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arXiv:2604.24675 · Physical review. E

Conformal invariance of the large- N limit of the O ( N ) universality class

Santiago Cabrera, Gonzalo De Polsi, Adam Rançon, Nicolás Wschebor

Published Jul 20, 2026• 0 citations
arXivHTMLPDF

Abstract

Conformal symmetry is expected to be realized in many equilibrium statistical mechanical systems at criticality. Although this is certainly true in two-dimensional systems, the three-dimensional case is subtler, and only a few proofs exist, only so in very specific cases. In this work, we give two proofs for the large-N limit of the O(N) universality class within the nonperturbative renormalization group framework: one functional, and one vertex-by-vertex in Fourier space. While doing so, we unveil how the theory is structured in order for conformal symmetry to be realized. As a consequence, we shed light on what to expect, on rather general grounds, for a theory to be conformally invariant.

DOI
10.1103/yznz-gcqs

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Conformal symmetry is expected to be realized in many equilibrium statistical mechanical systems at criticality. Although this is certainly true in two-dimensional systems, the three-dimensional case is subtler, and only a few proofs exist, only so in very specific cases. In this work, we give two proofs for the large-N limit of the O(N) universality class within the nonperturbative renormalization group framework: one functional, and one vertex-by-vertex in Fourier space. While doing so, we unveil how the theory is structured in order for conformal symmetry to be realized. As a consequence, we shed light on what to expect, on rather general grounds, for a theory to be conformally invariant.
The behavior of many critical phenomena at large distances is expected to be invariant under the full conformal group, rather than only isometries and scale transformations. When studying critical phenomena, approximations are often required, and the framework of the nonperturbative, or functional, renormalization group is no exception. The derivative expansion is one of the most popular approximation schemes within this framework, due to its great performance on multiple systems, as evidenced in the past few decades. Nevertheless, it has the downside of breaking conformal symmetry at a finite order. This breaking is not observed at the leading order of the expansion, denoted the local potential approximation, and it only appears once one considers, at least, the next-to-leading order of the derivative expansion [O(∂^{2})] when including composite operators. In this work, we study the constraints arising from conformal symmetry for the O(N) models using the derivative expansion at order O(∂^{2}). We explore various values of N and minimize the breaking of conformal symmetry to fix the nonphysical parameters of the approximation procedure. We compare our prediction for the critical exponents with those coming from a more usual procedure, known as the principle of minimal sensitivity.
Conformal invariance is expected to be an emergent property of many systems in their critical regime. However, approximation schemes generically spoil this property. This is in particular the case of the derivative expansion, a widely used approximation scheme in the framework of the functional renormalization group. In this article, we consider Ward identities associated with conformal invariance in the 3-d Ising universality class with truncations at order 4 (next-to-next-to-leading order) in the derivative expansion, with Z2Z_2Z2​ invariant composite operators. Our results confirm that the regulating functions which yield a small breaking of conformal invariance also present a small sensitivity of the universal critical exponents with the choice of this regulating function. We also show that, in the vicinity of regulator-parameter values for which the conformal constraints are best satisfied, the breaking of conformal invariance reduces as the order of the derivative expansion is increased, providing a new indication of the convergence of this approximation scheme.
Determining whether an additional local symmetry affects the universality class of a sta-tistical model is an important issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin re-versals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model is con-sistent with the mean field scenario. On the other hand, in the two-dimensional case –the one we focus on in this thesis – fluctuations are stronger and minimize the reliability of mean field predictions, as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions generically suggests that criticality is limited to zero temperature, and Monte Carlo studies for T → 0 showed a fast growth of the correlation length which made particularly hard to reach the asymp-totic limit and draw conclusions. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one – which should definitely occur for RP 1 ∼ O(2) – and mediated by ”disclination” defects has also been debated in numerical studies. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability and conformal field theory, models with local symmetries traditionally remained outside the range of application of these methods. In this thesis, however, we show how the renormalization group fixed points of the RP N −1 model and of its complex generalization, the CP N −1 model, can be accessed in an exact way. This is achieved in the scale invariant scattering framework which, as we review in the introductory part of the thesis, implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last few years new results for pure and disordered systems. In the last part of the thesis, we will exploit the generality of the scale invariant scattering method to progress with another long standing problem of two-dimensional criticality, namely that of spin clusters in Potts correlated percolation. It has been known for long time that the problem can be addressed considering two coupled Potts models, but the need to consider the number of states of one of these models as a continuous variable has severely limited the analytical or numerical study of the critical points. Also here, the ability of the scattering method to enforce conformal invariance for the internal symmetry characteristic of the universality class will allow us to obtain exact equations for the critical points and to determine their solutions in the relevant limits.
Field-theoretical calculations performed in an approximation scheme often present a spurious dependence of physical quantities on some unphysical parameters associated with the details of the calculation setup (such as the renormalization scheme or, in perturbation theory, the resummation procedure). In the present article, we propose to reduce this dependence by invoking conformal invariance. Using as a benchmark the three-dimensional Ising model, we show that, within the derivative expansion at order 4, performed in the nonperturbative renormalization group formalism, the identity associated with this symmetry is not exactly satisfied. The calculations which best satisfy this identity are shown to yield critical exponents which coincide to a high accuracy with those obtained by the conformal bootstrap. Additionally, this work gives a strong justification to the success of a widely used criterion for fixing the appropriate renormalization scheme, namely the principle of minimal sensitivity.
Abstract In the previous chapters we have dedicated ample space to the study of two-dimensional statistical systems at criticality, providing their exact solutions in terms of conformal field theories. In this chapter we start investigating the deformations of conformal field theories that move the statistical systems away from criticality. As pointed out in Chapter 8, in the renormalization group approach the characterization of the universality classes must include, in addition to the conformal theory of the fixed points, also the description of the scaling region nearby.
We show that for conformally invariant two-dimensional systems, the amplitude of the finite-size corrections to the free energy of an infinitely long strip of width L at criticality is linearly related to the conformal anomaly number c, for various boundary conditions. The result is confirmed by renormalization-group arguments and numerical calculations. It is also related to the magnitude of the Casimir effect in an interacting one-dimensional field theory, and to the low-temperature specific heat in quantum chains.