A structural test for the conformal invariance of the critical 3d Ising model

Abstract

How can a renormalization group fixed point be scale invariant without being conformal? Polchinski (1988) showed that this may happen if the theory contains a virial current -- a non-conserved vector operator of dimension exactly $(d-1)$, whose divergence expresses the trace of the stress tensor. We point out that this scenario can be probed via lattice Monte Carlo simulations, using the critical 3d Ising model as an example. Our results put a lower bound ${\Delta }_V>5.0$ on the scaling dimension of the lowest virial current candidate $V$, well above 2 expected for the true virial current. This implies that the critical 3d Ising model has no virial current, providing a structural explanation for the conformal invariance of the model.

arXiv commentv2: 27pp; many explanations and some corrections added; Monte Carlo measurement reinterpreted as a lower bound; conclusions unchanged; v3: misprint in Eq. (2.5) corrected; v4: Eqs (1.4,1.5) and appendix B added, version to appear in JHEP; v5: Ref. [4] to 3d SAW conformal invariance added

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• Conformal invariance and composite operators: A strategy for improving the derivative expansion of the nonperturbative renormalization group
• Exploring instantons with spin-lattice systems
• Anomalous Dimensions in Hypercubic Theories
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• Disturbing news about the $d=2+\epsilon$ expansion