Conformal Invariance in the Long-Range Ising Model

TL;DR

This paper investigates whether the long-range Ising model at criticality exhibits conformal invariance, providing evidence through perturbative calculations and a proof based on a higher-dimensional defect theory.

Contribution

It offers the first proof of conformal invariance in the long-range Ising model to all orders in epsilon expansion, despite the absence of a stress tensor.

Findings

Correlation functions are consistent with conformal invariance.

Conformal invariance is proven to all orders in epsilon expansion.

Provides a detailed review of conformal invariance in the short-range Ising model.

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.