Near-critical Ising, sine-Gordon at the free fermion point, and bosonization

Abstract: In this article, we study the continuous correlations of the near-critical Ising model in two dimensions with plus boundary conditions, and prove that doubled correlation functions of primary fields (spin, disorder, fermions, energy) in the Ising model are given by correlation functions of the sine-Gordon model at the free fermion point. This is an instance of bosonization. The main ideas involve analyticity of correlation functions in a mass parameter in finite volume and proving that in a perturbative regime, the Taylor coefficients of the correlation functions match due to known bosonization results for the critical Ising model in terms of the Gaussian free field. The main techniques on the Ising side involve construction and precise estimates of certain massive holomorphic functions while on the sine-Gordon side, we control an iterated Mayer expansion with techniques going back to Brydges and Kennedy.

• The paper establishes exact equality between doubled Ising correlators and sine-Gordon trigonometric fields at β=4π using a rigorous bosonization dictionary.
• It employs analytic continuations and Taylor expansions to precisely control correlations and manage non-constant mass functions in simply-connected planar domains.
• The work bridges discrete spin systems with integrable bosonic models, providing rigorous benchmarks via Painlevé transcendent analysis in the infinite volume limit.

Bosonization and Scaling Limits: Near-Critical Ising, Sine-Gordon, and Fermionic Duality

Introduction and Motivation

This work establishes rigorous analytic connections between the near-critical two-dimensional (2D) Ising model with plus boundary conditions and the sine-Gordon model at $\beta=4\pi$ (the free fermion point). Specifically, it proves that doubled correlation functions of Ising primaries (spin, disorder, fermion, energy) in the near-critical regime coincide with correlation functions of trigonometric and derivative fields in the sine-Gordon theory at this point, thus realizing a concrete form of bosonization away from criticality. The construction relies on extending known massless bosonization correspondences and utilizing control over massive holomorphic observables and techniques from Mayer expansion analysis.

This result meticulously formalizes the physical intuition: the massive Ising field theory (for thermal perturbations) is equivalent, through bosonization, to the sine-Gordon theory at the special coupling $\beta = 4\pi$, including at and near the critical point. This rigorously connects discrete random spin systems, fermionic quantum field theory, and integrable bosonic models.

Main Results and Structure

Extension of Bosonization Beyond Criticality

For a general (possibly nonconstant) mass function $\alpha$ and a simply-connected planar domain with smooth boundary, the authors construct continuum analogues of Ising and double Ising correlation functions for all Ising primaries, including spin, disorder, Majorana/Dirac fermions, and energy densities, building on prior works at the critical point and for constant mass [BIVW, P, CIM23]. The precise bosonization “dictionary” relates insertion products in the double (two-copy) Ising theory to trigonometric and derivative correlators in the sine-Gordon model:

Double Ising Field	Sine-Gordon (at $\beta=4\pi$)
mass parameter $\alpha$	chemical potential $\rho = -\tfrac{4}{\pi}\alpha$
$\sigma\widetilde{\sigma}$	$\sqrt{2}\,:\!\cos(\sqrt{\pi}\varphi)\!:$
$\mu\widetilde{\mu}$	$\sqrt{2}\,:\!\sin(\sqrt{\pi}\varphi)\!:$
$\psi\widetilde{\psi}$	$2i\sqrt{\pi}\partial\varphi$
$[\psi^*\widetilde{\psi}^*]$	$-2i\sqrt{\pi}\bar{\partial}\varphi$
energy $+\widetilde{+}$	$4\,\vdots\!\cos(\sqrt{4\pi}\varphi)\!\vdots$

The primary claim is that (under smoothness/homogeneity requirements) the Ising model double correlators are exactly equal, as functions or generalized functions on disjoint configurations, to their sine-Gordon counterparts under this dictionary for general $\alpha$ [Theorem~1].

Analyticity and Taylor Expansion in Mass/Strength

The analysis proceeds by showing that both double Ising and smeared sine-Gordon correlation functions are analytic in the perturbation parameter (mass $m$ for Ising, chemical potential for sine-Gordon) near zero, with the Taylor coefficients given by explicit, combinatorially structured perturbative expansions. At $m=0$, this reproduces the massless correspondence with the Gaussian free field (GFF), and all coefficients correspond (by matching cumulants and operator product expansions) to the dictionary above.

The Taylor expansions are shown to converge in explicit neighborhoods, with coefficients controlled via norm and integrability bounds derived from a combination of complex analysis (massive Vekua equations for the Ising side) and Mayer/cluster expansion methods (for sine-Gordon, following Brydges-Kennedy, Balaban et al.).

Infinite Volume Limit and Explicit Solutions

By controlling boundary behavior and decorrelation, the authors further analyze the infinite-volume ($\Omega \to \mathbb{C}$) scaling limits for constant mass; specifically, the squared two-point (pure spin and disorder) Ising expectations correspond to integrals determined by Painlevé III transcendent solutions (see equation (1.13)), thus connecting to the integrable structure of these field theories and classical results on Ising scaling limits and form factors [WMTB].

Methodological Core

Ising Model Side

• Massive Holomorphic Observable Construction: Spin- and disorder-weighted two-point fermion and disorder-fermion correlators are characterized as unique solutions to Vekua-type equations ($\bar{\partial}f = -i\alpha\bar{f}$) on branched covers, with boundary and monodromy prescriptions matching the critical theory but with a real-valued mass function. These equations admit a representation-theoretic characterization (Pfaffian structure for multipoint fermionic correlators).
• Continuity and Analyticity in Mass: The Ising correlation functions, for smooth mass, are shown to be jointly (locally) analytic in $m$, with explicit recursive/expansive formulas for Taylor coefficients, involving cumulants of critical correlations and integrated insertions of $+\widetilde{+}$ (energy density) fields.

Sine-Gordon Side

• Heat Kernel Regularization: A rigorous construction of the finite-volume sine-Gordon model at and near $\beta=4\pi$ is given using heat kernel regularizations of the GFF and corresponding Wick-ordered exponential functions, including necessary subtractions to handle the known divergences (e.g., for $:\!\cos(\sqrt{4\pi}\varphi)\!:$, which requires an additional infinite subtracted normalization).
• Cluster/Mayer Expansion: Analyticity, correlation function existence, and Taylor/perturbative expansions are established using a multi-scale Mayer expansion along the lines of Brydges-Kennedy cluster expansion, controlling the radius of analyticity and convergence via precise bounds on the renormalized potential and partition function.
• Cumulant Structures: Taylor coefficients for the sine-Gordon correlations can be written explicitly as integrals/sums of cumulants (joint moments) of GFF correlations and insertions, matching the combinatorics of doubly-inserted Ising energy densities.

Significant Claims and Contrasts

Among the strong claims:

• Exact Equality of Doubled Ising and Sine-Gordon Correlators: The rigorous demonstration that all doubled Ising correlators, for general primary field insertions and smooth mass functions, are exactly (not merely in some scaling or weak limit) equal to the corresponding sine-Gordon correlators at $\beta=4\pi$, with explicit normalization, including for non-constant mass parameters [Theorem~1, Corollary~1].
• The construction simultaneously yields all the integration constants and normalization ambiguities that have historically appeared in bosonization procedures and Painlevé-III representations of Ising correlators.

Theoretical and Practical Implications

Probabilistic Verification of QFT Predictions

The results provide a direct probabilistic realization (in the continuum, away from the scaling limit of discrete models) of bosonization for the near-critical two-dimensional Ising model, including both the critical and off-critical (thermal perturbation) cases. This includes the analytic continuation in the mass/perturbation parameter, the existence and characterization of all primary correlations, and the explicit identification of form factors and large-distance asymptotics with those of the sine-Gordon theory.

Extension to General Mass and Boundary Dependence

Unlike earlier works where only constant mass (or translation-invariance) was treated, this analysis allows for spatially varying mass, offering access to more general inhomogeneous perturbations and their analytic properties. The construction also aligns with the combinatorial correspondence between Ising/dimer models and specializations of the sine-Gordon/Liouville field at critical or free fermion points.

Foundational for Further 2D Field Theory Constructions

The robust analytic, combinatorial, and PDE control over both sides of the correspondence paves the way for extending bosonization and duality constructions to other near-integrable 2D lattice models and their field-theoretic perturbations or couplings to curvature (as in Liouville theory, random planar maps, or inhomogeneous environments).

Benchmarks for Integrable Models

Finally, the explicit solutions in the infinite volume limit and their characterization via Painlevé equations provide a rigorous benchmark against which predictions from the integrable-quantum-field-theory program can be checked, including form factor expansions and exact $S$-matrix predictions.

Speculative Outlook for Future Development

• Beyond Free Fermion Point: The methods utilized suggest potential adaptation to analyze bosonization/boson-fermion duality for the sine-Gordon model away from $\beta=4\pi$, corresponding to interacting Thirring models (as in the famous Coleman correspondence).
• Extension to Dynamics and Non-Euclidean Geometries: The analytic control on mass and boundary conditions and the derivation in arbitrary simply-connected domains allows, in principle, for extension to dynamical models or those on random geometries (random planar maps, Liouville quantum gravity).
• Interacting Dimer/Ising and SLE Processes: Given the links to the dimer model and SLE-type results, these findings may anchor rigorous study of domain-wall curves, height functions, and other geometric observables under near-critical scaling.

Conclusion

This article achieves a mathematically rigorous extension of 2D bosonization results—connecting the near-critical Ising (and double Ising) model to the sine-Gordon model at $\beta=4\pi$—with full control over analytic structure, Taylor expansion, and the dictionary between all primary field correlators. The analysis bridges discrete statistical mechanics, conformal and integrable field theory, and stochastic analysis, providing both a foundation and a blueprint for further exploration of fermion-boson dualities in two dimensions.

Reference: "Near-critical Ising, sine-Gordon at the free fermion point, and bosonization" (2512.11304)