\sinh 2z \ (\cosω_1 + \cos ω_2 + \cosω_3)] {\rm{d}}ω_1{\rm{d}}ω_2{\rm{d}}ω_3, α\in[\sqrt{2},\sqrt{3}]EPF3D​​=2π31​∫0π​∫0π​∫0π​log[2(2cosh32z+3sinh22z+2)21​−2αsinh2z (cosω1​+cosω2​+cosω3​)]dω1​dω2​dω3​,α∈[2​,3​] (where z=εkTz=\fracε{kT}z=kTε​, εεε the interaction energy, TTT the temperature, and kkk Boltzmann constant). When α=2α=\sqrt{2}α=2​, this EPF is consistent well numerically with the result from high temperature expansions by Guttmann and Enting (1993). The specific heat from this EPF approaches infinity non-logarithmically at the critical temperature TcT_cTc​. εkTc=cosh⁡−1[14(17−317)]/2≈0.277212\fracε{kT_c}=\cosh^{-1}[\frac{1}{4} (17-3\sqrt{17})]/2\approx 0.277212kTc​ε​=cosh−1[41​(17−317​)]/2≈0.277212, which is greater than 0.221654 from the recent Monte Carlo study.