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arXiv:1804.08374

Exact critical exponents for vector operators in the 3d Ising model and conformal invariance

Gonzalo De Polsi, Matthieu Tissier, Nicolás Wschebor

Published Apr 23, 2018• 5 citations
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Abstract

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 $-1$. 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 $d=2$ and $d=4$. Moreover, we prove that the operator previously considered in Monte-Carlo simulations has also scaling dimension exactly $3$ in any dimension.

DOI
10.48550/arxiv.1804.08374

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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.
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.
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.
Thanks to the impressive progress of conformal bootstrap methods we have now very precise estimates of both scaling dimensions and operator product expansion coefficients for several 3D universality classes. We show how to use this information to obtain similarly precise estimates for off-critical correlators using conformal perturbation. We discuss in particular the $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\epsilon}(r)\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\sigma}(r)\ensuremath{\epsilon}(0)⟩$ two-point functions in the high and low temperature regimes of the 3D Ising model and evaluate the leading and next to leading terms in the $s={\mathrm{tr}}^{{\mathrm{\ensuremath{\Delta}}}_{t}}$ expansion, where ttt is the reduced temperature. Our results for $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$ agree both with Monte Carlo simulations and with a set of experimental estimates of the critical scattering function.
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 ) .