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arXiv:1501.01776 · Physical review. E

Scale invariance implies conformal invariance for the three-dimensional Ising model

Bertrand Delamotte, Matthieu Tissier, Nicolás Wschebor

Published Jan 25, 2016• 42 citations
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Abstract

Using the Wilson renormalization group, we show that if no integrated vector operator of scaling dimension -1 exists, then scale invariance implies conformal invariance. By using the Lebowitz inequalities, we prove that this necessary condition is fulfilled in all dimensions for the Ising universality class. This shows, in particular, that scale invariance implies conformal invariance for the three-dimensional Ising model.

DOI
10.1103/physreve.93.012144

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Using the Wilson renormalization group, we show that if no integrated vector operator of scaling dimension -1 exists, then scale invariance implies conformal invariance. By using the Lebowitz inequalities, we prove that this necessary condition is fulfilled in all dimensions for the Ising universality class. This shows, in particular, that scale invariance implies conformal invariance for the three-dimensional Ising model.
It is widely expected that the realization of scale invariance in the critical regime implies conformal invariance for a large class of systems. This is known to be true if there exist no integrated operator which transforms like a vector under rotations and which has scaling dimension −1-1−1. In this article we give exact expressions for the critical exponents of some of these vector operators. In particular, we show that one operator has scaling dimension exactly 3 in any space dimension. This operator turns out be the leading operator at least in d=2d=2d=2 and d=4d=4d=4. Moreover, we prove that the operator previously considered in Monte-Carlo simulations has also scaling dimension exactly 333 in any dimension.
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 Δ 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.
In a recent preprint [ArXiv 1802.02319], Meneses et al. challenge our proof that scale invariance implies conformal invariance for the three-dimensional Ising model [B. Delamotte, M. Tissier and N. Wschebor, Phys. Rev. E 93 (2016), 012144.]. We refute their arguments. We also point out a mistake in their one-loop calculation of the dimension of the vector operator VμV_μVμ​ of lowest dimension which is not a total derivative.
We investigate a critical Ising-like model in the curved geometry ${S}^{2}\ifmmode\times\else\texttimes\fi{}{R}^{1}$ obtained by a conformal mapping of the infinite 3D space R3{R}^{3}R3. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as Xh=0.5178(12){X}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.5178(12)Xh​=0.5178(12) and Xt=1.423(19){X}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.423(19)Xt​=1.423(19), respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.
Recent advances in conformal field theory and critical phenomena have focused on the characterization of boundary or defects in a conformally invariant system. In this Letter, we study the critical behavior of the three-dimensional Ising universality class in the presence of a surface, realizing the ordinary, the special, and the normal universality classes. By combining high-precision Monte Carlo simulations of an improved model, where leading scaling corrections are suppressed, with a finite-size scaling analysis informed by conformal field theory, we determine unbiased, accurate estimates of universal boundary operator product expansion coefficients of experimental relevance. Furthermore, we improve the value of the scaling dimension of the surface field at the special transition by the estimate Δ ̂ σ = 0.3531 ( 3 ) .
Using a continuous cluster Monte Carlo algorithm, we investigate the critical three-dimensional Ising model in its anisotropic limit. From the ratio of the magnetic correlations in the strong- and the weak-coupling directions, we determine the length ratio relating the isotropic Ising model and the anisotropic limit. On this basis, we simulate the critical Ising model on a spherocylinder S2 x R1, i.e., a curved geometry obtained from a conformal mapping of the infinite space R3. From correlation lengths along the spherocylinder, combined with the prediction of conformal invariance, we estimate the magnetic and thermal scaling dimensions as X(h)=0.5182(6) and X(t)=1.419(7), respectively. The behavior of the Binder cumulant is also determined in the limit of an infinitely long spherocylinder. Next, free boundary conditions are imposed on the equators of the spherocylinder, and thus the geometry S1 x S+ x R1 is obtained. The surface magnetic scaling dimension is estimated as X(s)(h)=1.263(5). The consistency of the aforementioned estimations and existing results confirms that the three-dimensional Ising model is conformally invariant. Further, the precision of these results reveals that, as in two dimensions, conformal mappings provide a powerful tool to investigate critical phenomena. With the continuous cluster algorithm, we also perform simulations of systems inside a conventional solid cylinder. The surface magnetic correlation length differs, within the estimated error margin, by a factor pi/2 from that along a half spherocylinder S1 x S+ x R1 with the same radius.