論文の公開元へ

書き出し

Refer/BibIX

RIS

BibTeX

TSV

An Automated Generation of Bootstrap Equations for Numerical Study of Critical Phenomena

呉, 孟超東京大学 DOI:10.15083/0002004682

2022.06.22

概要

as δS = ∫ ddx(hσ + tϵ). From the dimensional analysis, the scalings for t, h are ∆t = d − ∆є, ∆h = d − ∆σ and the critical exponents are related with the scaling dimensions of σ, ϵ by
β = ∆σ/d − ∆є, (2)

ν = 1/ d − ∆є. (3)

A conformal field theory (CFT) is a quantum field theory (QFT) with conformal symmetry, which has Euclidean or Poincaré group as a subgroup and is enhanced by scale invariance. The scale invariance arises, e.g., in the critical phenomena of lattice theory like the O(N ) vector model or the Ising model on the d-dimensional lattice. Rescaling of a field theory generates a flow of Wilson’s renormalization group (RG), and the critical points, which are the fixed points of the flow, are scale invariant. A conformal transformation x → xr is an x-dependent local rescaling ω(x) > 0 with a rotation Λµ (x) ∈ O(d)
∂xrµ µ/∂xν = ω(x)Λ ν(x), (1)
and the conformal symmetry is a natural generalization of the Euclidean symmetry, which is a fundamental spacetime symmetry of a usual QFT. In general, a scale invariant theory is also conformal invariant.

It is well-known that the RG fixed point of φ4 theory in 3 dimensions, the Ising model in 3 di- mensions at the critical temperature, and the behavior of water at the critical point are described by the same CFT. This property at the IR limit is called the critical universality, and it is conjectured that any theory with the same symmetry reaches the same CFT at the continuum limit regardless of its microscopic details. The Ising model has Z2 symmetry which flips all spins. Other impor- tant examples are the O(N ) vector models, which have O(N ) symmetry which rotates all spins. The O(2) model describes the superfluid transition in 4He and measurements of specific heat was done in the space shuttle STS-52.

The scaling dimensions ∆ of operators in a CFT are related with the critical exponents. Take a lattice theory with spin field φ around the critical temperature T ≈ Tc. The critical exponents β, ν are defined by the power law of the following quantities:
• the magnetization: m ∼ (−t)β,
• the correlation length: ξ ∼ |t|−ν,
where t = (T − Tc)/Tc is the reduced temperature, h is the magnetic field coupling with φ. Let the deformation of the Hamiltonian from the critical point be described by primary operators σ, ϵ

The observable quantities in a CFT are correlation functions ⟨O1(x1) · · · On(xn)⟩; the S ma- trix is not well-defined in a CFT due to the Coleman-Mandula’s theorem. We later show that all correlation functions are determined by the CFT data { ∆i, λijk } in a unitary CFT, and a four- point function ⟨φ1φ2φ3φ4⟩ in a CFT can be constructed from three-point functions, but in more than one way, depending on how to group the four operators for the operator product expansions (OPEs): (φ1φ2)(φ3φ4), (φ1φ3)(φ2φ4) or (φ1φ4)(φ2φ3) shown in Figure 1. The bootstrap equation expresses the equality of the four-point function computed in these different decompositions, and is one of the fundamental consistency conditions of a CFT. The conformal bootstrap program aims to solve the CFT data non-perturbatively from the bootstrap equation with the knowledge about the global symmetry of the theory and without a microscopic description such as a Lagrangian.

Figure 1: OPE in a four-point function

The bootstrap equation has been known for almost a third of a century; for other early papers, we refer the reader to the footnote 5 of [1]. It is particularly powerful in d = 2 where the conformal group is infinite dimensional and generated by the Virasoro algebra, and was already successfully applied for the study of 2d CFT in 1984. Its application to CFTs in dimension higher than two had to wait until 2008, where the seminal paper [2] showed that a clever rewriting into a form wherelinear programming was applicable allowed us to extract detailed numerical information from the bootstrap equation. The technique was rap idly developed by many groups and has been applied to many systems. By now, we have many good introductory review articles on this approach, which make the entry to this fascinating and rapidly growing subject easier.

Previously, the study of a CFT has been done by the Monte Carlo (MC) simulation on lattice theory, the direct measurements of physical system at critical point and perturbative loop diagram computations. The MC simulations cannot handle systems with infinite size, and appropriate corrections are needed to obtain results at infinite size limit from finite systems. The direct mea- surements was done in the space shuttle STS-52, and the results about the specific heat of liquid helium near the lambda point tell us the scaling dimension ∆є of the first O(2) invariant primary scalar ϵ in the three dimensional O(2) vector model:
∆є = 1.50946(22). (4)

Two primary examples of perturbation methods are the large-N expansion and the ϵ-expansion. Summarizing these results on the three dimensional Ising model, we have
∆σ = 0.5182(3), ∆є = 1.413(1). (5)

The conformal bootstrap applied for the same CFT shows that the relevant scaling dimensions
(∆σ, ∆є) must reside in a small island whose bounding box is,
∆σ = 0.5181489(10), ∆є = 1.412625(10). (6)

The assumption used is the unitarity of the theory, Z2 invariance and that σ, ϵ are the only relevant primary scalars, i.e., other primary scalars have ∆ ≥ d = 3. The tininess of the island (6) explains the critical universality; any CFT with the same symmetry and same number of relevant operators must have quite close scaling dimensions to those of the Ising model. The universality could be proven by showing that the island shrinks to the Ising point.

The developments in the conformal bootstrap have been helped by various computing tools. For example, we have SDPB, an efficient solver of semidefinite programming (SDP) designed for the conformal bootstrap, and SDP generators written in Python, namely PyCFTBoot and cboot, and in Julia, JuliBootS. There is also a Mathematica package to generate 4d bootstrap equations of arbitrary spin.

To study some CFT with bootstrap, the first step is to enumerate the physical properties of the system such as the global symmetry or the number of relevant primaries and then proceed along the path in Figure 2. The second step to write bootstrap equations has been ad hoc and done only by hand or with some help of Mathematica, and is automated by our autoboot. We have some libraries in some programming languages for the third step to convert to a computable optimization problem, and our qboot aims to handle more generic assumption on the spectrum. SDP solvers such as SDPA or SDPB complete the last step and from their results, we can get in- formation on the CFT. Now we can use free software in step 2, 3 and 4, and since our autoboot, qboot are designed to work smoothly with SDPB, a researcher can concentrate on the first step.

This thesis is based on a published paper [3] collaborated with Yuji Tachikawa, and a single- authored paper [4] to appear. The code is freely available at https://github.com/selpoG/ autoboot/ and https://github.com/selpoG/qboot/.

論文の公開元へ

参考文献

[1] M. Go and Y. Tachikawa, autoboot: A generator of bootstrap equations with global symmetry, JHEP 06 (2019) 084, arXiv:1903.10522 [hep-th]. https://github.com/selpoG/autoboot.

[2] M. Go, “qboot: A new generator of semidefinite programming from conformal bootstrap equations.” to appear; https://github.com/selpoG/qboot.

[3] K. G. Wilson and J. Kogut, The renormalization group and the ϵ expansion, Physics Reports 12 (1974) 75–199.

[4] K. G. Wilson, The renormalization group and critical phenomena, Rev. Mod. Phys. 55 (Jul, 1983) 583–600.

[5] A. M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381–383. [Pisma Zh. Eksp. Teor. Fiz.12,538(1970)].

[6] J. Polchinski, Scale and Conformal Invariance in Quantum Field Theory, Nucl. Phys. B303 (1988) 226–236.

[7] Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rept. 569 (2015) 1–93, arXiv:1302.0884 [hep-th].

[8] A. Dymarsky, Z. Komargodski, A. Schwimmer, and S. Theisen, On Scale and Conformal Invariance in Four Dimensions, JHEP 10 (2015) 171, arXiv:1309.2921 [hep-th].

[9] A. Pelissetto and E. Vicari, Critical phenomena and renormalization group theory, Phys. Rept. 368 (2002) 549–727, arXiv:cond-mat/0012164.

[10] F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, Bootstrapping the O(N ) Archipelago, JHEP 11 (2015) 106, arXiv:1504.07997 [hep-th].

[11] S. R. Coleman and J. Mandula, All Possible Symmetries of the S Matrix, Phys. Rev. 159 (1967) 1251–1256.

[12] R. Haag, J. T. Lopuszanski, and M. Sohnius, All Possible Generators of Supersymmetries of the s Matrix, Nucl. Phys. B88 (1975) 257. [,257(1974)].

[13] V. K. Dobrev, V. B. Petkova, S. G. Petrova, and I. T. Todorov, Dynamical Derivation of Vacuum Operator Product Expansion in Euclidean Conformal Quantum Field Theory, Phys. Rev. D13 (1976) 887.

[14] V. K. Dobrev, G. Mack, V. B. Petkova, S. G. Petrova, and I. T. Todorov, Harmonic Analysis on the N-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory, Lect. Notes Phys. 63 (1977) 1–280.

[15] D. Poland, S. Rychkov, and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 15002, arXiv:1805.04405 [hep-th].

[16] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B241 (1984) 333–380. [,605(1984)].

[17] R. Rattazzi, V. S. Rychkov, E. Tonni, and A. Vichi, Bounding Scalar Operator Dimensions in 4D CFT, JHEP 12 (2008) 031, arXiv:0807.0004 [hep-th].

[18] V. S. Rychkov and A. Vichi, Universal Constraints on Conformal Operator Dimensions, Phys. Rev. D80 (2009) 045006, arXiv:0905.2211 [hep-th].

[19] F. Caracciolo and V. S. Rychkov, Rigorous Limits on the Interaction Strength in Quantum Field Theory, Phys. Rev. D81 (2010) 085037, arXiv:0912.2726 [hep-th].

[20] D. Poland and D. Simmons-Duffin, Bounds on 4D Conformal and Superconformal Field Theories, JHEP 05 (2011) 017, arXiv:1009.2087 [hep-th].

[21] R. Rattazzi, S. Rychkov, and A. Vichi, Central Charge Bounds in 4D Conformal Field Theory, Phys. Rev. D83 (2011) 046011, arXiv:1009.2725 [hep-th].

[22] R. Rattazzi, S. Rychkov, and A. Vichi, Bounds in 4D Conformal Field Theories with Global Symmetry, J. Phys. A44 (2011) 035402, arXiv:1009.5985 [hep-th].

[23] A. Vichi, Improved Bounds for CFT’s with Global Symmetries, JHEP 01 (2012) 162, arXiv:1106.4037 [hep-th].

[24] D. Poland, D. Simmons-Duffin, and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 05 (2012) 110, arXiv:1109.5176 [hep-th].

[25] S. Rychkov, Conformal Bootstrap in Three Dimensions?, arXiv:1111.2115 [hep-th].

[26] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D86 (2012) 025022, arXiv:1203.6064 [hep-th].

[27] P. Liendo, L. Rastelli, and B. C. van Rees, The Bootstrap Program for Boundary CFTD, JHEP 07 (2013) 113, arXiv:1210.4258 [hep-th].

[28] S. El-Showk and M. F. Paulos, Bootstrapping Conformal Field Theories with the Extremal Functional Method, Phys. Rev. Lett. 111 (2013) 241601, arXiv:1211.2810 [hep-th].

[29] C. Beem, L. Rastelli, and B. C. van Rees, The = 4 Superconformal Bootstrap, Phys. Rev. Lett. 111 (2013) 071601, arXiv:1304.1803 [hep-th].

[30] F. Kos, D. Poland, and D. Simmons-Duffin, Bootstrapping the O(N ) vector models, JHEP 06 (2014) 091, arXiv:1307.6856 [hep-th].

[31] L. F. Alday and A. Bissi, The Superconformal Bootstrap for Structure Constants, JHEP 09 (2014) 144, arXiv:1310.3757 [hep-th].

[32] D. Gaiotto, D. Mazac, and M. F. Paulos, Bootstrapping the 3D Ising Twist Defect, JHEP 03 (2014) 100, arXiv:1310.5078 [hep-th].

[33] M. Berkooz, R. Yacoby, and A. Zait, Bounds on = 1 superconformal theories with global symmetries, JHEP 08 (2014) 008, arXiv:1402.6068 [hep-th]. [Erratum: JHEP01,132(2015)].

[34] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap II. C-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869, arXiv:1403.4545 [hep-th].

[35] Y. Nakayama and T. Ohtsuki, Approaching the conformal window of O(n) O(m) symmetric Landau-Ginzburg models using the conformal bootstrap, Phys. Rev. D89 (2014) 126009, arXiv:1404.0489 [hep-th].

[36] Y. Nakayama and T. Ohtsuki, Five dimensional O(N )-symmetric CFTs from conformal bootstrap, Phys. Lett. B734 (2014) 193–197, arXiv:1404.5201 [hep-th].

[37] L. F. Alday and A. Bissi, Generalized bootstrap equations for = 4 SCFT, JHEP 02 (2015) 101, arXiv:1404.5864 [hep-th].

[38] S. M. Chester, J. Lee, S. S. Pufu, and R. Yacoby, The = 8 superconformal bootstrap in three dimensions, JHEP 09 (2014) 143, arXiv:1406.4814 [hep-th].

[39] F. Kos, D. Poland, and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109, arXiv:1406.4858 [hep-th].

[40] F. Caracciolo, A. Castedo Echeverri, B. von Harling, and M. Serone, Bounds on OPE Coefficients in 4D Conformal Field Theories, JHEP 10 (2014) 020, arXiv:1406.7845 [hep-th].

[41] Y. Nakayama and T. Ohtsuki, Bootstrapping Phase Transitions in QCD and Frustrated Spin Systems, Phys. Rev. D91 (2015) 021901, arXiv:1407.6195 [hep-th].

[42] J.-B. Bae and S.-J. Rey, Conformal Bootstrap Approach to O(N ) Fixed Points in Five Dimensions, arXiv:1412.6549 [hep-th].

[43] C. Beem, M. Lemos, P. Liendo, L. Rastelli, and B. C. van Rees, The = 2 superconformal bootstrap, JHEP 03 (2016) 183, arXiv:1412.7541 [hep-th].

[44] S. M. Chester, S. S. Pufu, and R. Yacoby, Bootstrapping O(N ) vector models in 4 < d < 6 , Phys. Rev. D91 (2015) 086014, arXiv:1412.7746 [hep-th].

[45] D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 06 (2015) 174, arXiv:1502.02033 [hep-th]. https://github.com/davidsd/sdpb.

[46] N. Bobev, S. El-Showk, D. Mazac, and M. F. Paulos, Bootstrapping SCFTs with Four Supercharges, JHEP 08 (2015) 142, arXiv:1503.02081 [hep-th].

[47] S. M. Chester, S. Giombi, L. V. Iliesiu, I. R. Klebanov, S. S. Pufu, and R. Yacoby, Accidental Symmetries and the Conformal Bootstrap, JHEP 01 (2016) 110, arXiv:1507.04424 [hep-th].

[48] C. Beem, M. Lemos, L. Rastelli, and B. C. van Rees, The (2, 0) Superconformal Bootstrap, Phys. Rev. D93 (2016) 025016, arXiv:1507.05637 [hep-th].

[49] L. Iliesiu, F. Kos, D. Poland, S. S. Pufu, D. Simmons-Duffin, and R. Yacoby, Bootstrapping 3D Fermions, JHEP 03 (2016) 120, arXiv:1508.00012 [hep-th].

[50] D. Poland and A. Stergiou, Exploring the Minimal 4D = 1 SCFT, JHEP 12 (2015) 121, arXiv:1509.06368 [hep-th].

[51] M. Lemos and P. Liendo, Bootstrapping = 2 chiral correlators, JHEP 01 (2016) 025, arXiv:1510.03866 [hep-th].

[52] Y.-H. Lin, S.-H. Shao, D. Simmons-Duffin, Y. Wang, and X. Yin, = 4 superconformal bootstrap of the K3 CFT, JHEP 05 (2017) 126, arXiv:1511.04065 [hep-th].

[53] S. M. Chester, L. V. Iliesiu, S. S. Pufu, and R. Yacoby, Bootstrapping O(N ) Vector Models with Four Supercharges in 3 ≤ d ≤ 4, JHEP 05 (2016) 103, arXiv:1511.07552 [hep-th].

[54] S. M. Chester and S. S. Pufu, Towards bootstrapping QED3, JHEP 08 (2016) 019, arXiv:1601.03476 [hep-th].

[55] Y. Nakayama, Bootstrapping Critical Ising Model on Three-Dimensional Real Projective Space, Phys. Rev. Lett. 116 (2016) 141602, arXiv:1601.06851 [hep-th].

[56] C. Behan, PyCFTBoot: A flexible interface for the conformal bootstrap, Commun. Comput. Phys. 22 (2017) 1–38, arXiv:1602.02810 [hep-th].

[57] Y. Nakayama and T. Ohtsuki, Conformal Bootstrap Dashing Hopes of Emergent Symmetry, Phys. Rev. Lett. 117 (2016) 131601, arXiv:1602.07295 [cond-mat.str-el]. https://github.com/tohtsky/cboot.

[58] H. Iha, H. Makino, and H. Suzuki, Upper Bound on the Mass Anomalous Dimension in Many-Flavor Gauge Theories: a Conformal Bootstrap Approach, PTEP 2016 (2016) 053B03, arXiv:1603.01995 [hep-th].

[59] F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, Precision Islands in the Ising and O(N ) Models, JHEP 08 (2016) 036, arXiv:1603.04436 [hep-th].

[60] Y. Nakayama, Bootstrap Bound for Conformal Multi-Flavor QCD on Lattice, JHEP 07 (2016) 038, arXiv:1605.04052 [hep-th].

[61] A. Castedo Echeverri, B. von Harling, and M. Serone, The Effective Bootstrap, JHEP 09 (2016) 097, arXiv:1606.02771 [hep-th].

[62] Z. Li and N. Su, Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N ) Models, JHEP 04 (2017) 098, arXiv:1607.07077 [hep-th].

[63] Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, (2, 2) Superconformal Bootstrap in Two Dimensions, JHEP 05 (2017) 112, arXiv:1610.05371 [hep-th].

[64] J.-B. Bae, K. Lee, and S. Lee, Bootstrapping Pure Quantum Gravity in AdS3, arXiv:1610.05814 [hep-th].

[65] J.-B. Bae, D. Gang, and J. Lee, 3d = 2 minimal SCFTs from Wrapped M5-branes, JHEP 08 (2017) 118, arXiv:1610.09259 [hep-th].

[66] M. Lemos, P. Liendo, C. Meneghelli, and V. Mitev, Bootstrapping = 3 superconformal theories, JHEP 04 (2017) 032, arXiv:1612.01536 [hep-th].

[67] C. Beem, L. Rastelli, and B. C. van Rees, More = 4 superconformal bootstrap, Phys. Rev. D96 (2017) 046014, arXiv:1612.02363 [hep-th].

[68] D. Li, D. Meltzer, and A. Stergiou, Bootstrapping mixed correlators in 4D = 1 SCFTs, JHEP 07 (2017) 029, arXiv:1702.00404 [hep-th].

[69] M. Cornagliotto, M. Lemos, and V. Schomerus, Long Multiplet Bootstrap, JHEP 10 (2017) 119, arXiv:1702.05101 [hep-th].

[70] Y. Nakayama, Bootstrap Experiments on Higher Dimensional CFTs, Int. J. Mod. Phys. A33 (2018) 1850036, arXiv:1705.02744 [hep-th].

[71] A. Dymarsky, J. Penedones, E. Trevisani, and A. Vichi, Charting the Space of 3D CFTs with a Continuous Global Symmetry, arXiv:1705.04278 [hep-th].

[72] C.-M. Chang and Y.-H. Lin, Carving Out the End of the World Or (Superconformal Bootstrap in Six Dimensions), JHEP 08 (2017) 128, arXiv:1705.05392 [hep-th].

[73] G. F. Cuomo, D. Karateev, and P. Kravchuk, General Bootstrap Equations in 4D CFTs, JHEP 01 (2018) 130, arXiv:1705.05401 [hep-th].

[74] C. A. Keller, G. Mathys, and I. G. Zadeh, Bootstrapping Chiral CFTs at Genus Two, arXiv:1705.05862 [hep-th].

[75] M. Cho, S. Collier, and X. Yin, Genus Two Modular Bootstrap, arXiv:1705.05865 [hep-th].

[76] Z. Li and N. Su, 3D CFT Archipelago from Single Correlator Bootstrap, arXiv:1706.06960 [hep-th].

[77] A. Dymarsky, F. Kos, P. Kravchuk, D. Poland, and D. Simmons-Duffin, The 3D Stress-Tensor Bootstrap, JHEP 02 (2018) 164, arXiv:1708.05718 [hep-th].

[78] J.-B. Bae, S. Lee, and J. Song, Modular Constraints on Conformal Field Theories with Currents, JHEP 12 (2017) 045, arXiv:1708.08815 [hep-th].

[79] E. Dyer, A. L. Fitzpatrick, and Y. Xin, Constraints on Flavored 2D CFT Partition Functions, JHEP 02 (2018) 148, arXiv:1709.01533 [hep-th].

[80] C.-M. Chang, M. Fluder, Y.-H. Lin, and Y. Wang, Spheres, Charges, Instantons, and Bootstrap: a Five-Dimensional Odyssey, JHEP 03 (2018) 123, arXiv:1710.08418 [hep-th].

[81] M. Cornagliotto, M. Lemos, and P. Liendo, Bootstrapping the (A1, A2) Argyres-Douglas theory, JHEP 03 (2018) 033, arXiv:1711.00016 [hep-th].

[82] N. B. Agmon, S. M. Chester, and S. S. Pufu, Solving M-theory with the Conformal Bootstrap, JHEP 06 (2018) 159, arXiv:1711.07343 [hep-th].

[83] J. Rong and N. Su, Scalar CFTs and Their Large N Limits, JHEP 09 (2018) 103, arXiv:1712.00985 [hep-th].

[84] M. Baggio, N. Bobev, S. M. Chester, E. Lauria, and S. S. Pufu, Decoding a Three-Dimensional Conformal Manifold, JHEP 02 (2018) 062, arXiv:1712.02698 [hep-th].

[85] A. Stergiou, Bootstrapping Hypercubic and Hypertetrahedral Theories in Three Dimensions, JHEP 05 (2018) 035, arXiv:1801.07127 [hep-th].

[86] C. Hasegawa and Y. Nakayama, Three ways to solve critical φ4 theory on 4 ϵ dimensional real projective space: perturbation, bootstrap, and Schwinger-Dyson equation, Int. J. Mod. Phys. A33 (2018) 1850049, arXiv:1801.09107 [hep-th].

[87] P. Liendo, C. Meneghelli, and V. Mitev, Bootstrapping the Half-BPS Line Defect, JHEP 10 (2018) 077, arXiv:1806.01862 [hep-th].

[88] J. Rong and N. Su, Bootstrapping minimal = 1 superconformal field theory in three dimensions, arXiv:1807.04434 [hep-th].

[89] A. Atanasov, A. Hillman, and D. Poland, Bootstrapping the Minimal 3D SCFT, JHEP 11 (2018) 140, arXiv:1807.05702 [hep-th].

[90] C. Behan, Bootstrapping the Long-Range Ising Model in Three Dimensions, arXiv:1810.07199 [hep-th].

[91] S. R. Kousvos and A. Stergiou, Bootstrapping Mixed Correlators in Three-Dimensional Cubic Theories, arXiv:1810.10015 [hep-th].

[92] A. Cappelli, L. Maffi, and S. Okuda, Critical Ising Model in Varying Dimension by Conformal Bootstrap, JHEP 01 (2019) 161, arXiv:1811.07751 [hep-th].

[93] C. N. Gowdigere, J. Santara, and Sumedha, Conformal Bootstrap Signatures of the Tricritical Ising Universality Class, arXiv:1811.11442 [hep-th].

[94] Z. Li, Solving QED3 with Conformal Bootstrap, arXiv:1812.09281 [hep-th].

[95] D. Karateev, P. Kravchuk, M. Serone, and A. Vichi, Fermion Conformal Bootstrap in 4D, arXiv:1902.05969 [hep-th].

[96] E. El-Schowk, Solving conformal theory with bootstrap, in Lecture at the 9th Asian Winter School. 2015. http://home.kias.re.kr/MKG/h/AWSSPC2015/?pageNo=1006.

[97] J. D. Qualls, Lectures on Conformal Field Theory, arXiv:1511.04074 [hep-th].

[98] S. Rychkov, EPFL Lectures on Conformal Field Theory in D 3 Dimensions. SpringerBriefs in Physics. Springer, 2016. arXiv:1601.05000 [hep-th].

[99] D. Simmons-Duffin, The Conformal Bootstrap, in Proceedings, TASI 2015: Boulder, CO., USA, June 1-26, 2015, pp. 1–74. 2017. arXiv:1602.07982 [hep-th].

[100] A. Antunes, Numerical Methods in the Conformal Bootstrap, arXiv:1709.01529 [hep-th].

[101] 中山優, “高次元共形場理論への招待 3次元臨界Ising模型を解く”. SGCライブラリ. サイエンス社, October, 2019. https: //www.saiensu.co.jp/search/?isbn=978-4-7819-1460-2&y=2019.

[102] M. Hasenbusch, Finite size scaling study of lattice models in the three-dimensional Ising universality class, Phys. Rev. B 82 (Nov, 2010) 174433.

[103] M. Hasenbusch, Finite Size Scaling Study of Lattice Models in the Three-Dimensional Ising Universality Class, Phys. Rev. B82 (2010) 174433, arXiv:1004.4486 [cond-mat.stat-mech].

[104] J. A. Lipa, J. A. Nissen, D. A. Stricker, D. R. Swanson, and T. C. P. Chui, Specific heat of liquid helium in zero gravity very near the lambda point, Physical Review B 68 (Nov, 2003) 174518.

[105] S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, and A. Vichi, Carving out OPE space and precise O(2) model critical exponents, arXiv:1912.03324 [hep-th].

[106] K. Lang and W. Ruhl, The Critical O(N) sigma model at dimension 2 < d < 4 and order 1/n**2: Operator product expansions and renormalization, Nucl. Phys. B377 (1992) 371–401.

[107] A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N) vector model, Annals Phys. 249 (1996) 180–221, arXiv:hep-th/9410093.

[108] A. C. Petkou, C(T) and C(J) up to next-to-leading order in 1/N in the conformally invariant 0(N) vector model for 2 < d < 4, Phys. Lett. B359 (1995) 101–107, arXiv:hep-th/9506116.

[109] J. A. Gracey, Crossover exponent in O(N) phi**4 theory at O(1 / N**2), Phys. Rev. E66 (2002) 027102, arXiv:cond-mat/0206098.

[110] M. Moshe and J. Zinn-Justin, Quantum field theory in the large N limit: A Review, Phys. Rept. 385 (2003) 69–228, arXiv:hep-th/0306133.

[111] K. G. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240–243.

[112] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, p. 1054. Oxford, England: Clarendon Pr., 2002.

[113] P. Dey and A. Kaviraj, Towards a Bootstrap approach to higher orders of epsilon expansion, JHEP 02 (2018) 153, arXiv:1711.01173 [hep-th].

[114] W. Landry and D. Simmons-Duffin, Scaling the semidefinite program solver SDPB, arXiv:1909.09745 [hep-th].

[115] M. F. Paulos, JuliBootS: a Hands-On Guide to the Conformal Bootstrap, arXiv:1412.4127 [hep-th].

[116] M. Yamashita, K. Fujisawa, and M. Kojima, Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0), Optimization Methods and Software 18 (2003) 491–505, https://doi.org/10.1080/1055678031000118482.

[117] M. Yamashita, K. Fujisawa, K. Nakata, M. Nakata, M. Fukuda, K. Kobayashi, and K. Goto, A high-performance software package for semidefinite programs: SDPA 7,.

[118] M. Nakata, A numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD., in 2010 IEEE International Symposium on Computer-Aided Control System Design, pp. 29–34. Sep, 2010.

[119] B. Eick, H. U. Besche, and E. O’Brien, SmallGrp – The GAP Small Groups Library–, https://gap-packages.github.io/smallgrp/.

[120] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.0, https://www.gap-system.org.

[121] The Sage Developers, Sagemath, the Sage Mathematics Software System, http://www.sagemath.org.

[122] J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2007.

[123] S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 783–851, arXiv:hep-th/9712074.

[124] A. Bourget and J. Troost, The Conformal Characters, JHEP 04 (2018) 055, arXiv:1712.05415 [hep-th].

[125] J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher Spin Symmetry, J. Phys. A46 (2013) 214011, arXiv:1112.1016 [hep-th].

[126] V. Alba and K. Diab, Constraining conformal field theories with a higher spin symmetry in d=4, arXiv:1307.8092 [hep-th].

[127] V. Alba and K. Diab, Constraining conformal field theories with a higher spin symmetry in d > 3 dimensions, JHEP 03 (2016) 044, arXiv:1510.02535 [hep-th].

[128] D. Pappadopulo, S. Rychkov, J. Espin, and R. Rattazzi, OPE Convergence in Conformal Field Theory, Phys. Rev. D86 (2012) 105043, arXiv:1208.6449 [hep-th].

[129] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071, arXiv:1107.3554 [hep-th].

[130] E. Elkhidir, D. Karateev, and M. Serone, General Three-Point Functions in 4D CFT, JHEP 01 (2015) 133, arXiv:1412.1796 [hep-th].

[131] T. Levy and Y. Oz, Liouville Conformal Field Theories in Higher Dimensions, JHEP 06 (2018) 119, arXiv:1804.02283 [hep-th].

[132] M. S. Costa, J. Penedones, D. Poland, and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154, arXiv:1109.6321 [hep-th].

[133] F. A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B678 (2004) 491–507, arXiv:hep-th/0309180.

[134] F. A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [hep-th].

[135] F. A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B599 (2001) 459–496, arXiv:hep-th/0011040.

[136] A. L. Fitzpatrick, J. Kaplan, D. Poland, and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004, arXiv:1212.3616 [hep-th].

[137] Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140, arXiv:1212.4103 [hep-th].

[138] D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086, arXiv:1612.08471 [hep-th].

[139] S. El-Showk and M. F. Paulos, Extremal bootstrapping: go with the flow, JHEP 03 (2018) 148, arXiv:1605.08087 [hep-th].

[140] M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D87 (2013) 106004, arXiv:1303.1111 [hep-th].

[141] M. Hogervorst, H. Osborn, and S. Rychkov, Diagonal Limit for Conformal Blocks in d Dimensions, JHEP 08 (2013) 014, arXiv:1305.1321 [hep-th].

[142] J. Penedones, E. Trevisani, and M. Yamazaki, Recursion Relations for Conformal Blocks, JHEP 09 (2016) 070, arXiv:1509.00428 [hep-th].

[143] J. D. Dixon, Constructing representations of finite groups, in Groups and computation (New Brunswick, NJ, 1991), vol. 11 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 105–112. Amer. Math. Soc., Providence, RI, 1993.

[144] H. Osborn and A. C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311–362, arXiv:hep-th/9307010.

[145] P. Calabrese and P. Parruccini, Harmonic crossover exponents in O(n) models with the pseudo-epsilon expansion approach, Phys. Rev. B71 (2005) 064416, arXiv:cond-mat/0411027.

[146] M. Hasenbusch, Monte Carlo study of an improved clock model in three dimensions, arXiv e-prints (Oct, 2019) arXiv:1910.05916, arXiv:1910.05916 [cond-mat.stat-mech].

[147] M. Campostrini, M. Hasenbusch, A. Pelissetto, and E. Vicari, The Critical Exponents of the Superfluid Transition in He-4, Phys. Rev. B74 (2006) 144506, arXiv:cond-mat/0605083.

[148] F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602, arXiv:1307.3111 [hep-th].

参考文献をもっと見る

分野

大学

学位論文種類・取得年

言語

An Automated Generation of Bootstrap Equations for Numerical Study of Critical Phenomena

概要

関連論文

Exploration of Classical Integrable Systems Assisted by Neural Networks

Bosonization with background U(1) gauge ﬁeld and its realization in one-dimensional quantum many-body systems

Study of the sign problem based on the generalized Lefschetz thimble method using gradient flows without blow-up

New Methods in Strongly-Coupled Field Theories towards Quantum Gravity

Classification and Representation of Physical States by Neural Networks

参考文献

分野

大学

学位論文種類・取得年

言語

コピーが完了しました

URLをコピーしました

An Automated Generation of Bootstrap Equations for Numerical Study of Critical Phenomena

概要

関連論文

Exploration of Classical Integrable Systems Assisted by Neural Networks

Bosonization with background U(1) gauge ﬁeld and its realization in one-dimensional quantum many-body systems

Study of the sign problem based on the generalized Lefschetz thimble method using gradient flows without blow-up

New Methods in Strongly-Coupled Field Theories towards Quantum Gravity

Classification and Representation of Physical States by Neural Networks

参考文献