The singular free energy of the two-dimensional Ising universality class near criticality is governed by a single universal function: in Aharony-Fisher non-linear scaling variables, F(m,h)=m28πlog⁡m2+h16/15Φ(η)F(m,h)=\frac{m^2}{8\pi}\log m^2+h^{16/15}\Phi(\eta) with η=m/h8/15\eta=m/h^{8/15}, where FF is the vacuum energy density of the Ising field theory, the c=12c=\frac{1}{2} conformal field theory of a free Majorana fermion perturbed by its energy and spin fields. The scaling function Φ(η)\Phi(\eta) controls the equation of state and all crossover thermodynamics of the critical domain. It is known only through high-precision numerics (lattice corner-transfer-matrix and truncated free-fermion-space calculations, agreeing to about 10−610^{-6}) and through asymptotic expansions about the integrable axes η=0\eta=0 and ∣η∣→∞|\eta|\to\infty in which only a handful of coefficients are exactly known; its analytic structure at complex η\eta (Yang-Lee edge singularities, Langer branch cut) is also known only numerically. The problem is to give Φ(η)\Phi(\eta) a complete analytic characterization: a closed-form expression or exact specification valid for general η\eta, together with the exact singularity structure in the complex η\eta-plane, replacing the current numerics-only and asymptotic-only descriptions.

Background

The two-dimensional Ising model is the paradigmatic exactly solved model of statistical mechanics: in zero magnetic field its free energy was computed by Onsager and its spontaneous magnetization by Yang, and the scaling-region correlation functions of Wu, McCoy, Tracy and Barouch (WMTB) are expressed through integral quantities involving solutions of a Painleve III equation (one of the six classical nonlinear ordinary differential equations with the Painleve property that their solutions have no movable singularities other than poles). In a nonzero magnetic field, however, no exact solution of the lattice model exists. The modern formulation is through scaling theory. Near the critical point, the scaling and universality hypotheses predict that the singular part of the free energy is a universal function F(m,h)F(m,h) of two non-linear scaling variables in the sense of Aharony and Fisher: analytic reparametrizations of the temperature deviation Δβ\Delta\beta and the field HH, namely m=O(Δβ)+O((Δβ)3)+O(H2)+…m=O(\Delta\beta)+O((\Delta\beta)^3)+O(H^2)+\dots and h=O(H)+H O(Δβ)+O(H3)+…h=O(H)+H\,O(\Delta\beta)+O(H^3)+\dots, chosen so that corrections to scaling are analytic. In two dimensions F(m,h)F(m,h) has a direct quantum-field-theory meaning: it is the vacuum energy density of the Ising Field Theory (IFT), the conformal field theory of a free massless Majorana fermion (central charge c=12c=\frac{1}{2}) perturbed by its two relevant primary fields, the energy density ϵ\epsilon (scaling dimension 11) with coupling proportional to mm, and the spin σ\sigma (scaling dimension 18\frac{1}{8}) with coupling hh of mass dimension 158\frac{15}{8}. The universal content is isolated as

F(m,h)=m28πlog⁡m2+h16/15 Φ(η),η=mh8/15,F(m,h)=\frac{m^2}{8\pi}\log m^2+h^{16/15}\,\Phi(\eta),\qquad \eta=\frac{m}{h^{8/15}},

with the normalization F(m,0)=m28πlog⁡m2F(m,0)=\frac{m^2}{8\pi}\log m^2. Here Φ(η)\Phi(\eta) is a universal scaling function of the single dimensionless scaling parameter η\eta; it controls the equation of state, magnetization, susceptibility and all crossover thermodynamics of every model in the 2D Ising universality class in the critical domain.

The IFT is integrable on the two axes of the (m,h)(m,h) plane: at h=0h=0 it is a free massive fermion (Onsager, Yang, WMTB), and at m=0m=0 (the critical isotherm in a field) it is Zamolodchikov's E8E_8 integrable theory with eight stable particles. At a generic point (m,h)(m,h), which is exactly what Φ(η)\Phi(\eta) at generic η\eta probes, the IFT is believed to be non-integrable, and this is the basic obstruction to an exact solution. What is known analytically are expansions about these axes: for ∣η∣→∞|\eta|\to\infty on the T<TcT<T_c side, Φlow(η)=η2∑kG~k η−15k/8\Phi_{\rm low}(\eta)=\eta^2\sum_k \tilde{G}_k\,\eta^{-15k/8}; on the T>TcT>T_c side, Φhigh(η)=η2∑kG2k∣η∣−30k/8\Phi_{\rm high}(\eta)=\eta^2\sum_k G_{2k}|\eta|^{-30k/8}; and for small η\eta, Φ(η)=−η28πlog⁡η2+∑kΦkηk\Phi(\eta)=-\frac{\eta^2}{8\pi}\log\eta^2+\sum_k \Phi_k\eta^k. Only a handful of these coefficients are exactly known: G~1\tilde{G}_1 has a simple explicit form, G2G_2 and G~2\tilde{G}_2 are given by WMTB Painleve III quadratures (evaluated to 50 digits by Orrick, Nickel, Guttmann and Perk), Φ0\Phi_0 was obtained by Fateev from the integrable E8E_8 theory, and Φ1\Phi_1 by Fateev, Lukyanov, Zamolodchikov and Zamolodchikov; the remaining coefficients were computed numerically by Fonseca and Zamolodchikov (FZ) using the Truncated Free Fermion Space Approach (TFFSA, a Hamiltonian truncation in the free-fermion Fock basis of the critical Ising CFT), and independently by the lattice corner-transfer-matrix (CTM) calculations of Mangazeev et al. [14] (the corner transfer matrix is Baxter's representation of the contribution of one quadrant of the lattice to the partition function; that work computes the scaling limit by a variational renormalization-group treatment of this object), the two agreeing to all six significant digits quoted by FZ (the lattice data have a relative spread of about 10−610^{-6}). FZ also studied Φ(η)\Phi(\eta) at complex η\eta: they determined the discontinuities across the branch cuts associated with the Yang-Lee edge singularities (the accumulation of Lee-Yang zeros of the partition function at complex magnetic field, whose leading behavior is governed by the non-unitary Yang-Lee minimal conformal field theory) and with Langer's essential singularity of the metastable branch, and they conjectured 'extended analyticity': that the Yang-Lee branching point is the nearest singularity under Langer's branch cut. Despite all this, as Mangazeev et al. [14] note, 'its complete analytic characterization is still lacking': no closed-form expression or exact defining equations are known for Φ(η)\Phi(\eta) at general η\eta, the known series are asymptotic, and the complex-η\eta singularity structure is known only numerically.

Problem Statement

Give a complete analytic characterization of the universal scaling function Φ(η)\Phi(\eta) of the two-dimensional Ising model in a magnetic field, defined through the scaling-limit free energy (the vacuum energy density of the Ising field theory, the c=12c=\frac{1}{2} conformal field theory of a free Majorana fermion perturbed by the energy field ϵ\epsilon and the spin field σ\sigma):

F(m,h)=m28πlog⁡m2+h16/15 Φ(η),η=mh8/15,F(m,h)=\frac{m^2}{8\pi}\log m^2+h^{16/15}\,\Phi(\eta),\qquad \eta=\frac{m}{h^{8/15}},

with the normalization F(m,0)=m28πlog⁡m2F(m,0)=\frac{m^2}{8\pi}\log m^2, where mm and hh are the Aharony-Fisher non-linear scaling variables for temperature and field.

A complete characterization consists of the following, established with proof:

  1. An analytic determination of Φ(η)\Phi(\eta) valid for general η\eta: a closed-form expression, or an exact specification (for instance an exact system of differential, Riemann-Hilbert, or Bethe-ansatz-type equations with a uniqueness theorem) whose solution is Φ(η)\Phi(\eta), going beyond the known asymptotic expansions about η=0\eta=0 and ∣η∣→∞|\eta|\to\infty and valid across the whole critical domain, including the crossover regions ∣η∣∼1|\eta|\sim 1 where currently only numerical data exist.

  2. The analytic structure of Φ(η)\Phi(\eta) in the complex η\eta-plane: the exact locations and natures of its singularities, in particular the Yang-Lee edge singularities and the Langer branch cut, settling (proving or refuting) the Fonseca-Zamolodchikov extended-analyticity conjecture.

These two deliverables jointly constitute one objective: replacing the numerically mapped function Φ(η)\Phi(\eta) by an analytically characterized universal function. Deliverable 2 follows from, or together with, deliverable 1 (the singularity structure of a closed form can be read off); a resolution of deliverable 2 alone would be a major advance but not the complete characterization requested. Two further limitations flagged by Mangazeev et al. [14], a first-principles derivation directly from lattice models of the subleading (irrelevant-operator) corrections used in lattice-to-field-theory matching, and the development of high-order small-τ\tau susceptibility expansions for the triangular lattice analogous to the square-lattice results of Orrick, Nickel, Guttmann and Perk, are linked auxiliary problems with independent completion boundaries; they are recorded here rather than folded into this core objective.

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