Thomas Wyse Jackson Supervised by Dmytro Volin

The Conformal Bootstrap in 3 Dimensions Trinity College Dublin Coláiste na Tríonóide, Baile Átha Cliath Thomas Wyse Jackson The University of Dublin Supervised by Dmytro Volin Abstract Crossing Symmetry Conformal field theory allows the study of scale free systems, and one of the largest We can extend correlation functions to three and four points. This is done by taking the unsolved problems of CFT is the 3 dimensional Ising model at the critical temperature. correlators of two of the operators, reducing it to one operator, and repeating this process. The ‘Conformal Bootstrap’ is a new method developed which can make strong This means is that there can be an ambiguity in the order one takes these operations. statements about the low dimension, low spin fields which contain the most important 풪 푥 풪 푥 풪 푥 풪 푥 = 풪 푥 풪 푥 풪 푥 풪 푥 = 풪 푥 풪 푥 풪 푥 풪 푥 contributions to the theory. We review the constraints of conformal field theory and 1 2 3 4 1 2 3 4 1 3 2 4 the method of the conformal bootstrap to gain an insight into solving these type of This is called crossing symmetry [1]. This symmetry creates a constraint for solving four systems. point identical scalar correlators. We find the scaling dimensions of the primary fields of the three dimensional Ising model at critical temperature to be Δ휎 ∈ [0.5170, 0.5249], Δ휖 ∈ [1.409,1.455]. We discuss other results which present themselves from the simulations, and then discuss questions which arise and where this research could be carried on. Critical Phase Phenomena If you heat a ferromagnet, as you get close to the critical temperature, Tc • Magnetisation drops • Isothermal susceptibility → ∞ Figure 3. A demonstration of crossing symmetry in the four point correlator These are examples of critical phase phenomena, and second order phase transitions. This symmetry can be reshaped into equations which we call the ‘conformal blocks’. These occur in scale free systems. In a scale free quantum field theory, the ‘scaling These conformal blocks describe the spectrum of primary fields within the theory. dimension’ Δ refers to how an operator or field reacts to a dilation 푥 → 휆푥. These scaling dimensions uniquely characterise the system. The Conformal Bootstrap The conformal bootstrap was suggested in Polyakov’s 1970 paper [2]. It was only in 2008 however, that a way of properly computing it was devised [3]. We only 퐿 → 2퐿 consider low dimension and low spin states which make the most important contributions. We attempt to devise a functional 훼 to act on the conformal blocks to satisfy [4] 푖 훼푖 푣 푖 = 0, ∀푖≠0훼푖 ≥ 0, 훼0 = 1 퐴 → 4퐴 = 22퐴 푉 → 8푉 = 23 Where 푣 푖 is the truncation of the conformal blocks. ΔA = 2 ΔV = 3 If we can find a possible functional 훼, then that conformal block does not obey our unitary bounds [1], and that spectrum of fields is not possible. Figure 1. An example of scaling dimensions for area and volumes. Due to the property of ‘universality’, different systems can have the same scaling Results dimensions for low energies. • 3 dimensional Ising model at critical temperature Graphing Δ휎 against Δ휖 in figure 4, we recreate • Water at the liquid-vapour critical point the expected behaviour from [5]. The ‘kink’ gives the scaling dimensions of the Ising model. The kink is not easy to measure, so we use Conformal Field Theory central charge minimisation [6] - figure 5, to Conformal Field theory is a scale free theory. find the dimensions more precisely. We found Angles are preserved under transformations. 1. 2. Δ휎 ∈ [0.5170, 0.5249], Δ휖 ∈ 1.409,1.455 . We have four possible transformations. The most recent measurements were found to • Translations be Δ휎 = 0.518154 15 , Δ휖 = 1.41267(13) [7] 1. Translation • Dilations 3. 4. Figure 4. The upper value for Δ휖 for a given Δ휎 2. Dilation • Rotations 3. Rotation 4. Special Conformal • Special Conformal Transformations (SCT). Transformation These transformations have generators which can act upon the states and fields of Figure 2. The four transformations in Conformal Field Theory our theory. Primary and Descendant States The harmonic oscillator in quantum mechanics has the ground, and excited states. In Conformal Field theory, similar to the ground state, we have primary states (풪). These primary states satisfy Figure 5. Charge minimisation to find the Ising dimensions Figure 6. The OPE coefficients for the second primary field When we numerically calculate the four point correlator of the 퐾 풪 = 0 K = −i 2x 푥휈휕 − 푥2휕 휇 휇 휇 ν 휇 lowest dimension primary field ⟨휎휎휎휎⟩ , we find that the 퐾휇 is the generator of the special conformal transformation. contribution of the second lowest dimension field 휖, becomes less We can define the dimension of the state by acting on it with the dilation generator 퐷. important – figure 6. 휇 The kink in this graph occurs close to the Ising model dimensions. D 풪 = Δ 풪 퐷 = −i푥 휕휇 Figures 7. feature a few extra notes. 7i shows that the central charge When the momentum generator 푃 acts on the primary states, we get descendant states. 휇 reaches a maximum, then decreases, suggesting further minima. These descendant states have a dimension of 7ii demonstrates a singularity occurring at Δ휎 = 1. This is discussed Δ = Δ + 푁 Figure 7i & 7ii. Further 휎 discussion of results in the report. Δ휎 is the dimension of the primary state N is the number of momentum generators acting on the primary state. Conclusion • We were able to recreate the knee like behaviour found in [5]. Correlation Functions • The primary fields 3D Ising model at critical temperature were found to have scaling dimension Δ휎 ∈ [0.5170, 0.5249], Δ휖 ∈ 1.409,1.455 using central charge minimization. Multiplication is complicated in QFT, as we can’t deal with infinite precision. • The coefficient of the second primary field within the four point scalar correlator Without proper treatment, you can get divergences and incorrect answers. becomes less important as the primary field becomes higher dimensioned. We instead use operator products or correlation functions. • There are possibly more minima of central charge corresponding to further systems. 퐴 푥 퐵(푦) = 퐶푖 푥 − 푦 퐷푖 푥 − 푦 • There are possible issues with the software or theory at Δ휎 = 1. 푖 Where 퐶푖 푥 − 푦 are operator product expansion (OPE) coefficients of each of the new References [1] – David Simmons-Duffin. ”TASI Lectures on the Conformal Bootstrap”. [5] – Sheer El-Showk, Miguel F. Paulos, David Poland, Slava Rychkov, David operators 퐷푖 푥 − 푦 . These new operators are other primary fields in our theory. arXiv:1602.07982 [hep-th] Simmons-Duffin, Alessandro Vichi ”Solving the 3D Ising Model with the [2] – Alexander Markovich Polyakov ”Nonhamiltonian approach to conformal Conformal Bootstrap”. Phys. Rev. D 86, 025022 (2012). arXiv:1203.6064 [hep-th] We choose a basis such that we get a diagonal matrix, which greatly simplifies our quantum field theory”. Zh. Eksp. Teor. Fiz. 66, 2342 (1974). [6] – Sheer El-Showk, Miguel F. Paulos, David Poland, Slava Rychkov, David problems later on. [3] – Riccardo Rattazzi, Vyacheslav Slava Rychkov, Erik Tonni, and Alessandro Simmons-Duffin, Alessandro Vichi. ”Solving the 3D Ising Model with the Vichi. ”Bounding scalar operator dimensions in 4D CFT”. JHEP. 12, 031 (2008). Conformal Bootstrap II. c-Minimization and Precise Critical Exponents”. J. Stat. 훿푖푗 arXiv:0807.0004 [hep-th] Phys. 157, 869-914 (2014) 풪푖 푥 풪푗(푦) = [4] – Miguel F. Paulos. ”JuliBootS: a hands-on guide to the conformal bootstrap” [7] – Filip Kos, David Poland, David Simmons-Duffin, Alessandro Vichi ”Precision 푥 − 푦 arXiv:1412.4127 [hep-th] Islands in the Ising and O(N) Models”. arXiv:1603.04436 [hep-th] Final Year Maths Projects 2016 April 7th [email protected] http://www.maths.tcd.ie/~wysejact/.

Load more