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Journal Club for Condensed Matter Physics

Deep Understanding Achieved on the 3d Ising Model

Published Apr 30, 2014• 1 citation
Publisher

Abstract

Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. Authors: Sheer El-Showk, Miguel F. Paulos, David Poland, Slava Rychkov, David Simmons-Duffin, Alessandro Vichi. arXiv:1403.4545(2014) Recommended with a commentary by Leo P. Kadanoff, University of Chicago and The Perimeter Institute. |View Commentary| DOI: 10.36471/JCCM_April_2014_02 https://doi.org/10.36471/JCCM_April_2014_02

DOI
10.36471/jccm_april_2014_02

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Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. Authors: Sheer El-Showk, Miguel F. Paulos, David Poland, Slava Rychkov, David Simmons-Duffin, Alessandro Vichi. arXiv:1403.4545(2014) Recommended with a commentary by Leo P. Kadanoff, University of Chicago and The Perimeter Institute. |View Commentary| DOI: 10.36471/JCCM_April_2014_02 https://doi.org/10.36471/JCCM_April_2014_02
Herdeiro and Doyon [Phys. Rev. E 94, 043322 (2016)2470-004510.1103/PhysRevE.94.043322] introduced a numerical recipe, dubbed uv sampler, offering precise estimations of the conformal field theory (CFT) data of the planar two-dimensional (2D) critical Ising model. It made use of scale invariance emerging at the critical point in order to sample finite sublattice marginals of the infinite plane Gibbs measure of the model by producing holographic boundary distributions. The main ingredient of the Markov chain Monte Carlo sampler is the invariance under dilation. This paper presents a generalization to higher dimensions with the critical 3D Ising model. This leads to numerical estimations of a subset of the CFT data-scaling weights and structure constants-through fitting of measured correlation functions. The results are shown to agree with the recent most precise estimations from numerical bootstrap methods [Kos, Poland, Simmons-Duffin, and Vichi, J. High Energy Phys. 08 (2016) 03610.1007/JHEP08(2016)036].
We explain how the axioms of Conformal Field Theory are used to make predictions about critical exponents of continuous phase transitions in three dimensions, via a procedure called the conformal bootstrap. The method assumes conformal invariance of correlation functions, and imposes some relations between correlation functions of different orders. Numerical analysis shows that these conditions are incompatible unless the critical exponents take particular values, or more precisely that they must belong to a small island in the parameter space.
These lectures were given at the Weizmann Institute in the spring of 2019. They are intended to familiarize students with the nuts and bolts of the numerical bootstrap as efficiently as possible. After a brief review of the basics of conformal field theory in d>2 spacetime dimensions, we discuss how to compute conformal blocks, formulate the crossing equations as a semi-definite programming problem, solve this problem using SDPB on a personal computer, and interpret the results. We include worked examples for all steps, including bounds for 3d CFTs with Z2 or O(N) global symmetries. Each lecture includes a problem set, which culminate in a precise computation of the 3d Ising model critical exponents using the mixed correlator Z2 bootstrap. A Mathematica file is included that transforms crossing equations into the proper input form for SDPB.
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.