We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field $h$, 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 $2.6\%$, has no spontaneous magnetization, and gives an effective critical-isotherm exponent $δ\to 1$. 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 $βh = iπ/2$) 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 $0.15\%$ for all $β$ and $h$, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm $M\propto h^{1/δ}$ with $δ= 15.2$ and amplitude $1.055$, compared with the exact $δ=15$ and $1.059$. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, $δ=15$) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to $4\times 10^{-5}$.
We study the free energy of the classical square-lattice Ising model in a uniform real magnetic field h, 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%, has no spontaneous magnetization, and gives an effective critical-isotherm exponent δ→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/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% for all β and h, reproduces Yang's spontaneous magnetization exactly, has a positive specific heat, and yields the critical isotherm M∝h1/δ with δ=15.2 and amplitude 1.055, compared with the exact δ=15 and 1.059. A refined form, which builds in additional input from Ising field theory (including the exact critical isotherm, δ=15) and a few constants adjusted to transfer-matrix and TRG data, matches the TRG free energy to 4×10−5.
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=logsinh2K, so that the reduced free energy decomposes as 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 1.915×10−5 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,…. 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), η=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∝∣H∣1/15, and the two one-sided limits M(H→0±)=±M0 without separately fitting magnetization branches. Finally, finite size and finite field are combined in the scaling variables xt=Lm and xh=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). The resulting lattice free energy closes in the compact form ΦLglue(K,H)=Φlatglue(K,H)+L2Yglue(Lm,L15/8h)−Ybulk(Lm,L15/8h)+O(L−4) where m=m(r,H) and h=h(r,H) are the nonlinear scaling fields and 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, while the thermodynamic lattice free energy remains the previously constructed Φlatglue. On eleven transfer-matrix points withheld from the construction, the blind free-energy test gives RMS absolute error 3.90×10−4 and maximum absolute error 9.55×10−4 in 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.
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858, comparable to the exponent 4/7≈0.5714 found from renormalization arguments.
The free energy of the Ising model on the square lattice with alternating interactions in both horizontal and vertical directions is exactly derived. This model is distinct from the checkerboard Ising model. The result includes Onsager’s free energy as a special case, and also includes Lee–Yang’s free energy with an imaginary field, and relates these two solutions via continuous parameters. The result includes a generalization of Lee–Yang’s result to cases with four different couplings. It is also derived that each imaginary magnetic field iπ/2 applied to a lattice site corresponds to a single frustrated square in its dual lattice.
The one-dimensional Ising chain with inverse-square interactions is a paradigmatic long-range system: it exhibits the unusual Thouless transition, with a discontinuity in the spontaneous magnetization, yet its full field-dependent thermodynamics remains unknown in closed form. We develop the high-temperature expansion of its free energy while retaining the dependence on a uniform magnetic field at every order. We prove that, after a natural rescaling of the coupling, each coefficient is a rational polynomial in the squared hyperbolic tangent of the reduced field. Remarkably, at each order n the field-dependent free-energy coefficient of a finite, periodically repeated ring is an exact polynomial of degree at most n in L−2. The infinite-chain coefficient is its constant term and can therefore be reconstructed from n+1 finite rings without numerical extrapolation. Using this reconstruction, we determine the full field-dependent expansion through order 26: 26 polynomials containing 377 rational coefficient entries. We further derive exact formulas, valid at every order, for the coefficients of the two highest powers of the field variable using matching determinants and a cycle-Laplacian reduction. These results reveal an unexpected algebraic structure in the microscopic thermodynamics of the inverse-square chain and provide high-order information about its magnetic response.
We study the complex-temperature phase diagram of the square-lattice Ising model for nonzero external magnetic field H, i.e., for 0\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}\ensuremath{\infty}, where \ensuremath{\mu}=${\mathit{e}}^{\mathrm{\ensuremath{-}}2\mathrm{\ensuremath{\beta}}\mathit{H}}$. We also carry out a similar analysis for -\ensuremath{\infty}\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}0. The results for the interval -1\ensuremath{\le}\ensuremath{\mu}\ensuremath{\le}1 provide a way of continuously connecting the two known exact solutions of this model, viz., at \ensuremath{\mu}=1 (Onsager, Yang) and \ensuremath{\mu}=-1 (Lee and Yang). Our methods include numerical calculations of complex-temperature zeros of the partition function and an analysis of low-temperature series expansions. For real nonzero H, the inner branch of a lima\ifmmode \mbox{\c{c}}\else \c{c}\fi{}on bounding the FM phase breaks and forms two complex-conjugate arcs. We study the singularities and associated exponents of thermodynamic functions at the endpoints of these arcs. For \ensuremath{\mu}0, there are two line segments of singularities on the negative and positive u axis, and we carry out a similar study of the behavior at the inner endpoints of these arcs, which constitute the nearest singularities to the origin in this case. Finally, we also determine the exact complex-temperature phase diagrams at \ensuremath{\mu}=-1 on the honeycomb and triangular lattices and discuss the relation between these and the corresponding zero-field phase diagrams. \textcopyright{} 1996 The American Physical Society.
For the two-dimensional ferromagnetic Ising critical point, I show that the known values of the critical exponents imply the absence of logarithms of the reduced temperature in the leading contributions to any field derivative of the free energy at zero magnetic field. For the square-lattice Ising antiferromagnet in a weak magnetic field, I compute the critical line Tc(H)=Tc0(1-0.038 023 259H2) and the leading contribution to the susceptibility \ensuremath{\chi}=0.014 718 006 6H2ln(1/\ensuremath{\Vert}t\ensuremath{\Vert}), where t is the reduced temperature.