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Energy field of critical Ising model and examples of singular fields in QFT

Christophe Garban, Antti Kupiainen

TL;DR

The paper demonstrates that three natural QFT field measures are singular with respect to their natural base measures: the near-critical 2d Ising scaling limit in the β-direction, the hierarchical Sine-Gordon field relative to the hierarchical GFF for $β$ up to the BKT point, and the hierarchical $Φ^4_3$ field relative to the hierarchical GFF. The authors develop a tangible RG-based framework using effective potentials and mesoscopic observables to detect singularity across scales, yielding rigorous evidence for non-existence of certain fields as random Schwartz distributions in the plane. They provide detailed constructions for the hierarchical SG and hierarchical $Φ^4_3$ models, including convergence of effective potentials and explicit singular events, and discuss the near-critical Potts case as a contrasting scenario with potential existence of energy fields. The results offer a natural, down-to-earth method for identifying singular behavior at all scales, with potential applicability to other quantum field theory settings where effective potentials are under precise control.

Abstract

The goal of this paper is to prove singularity of three natural fields in QFT with respect to their natural base measure. The fields we consider are the following ones: (1) The near-critical limit of the $2d$ Ising model (in the $β$-direction) is locally singular w.r.t the critical scaling limit of $2d$ Ising. (N.B. In the $h$-direction it is not locally singular). (2) The $2d$ Hierarchical Sine-Gordon field is singular w.r.t the $2d$ hierarchical Gaussian Free Field for all $β\in[β_{L^2}, β_{BKT})$. (3) The Hierarchical $Φ^4_3$ field is singular w.r.t the $3d$ hierarchical GFF. Item (1) gives the first strong indication that the energy field of critical $2d$ Ising model does not exist as a random Schwarz distribution on the plane. Item (2) has been proved to be singular for the non-hierarchical $2d$ Sine-Gordon sufficiently far from the BKT point in [GM24] while item (3) is proved to be singular for the non-hierarchical $3d$ $Φ^4_3$ field in [BG21, OOT21, HKN24]. We believe our way to detect a singular behaviour at all scales is very much down to earth and may be applicable in all settings where one has a good enough control on the so-called effective potentials.

Energy field of critical Ising model and examples of singular fields in QFT

TL;DR

The paper demonstrates that three natural QFT field measures are singular with respect to their natural base measures: the near-critical 2d Ising scaling limit in the β-direction, the hierarchical Sine-Gordon field relative to the hierarchical GFF for up to the BKT point, and the hierarchical field relative to the hierarchical GFF. The authors develop a tangible RG-based framework using effective potentials and mesoscopic observables to detect singularity across scales, yielding rigorous evidence for non-existence of certain fields as random Schwartz distributions in the plane. They provide detailed constructions for the hierarchical SG and hierarchical models, including convergence of effective potentials and explicit singular events, and discuss the near-critical Potts case as a contrasting scenario with potential existence of energy fields. The results offer a natural, down-to-earth method for identifying singular behavior at all scales, with potential applicability to other quantum field theory settings where effective potentials are under precise control.

Abstract

The goal of this paper is to prove singularity of three natural fields in QFT with respect to their natural base measure. The fields we consider are the following ones: (1) The near-critical limit of the Ising model (in the -direction) is locally singular w.r.t the critical scaling limit of Ising. (N.B. In the -direction it is not locally singular). (2) The Hierarchical Sine-Gordon field is singular w.r.t the hierarchical Gaussian Free Field for all . (3) The Hierarchical field is singular w.r.t the hierarchical GFF. Item (1) gives the first strong indication that the energy field of critical Ising model does not exist as a random Schwarz distribution on the plane. Item (2) has been proved to be singular for the non-hierarchical Sine-Gordon sufficiently far from the BKT point in [GM24] while item (3) is proved to be singular for the non-hierarchical field in [BG21, OOT21, HKN24]. We believe our way to detect a singular behaviour at all scales is very much down to earth and may be applicable in all settings where one has a good enough control on the so-called effective potentials.

Paper Structure

This paper contains 23 sections, 22 theorems, 305 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results.
  3. Motivation: non-existence of fields (in the sense of distributions).
  4. Short introduction to the main objects.
  5. Main ideas of proofs.
  6. Near-critical Ising model
  7. (Near)-singularity at the discrete level.
  8. Existence of subsequential scaling limits.
  9. Singularity under $\mathrm{CLE}_{16/3}$ scaling limit.
  10. Singularity under $\mathrm{CLE}_3$ scaling limit ?
  11. On the singularity of near-critical $\mathrm{SLE}_{16/3}$ ?
  12. Near-critical Potts model, $q\in \{3,4\}$
  13. Singularity for hierarchical Sine-Gordon model away from the $L^2$ phase
  14. Setup and definition of the hierarchical Sine-Gordon field.
  15. Analysis of the RG flow.
  16. ...and 8 more sections

Key Result

Lemma 2.1

For any fixed $\lambda \in \mathbb{R}$ (including $\lambda=0$ which is the critical measure), Furthermore, the constants involved in $\asymp$ are uniform over $\lambda \in K$ for any compact set $K\in \mathbb{R}$ (but do depend on the choice of the compact set $K$).

Figures (1)

  • Figure 1: Illustration in a simpler setting of our main difficulty to identify singularity. The first line represents the sequence of measure $\{\mu_n\}$ on the unit interval $[0,1]$ with Radon-Nikodim derivative $d \mu_n(x) = (\sum_{i=0}^{2^n-1} 2^{2n} 1_{[i 2^{-n}, i 2^{-n} + 2^{-3n}]} )(x) dx$. These measures are more and more singular as $n\to \infty$ with respect to Lebesgue measure. Yet, their limiting measure is nothing but the Lebesgue measure on $[0,1]$. We thus need to prove that the three QFT fields we are interested in behave more like the second line with fractal behaviour. Namely the singularity spotted on an $\varepsilon$ regularisation remains after taking the UV limit in the RG flow.

Theorems & Definitions (47)

  • Remark 1
  • Example 1
  • Example 2
  • Lemma 2.1
  • Remark 2
  • Lemma 2.2
  • Proposition 2.3
  • Conjecture 1
  • Proposition 2.4
  • Remark 3
  • ...and 37 more