A structural test for the conformal invariance of the critical 3d Ising model
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Springer
Received: February 6, Revised: April 5, Accepted: April 11, Published: April 17,
2019 2019 2019 2019
Sim˜ ao Meneses,a Jo˜ ao Penedones,b Slava Rychkov,c,d,e J.M. Viana Parente Lopesa and Pierre Yvernaye a
Centro de F´ısica das Universidades do Minho e Porto, Departamento de Engenharia F´ısica, Faculdade de Engenharia and Departamento de F´ısica e Astronomia, Faculdade de Ciˆencias, Universidade do Porto, 4169-007 Porto, Portugal b ´ Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Rte de la Sorge, BSP 728, CH-1015 Lausanne, Switzerland c ´ Institut des Hautes Etudes Scientifiques, 35 rte de Chartres, Bures-sur-Yvette, France d D´epartement de Physique, Ecole Normale Sup´erieure, 24 rue Lhomond, Paris, France e Theoretical Physics Department, CERN, route de Meyrin, Geneva, Switzerland
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 ∆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. Keywords: Conformal and W Symmetry, Conformal Field Theory, Lattice Quantum Field Theory, Renormalization Group ArXiv ePrint: 1802.02319 Dedicated to the memory of Joe Polchinski (1954–2018).
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP04(2019)115
JHEP04(2019)115
A structural test for the conformal invariance of the critical 3d Ising model
Contents 1
2 Lattice setup 2.1 Boundary conditions 2.2 Lattice operator 2.3 Matching of the lattice operator with critical point operators 2.4 Choice of Monte Carlo algorithm
3 3 4 4 7
3 Results
7
4 Discussion and conclusions
10
A Theoretical expectations for the dimension of V A.1 Four dimensions A.1.1 Evanescent operators A.2 Two dimensions
12 12 14 15
lat is not a total lattice derivative B Why Oµ
16
C Comments on operator matching C.1 Matching in the lattice spin model C.2 Matching in the lattice field theory
17 18 20
D Possible boundary conditions D.1 Gluing b.c. D.2 Changing the strength of boundary interactions
22 23 23
E Heuristic optimization of boundary conditions
24
1
Introduction
It is believed that the critical point of the 3d ferromagnetic Ising model is conformally invariant. One strong piece of evidence is the excellent agreement between the critical exponents extracted from experiments and Monte Carlo simulations and from the conformal bootstrap [1–6]. Conformal invariance has been also checked directly on the lattice, by verifying functional constraint
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