# Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

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Referee this paper: arXiv:1403.4545 by S. El-Showk, M.F. Paulos, D. Poland, (show more)

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This 2014 paper by S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi  uses ''the conformal bootstrap to perform a precision study of the operator spectrum of the critical 3d Ising model. We conjecture that the 3d Ising spectrum minimizes the central charge c in the space of unitary solutions to crossing symmetry. Because extremal solutions to crossing symmetry are uniquely determined, we are able to precisely reconstruct the first several $Z_2$-even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension $\Delta_\sigma=0.518154(15)$, and find strong numerical evidence that operators decouple from the spectrum as one approaches the 3d Ising point. We compare this behavior to the analogous situation in 2d, where the disappearance of operators can be understood in terms of degenerate Virasoro representations.''

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paper authored Mar 18, 2014 to cond-mat
edited Aug 31, 2014

I found this very positive commentary by L. Kadanoff.

I found this talk given by David Simmons-Duffin (one of the authors) at KITP on the subject.

Perhaps we can ask Kadanoff to import his text as a review?

Here is a quora answer by David Simons-Duffin on how to read the two papers.

## 1 Review

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The 3-dimensional Ising model is a highly important model field theory because many realistic thermodynamic systems - among them all real fluids - are believed to belong to the universality class of this model. (See, e.g., the discussion and references in my paper here.)

The paper under review is an excellent paper, using the relations between conformal field theory and field theories at critical points and newly developed theoretical tools (a variant of the bootstrap program) to calculate the critical exponents of the 3-dimensional Ising model to an accuracy significantly exceeding the previously best calculations (which were obtained by Monte Carlo methods). This shows that the theoretical methods of operator-based quantum field theory are now developed to a very high level.

The techniques are also applied to the 2-dimensional Ising model, which is exactly solvable, with similarly good numerical results.

I already reviewed  a companion paper which contains the theoretical results underlying the present paper. I will complete this review in due time by describing the numerical methods, why they work, and what sort of results are obtained. (But because of traveling I must pause for more than week; so be patient.)

reviewed Sep 16, 2014 by (14,009 points)
edited Sep 20, 2014
looking forward to your review. This should really motivate me to start reading some OPE and CFT stuff.