# 臺灣博碩士論文加值系統

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DOI:10.6342/NTU202002223

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 格拉斯曼是一個發展完善的數學結構，他可以用於計算微擾規範場論的散射幅度。在最近，該工具也被發現可以運用在平面易辛模型。本文討論了正交格拉斯曼的細胞結構與平面易辛模型之間的等價關係。我們提出了一種以微觀結構發法，來建立兩者之間的對應關係。由兩者之間的等效性，使我們可以引入兩種新遞歸方法來計算易辛網絡的關聯函數。第一種基於對偶變換，此種變換生成易辛模網絡屬於格拉斯曼中同一的細胞。我們可以用這種變換來解碎形晶格，其中遞歸公式成為有效耦合常數的精確重整化群方程。對於第二個，我們使用黏合方法，其中每次迭代將原始晶格的大小加倍。這導致更有效計算關聯函數，其中複雜度相對於辛模晶格點的數量呈對數比例增長。
 Grassmannian is a well-developed mathematical structure to compute the scattering amplitudes of perturbative gauge theories. Recently, a new connection of this tool to the planar Ising model has been revealed. This thesis discusses the equivalence between planar Ising networks and cells in the positive orthogonal Grassmannian. We propose a microscopic construction based on amalgamation, which establishes the correspondence for any planar Ising network. The equivalence allows us to introduce two recursive methods for computing correlators of Ising networks. The first is based on duality moves, which generate networks belonging to the same cell in the Grassmannian. This leads to fractal lattices, where the recursion formulas become the exact RG equations of the effective couplings. For the second, we use amalgamation, where each iteration doubles the size of the seed lattice. This leads to an efficient way of computing the correlator where the complexity scales logarithmically with respect to the number of spin sites.
 致謝i中文摘要iiiAbstract vContents viiList of Figures ixList of Tables xiii1 Introduction 12 Ising Model 32.1 Introduction to Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Planar Ising Model in the Disk . . . . . . . . . . . . . . . . . . . . . . . 33 Introduction to the Positive Grassmannian and the Positive Orthogonal Grassmannian 73.1 Definition of Positive Grassmannian (Gr≥0) . . . . . . . . . . . . . . . . 73.2 Positive Orthogonal Grassmannian (OG≥0) . . . . . . . . . . . . . . . . 83.3 Cell Structure of Grassmannian . . . . . . . . . . . . . . . . . . . . . . . 94 Correspondence between Planar Ising Model and Cell of Positive Orthogonal Grassmannian 154.1 Embedding Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Example: Free Edge Network and Its Variants . . . . . . . . . . . . . . . 165 Fundamental Move in Planar Ising Model 195.1 From Planar Ising Model to Bipartite Graph . . . . . . . . . . . . . . . . 195.2 Duality move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Why Planar Ising models Lives in OG≥0? 256.1 Amalgamation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2 Microscopic Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Phase Transition and Renormalization Group 317.1 Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.2 Renormalization Group and Scale Transformation . . . . . . . . . . . . . 337.3 Fractal Lattices and Recursion via Duality Transformation . . . . . . . . 357.3.1 Fractal Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.3.2 Recursion via Duality Transformation . . . . . . . . . . . . . . . 367.4 Fractal Lattice and Non-fractal Lattice and Recursion through Amalgamation. . 437.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 Conclusion and Outlook 51A Example of the Correspondence 53A.1 Example 1: Triangle network . . . . . . . . . . . . . . . . . . . . . . . . 53A.2 Example 2: Square network . . . . . . . . . . . . . . . . . . . . . . . . . 55B Derivation for identification two spin sites 57Bibliography 59
 [1] N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov and J. Trnka, “Grassmannian Geometry of Scattering Amplitudes,” arXiv:1212.5605 [hep-th].[2] Y. T. Huang and C. Wen, “ABJM amplitudes and the positive orthogonal Grassmannian,” JHEP 1402, 104 (2014) [arXiv:1309.3252 [hep-th]].[3] N. Arkani-Hamed and J. Trnka, “The Amplituhedron,” JHEP 1410, 030 (2014) [arXiv:1312.2007 [hep-th]].[4] N. Arkani-Hamed, T-z Huang, and Y-t Huang: N. Arkani-Hamed, Y-t Huang and Shu-Heng Shao, To appear.[5] R. K. Pathria and Paul D. Beale, “Statistical Mechanics,”[6] “Ising model and the positive orthogonal Grassmannian”, P. Galashin, and P. Pylyavskyy, arXiv:1807.03282.[7] S. Lee, “Yangian Invariant Scattering Amplitudes in Supersymmetric Chern-Simons Theory,” Phys. Rev. Lett. 105, 151603 (2010) [arXiv:1007.4772 [hep-th]].[8] Higher-point functions can be written as products of two-point functions, and thus the correspondence generalizes easily [6].[9] “The Planar Ising Model and Total Positivity” M. Lis, Journal of Statistical Physics, 166(1): 72-89, 2017 [arXiv:1606.06068 [math]][10] For planar networks, the four-point function can be recast as a sum of products of two-point functions, see, “Correlation-function identities for general planar Ising systems” J. Groeneveld and R.J. Boel and P.W. Kasteleyn, Physica A: Statistical Mechanics and its Applications, 93 (1) 138-154, 1978.[11] Y. t. Huang, C. Wen, and D. Xie, “The Positive orthogonal Grassmannian and loop amplitudes of ABJM,” J. Phys. A 47, no. 47, 474008 (2014) [arXiv:1402.1479 [hepth]].[12] The bubble reductions are the “decoration transformation” used by Naya, and the triangle move is the “start-triangle transformation” found by Onsager, see: “On the Spontaneous Magnetizations of Honeycomb and Kagom´e Ising Lattices”, S. Naya, Progress of Theoretical Physics, 11 53 (1954), and “Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition”, L. Onsager, Phys. Rev. 65,117 (1944).[13] H. Elvang and Y. t. Huang, arXiv:1308.1697 [hep-th].[14] Y. Gefen, B. B. Mandelbrot and A. Aharony “Critical Phenomena on Fractal Lattices,” Phys. Rev. Lett. 45, 855 (1980)[15] Y. Gefen, A. Aharony, Y. Shapir and B. B. Mandelbrot, “Phase transitions on fractals.II. Sierpinski gaskets,” Journal of Physics A: Mathematical and General, Volume 17, Issue 2, pp. 435-444 (1984)[16] Hidetoshi Nishimon and Gerardo Ortiz “Elements of Phase Transitions and Critical Phenomena,” OXFORD University Press 45, (2011)[17] Michael Aizenman, Hugo Duminil-Copin, Vincent Tassion, and Simone Warzel “Emergent Planarity in two-dimensional Ising Models with finite-range Interactions,” arXiv:1801.04960v2 [math-ph]
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