On the Wilson-Fisher fixed point in the limit of integer spacetime dimensions

2026/01/27 by Bernardo Zan · 1 voice
Physics and Astronomy · #hep-th #cond-mat.stat-mech

paper · pdf · doi:10.21468/scipostphys.20.5.148

arxiv published 2026/01/27 · arxiv updated 2026/05/27

Abstract

The Wilson-Fisher fixed point defines a continuous family of interacting conformal field theories in non-integer dimensions. In integer dimensions, it is widely believed to lie in the same universality class as the critical Ising model. In this work, we revisit the identification between the Wilson-Fisher fixed point at integer dimensions and the Ising CFT. We argue that a literal equality between the two theories is incompatible with the emergence of Virasoro symmetry in two dimensions. Instead, we propose that the Ising model emerges only as a subsector of the Wilson-Fisher fixed point. We support this scenario through a detailed study of the two-dimensional O(n) model and by examining operators transforming in irreducible representations of the orthogonal group whose multiplicities become negative for integer values of the spacetime dimension. Finally, we comment on the implications of these results for attempts to construct a d=2+ε expansion starting from exact two-dimensional data.

Citations

• Long-range minimal models
• Phases of 2d Gauge Theories and Symmetric Mass Generation
• Disturbing news about the d=2+ε expansion
• The tricritical Ising CFT and conformal bootstrap
• Bootstrapping the 3d Ising stress tensor
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• Quantum Groups as Global Symmetries II. Coulomb Gas Construction
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• Critical Ising Model in Varying Dimension by Conformal Bootstrap
• Bootstrapping the long-range Ising model in three dimensions
• Minimally subtracted six loop renormalization of O(n)-symmetric ϕ⁴ theory and critical exponents
• A scaling theory for the long-range to short-range crossover and an infrared duality
• Long-range critical exponents near the short-range crossover
• A Mellin space approach to the conformal bootstrap
• Conformal Bootstrap in Mellin Space
• The Random-Bond Ising Model in 2.01 and 3 Dimensions
• Precision Islands in the Ising and O(N) Models
• The Epsilon-Expansion from Conformal Field Theory
• A Cheap Alternative to the Lattice?
• Conformal Field Theories in Fractional Dimensions
• Two Dimensional Renormalization Group Flows in Next to Leading Order
• Logarithmic conformal field theories as limits of ordinary CFTs and some physical applications
• The unitary representations of the Poincare group in any spacetime dimension
• Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions
• Conformal Four Point Functions and the Operator Product Expansion
• Conformal Field Theory
• On Reducible but Indecomposable Representations of the Virasoro Algebra
• Indecomposable Fusion Products
• The structure of the spectrum of anomalous dimensions in the N-vector model in 4-epsilon dimensions
• On Calculation of 2+\veps RG Functions in the Gross Neveu Model from Large N Expansions of Critical Exponents
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• Critical Exponents in 3.99 Dimensions
• Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition

Discussions

• basically where i'm at after arxiv.org/abs/2505.21611 and arxiv.org/abs/2601.19977 [bsky, 0 points, 0 comments]

Related

• Disturbing news about the d=2+ε expansion
• Unitarity violation at the Wilson-Fisher fixed point in 4-epsilon dimensions
• Critical O(N) Models in 6-ε Dimensions
• Generalized Wilson-Fisher critical points from the conformal OPE
• New Conformal Field Theories with Anomalous Dimensions