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Conformal Invariance in the Long-Range Ising Model

Miguel F. Paulos, Slava Rychkov, Balt C. van Rees, Bernardo Zan

TL;DR

The paper investigates whether the critical point of the long-range Ising model (LRI) possesses conformal invariance. It first provides an ε-expansion framework around σ = (d+ε)/2, showing nontrivial cross-correlators vanish up to second order and that the φ^n scaling dimensions acquire ε-dependent anomalous dims; the authors then prove conformal invariance to all orders by reformulating the LRI as a defect theory in a higher-dimensional space, enabling a Ward-identity-based argument analogous to the Wilson–Fisher fixed point. They also review conformal invariance for the Gaussian LRI phase and the short-range Ising model, including AdS/CFT and Caffarelli–Silvestre perspectives, to build a coherent picture. The results imply that LR lattice models with ε-based perturbations are conformally invariant at criticality, with potential implications for conformal bootstrap approaches and nonperturbative explorations. A notable outcome is the identification of a shadow-like relation Δ_{φ^3} = Δ_φ + σ, stemming from the nonlocal equation of motion, and the discussion of potential discontinuities when transitioning to the short-range limit.

Abstract

We consider the question of conformal invariance of the long-range Ising model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d-dimensional short-range Ising model is also included and may be of independent interest.

Conformal Invariance in the Long-Range Ising Model

TL;DR

The paper investigates whether the critical point of the long-range Ising model (LRI) possesses conformal invariance. It first provides an ε-expansion framework around σ = (d+ε)/2, showing nontrivial cross-correlators vanish up to second order and that the φ^n scaling dimensions acquire ε-dependent anomalous dims; the authors then prove conformal invariance to all orders by reformulating the LRI as a defect theory in a higher-dimensional space, enabling a Ward-identity-based argument analogous to the Wilson–Fisher fixed point. They also review conformal invariance for the Gaussian LRI phase and the short-range Ising model, including AdS/CFT and Caffarelli–Silvestre perspectives, to build a coherent picture. The results imply that LR lattice models with ε-based perturbations are conformally invariant at criticality, with potential implications for conformal bootstrap approaches and nonperturbative explorations. A notable outcome is the identification of a shadow-like relation Δ_{φ^3} = Δ_φ + σ, stemming from the nonlocal equation of motion, and the discussion of potential discontinuities when transitioning to the short-range limit.

Abstract

We consider the question of conformal invariance of the long-range Ising model at the critical point. The continuum description is given in terms of a nonlocal field theory, and the absence of a stress tensor invalidates all of the standard arguments for the enhancement of scale invariance to conformal invariance. We however show that several correlation functions, computed to second order in the epsilon expansion, are nontrivially consistent with conformal invariance. We proceed to give a proof of conformal invariance to all orders in the epsilon expansion, based on the description of the long-range Ising model as a defect theory in an auxiliary higher-dimensional space. A detailed review of conformal invariance in the d-dimensional short-range Ising model is also included and may be of independent interest.

Paper Structure

This paper contains 30 sections, 187 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Field-theoretical setup
  3. Tests of conformal invariance
  4. < phi phi3 >
  5. Explicit computation
  6. General argument
  7. < phi2 phi4 >
  8. Conformal invariance of the SRI critical point
  9. Scale and conformal invariance breaking effects
  10. A proof which is too quick, and a paradox
  11. Correct proof
  12. SRI conformal invariance beyond perturbation theory
  13. Conformal invariance of the gaussian phase
  14. Direct argument
  15. Argument from correlation functions
  16. ...and 15 more sections

Figures (2)

  • Figure 1: The three phases for the LRI critical point. The boundaries are $\sigma=d/2$ (straight solid) and $\sigma=2-\eta_{\rm SRI}(d)$ (curved dashed, interpolated using the known exact $\eta_{\rm SRI}(d)$ for $d=1,2,4$ and numerical values from the $\epsilon$-expansion LeGuillou:1987ph and the conformal bootstrap El-Showk:2013nia for a few intermediate $d$). Here we will be working near the boundary $\sigma=d/2$.
  • Figure 2: The horizontal, vertical, and diagonal straight lines show the location of the poles in (\ref{['eq:Mst']}). In this plot $d=3$ and $\epsilon=0.3$, but the same picture is valid for all $d$ of interest and $\epsilon\ll 1$. Only the leading poles are shown; each line is accompanied by a series of parallel lines spaced by 1 in the direction shown by short arrows. The real parts $(s_0,t_0)$ of the integration contours in the MB representation (\ref{['eq:MB']}) must lie in the small red triangle. On the other hand $(s_0,t_0)$ must lie in the large grey triangle for (\ref{['eq:convord']}) to be satisfied. The dashed arrow shows a possible contour shift, crossing two horizontal pole lines at $t=-\frac{d}{4}+\frac{3\epsilon}{4}$ and $t=0$.