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A structural test for the conformal invariance of the critical 3d Ising model. (English) Zbl 1415.81090

Summary: How can a renormalization group fixed point be scale invariant without being conformal? J. Polchinski [“Scale and conformal invariance in quantum field theory”, Nucl. Phys., B 303, No. 2, 226–236 (1988; doi:10.1016/0550-3213(88)90179-4)] 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.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T25 Quantum field theory on lattices
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
81T17 Renormalization group methods applied to problems in quantum field theory

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References:

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