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Thesis (Complete) UvA-DARE (Digital Academic Repository) Diverse methods for integrable models Fehér, G. Publication date 2017 Document Version Final published version License Other Link to publication Citation for published version (APA): Fehér, G. (2017). Diverse methods for integrable models. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:23 Sep 2021 Diverse methods for integrable models ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. ir. K.I.J. Maex ten overstaan van een door het College voor Promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op dinsdag 28 november 2017, te 14.00 uur door Gyorgy¨ Feher´ geboren te Boedapest, Hongarije Promotiecommissie: Promotor: Prof. dr. B.Nienhuis Universiteit van Amsterdam Overige leden: Prof. dr. J.V.Stokman Universiteit van Amsterdam Prof. dr. C.J.M.Schoutens Universiteit van Amsterdam Prof. dr. J.-S.Caux Universiteit van Amsterdam dr. V.Gritsev Universiteit van Amsterdam Prof. dr. N.Reshetikhin University of California, Berkeley, Universiteit van Amsterdam Prof. dr. J.-B.Zuber Universit´ePierre et Marie Curie Prof. dr. P.Zinn-Justin University of Melbourne Faculteit der Natuurwetenschappen, Wiskunde en Informatica List of Publications This thesis is based on the following publications: [1] G. Z. Feher and B. Nienhuis. Currents in the dilute O(n = 1) model. 2015, math-ph/1510.02721., preprint [2] Gyorgy Z. Feher, Paul A. Pearce, and Alessandra Vittorini-Orgeas. One- dimensional sums and finitized characters of 2 2 fused RSOS models. 2016, unpublished × [3] Jan de Gier, Gyorgy Z. Feher, Bernard Nienhuis, and Magdalena Rusaczonek. Integrable supersymmetric chain without particle conservation. 2015, math-ph/1510.02520, J. Stat. Mech. (2016) 023104 v Contents 1 Introduction 1 2 Current in dilute loop model 7 2.1 Introduction of loop models . 7 2.2 Dilute O(n) loop model . 12 2.3 Inhomogeneous weights . 15 2.3.1 Specialization of loop weights to n = 1 . 18 2.4 Vector space of link patterns . 22 2.5 Double row transfer matrix . 31 2.6 The quantum Knizhnik – Zamolodchikov equations . 34 2.7 Definition of spin-1 current and main result . 35 2.7.1 Main result . 36 2.8 Recursion relations . 37 2.8.1 Fusion equation . 38 2.8.2 Boundary fusion equation . 40 2.8.3 The missing recursion by size two . 43 2.9 Symmetries . 43 2.9.1 Partial symmetry of X and Y . 44 2.9.2 Full symmetry of Y ..................... 45 2.10 Proof of the main result . 48 2.10.1 Proof for the Y current . 48 2.10.2 Proof for the X current . 50 2.11 Conclusion . 51 3 Fused RSOS models and finitized characters 53 3.1 Introduction . 53 3.2 Minimal models and higher-level minimal coset models . 55 3.3 RSOS models . 57 vii 3.3.1 Fused RSOS models and fused weights . 59 2 2 case . 61 3 × 3 case . 63 3.3.2 Corner× transfer matrix . 63 3.3.3 RSOS paths, and shaded bands . 65 3.3.4 Low temperature limit and local energy functions . 67 1 1 local energy function . 68 2 × 2 local energy function . 69 3 × 3 local energy function . 72 3.3.5 One-dimensional× sums and energy statistics . 77 Groundstates and finitized characters . 78 3.4 RSOS(m, 2m + 1)2×2 sequence . 80 3.4.1 Adjacency graph for RSOS(m, 2m + 1)2×2 models . 80 3.4.2 Bijection between the JM and 2 2 RSOS paths . 82 3.5 Finitized characters . .× . 85 3.5.1 Conjectured finitized branching functions corresponding to π n n fused RSOS models at 0 < λ < n . 85 3.5.2 Conjectured× finitized minimal characters corresponding to π 2 2 fused RSOS models at 2 < λ < π . 87 3.6 Conclusion× and outlook . 90 4 SUSY fermion chain 91 4.1 Introduction . 91 4.2 Definition of the model . 93 4.2.1 Domain walls . 95 4.3 Solution by Bethe ansatz . 96 4.3.1 Bethe equations . 96 4.3.2 Two domain walls, no odd domain wall (m, k) = (2, 0) . 98 4.3.3 Two domain walls, one odd domain wall (m, k) = (2, 1) . 100 4.3.4 Arbitrary number of domain walls, no odd domain wall (m, k = 0) . 103 4.3.5 Arbitrary number of domain walls, one odd domain wall (m, k = 1) . 104 4.3.6 Arbitrary number of domain walls, two odd domain walls (m, k = 2) . 106 4.3.7 Arbitrary number of domain walls, arbitrary number of odd domain walls (m = 2n, k) . 107 4.4 Symmetries . 108 4.4.1 Supersymmetry . 109 4.4.2 Domain wall number conversation and translation symmetry110 4.4.3 Fermion parity symmetry . 110 4.4.4 Fermion–hole symmetry . 111 4.4.5 Domain wall–non-domain wall symmetry . 111 4.4.6 Shift symmetry . 112 4.4.7 Reflection symmetry of the spectrum for L = 4n . 112 4.4.8 Antiperiodic boundary conditions and reflection symmetry of the spectrum for L = 4n 2 . 113 − 4.4.9 Zero mode Cooper pairs . 114 4.5 Completness of Bethe equations, exact solutions . 115 4.5.1 L = 6, full spectrum . 117 Λ = 6 eigenvalue . 117 Other eigenvalues . 118 4.5.2 L = 10, Λ = 6 eigenvalue . 118 4.5.3 The groundstate and first excited state for L = 4n . 119 4.6 Conclusion and outlook . 121 5 Conclusion 123 A Current in dilute loop model 127 A.1 Derivation of K-matrix . 127 A.2 Dilute O(1) loop model as percolation . 129 A.3 Normalization of the T -matrix . 131 A.4 L = 1 GS and X current . 133 A.5 Proof of the fusion equation . 134 A.6 OBC K-matrix construction . 136 B Fused RSOS models and finitized characters 141 B.1 Coset construction . 141 B.2 Characters of higher fusion level coset models . 144 B.3 3 3 fused RSOS face weights . 146 × C SUSY fermion chain 149 C.1 GS degeneracy for L = 4n . 149 Summary 153 Samenvatting 159 Contribution to publications 165 Bibliography 167 List of Figures 2.1 Typical dense loop configuration. 8 2.2 Six-vertex model. 9 2.3 Spin-1 property of the current. 11 2.4 Typical dilute loop configurations with different boundary conditions. 14 2.5 Rapidities with OBC. 16 2.6 Mapping loop configuration to dilute link pattern. 22 2.7 Dilute link pattern, L = 2. 23 2.8 Some examples of R, and K-matrix elements acting on dilute link patterns. 24 2.9 Non equivalent paths. 46 2.10 Path and mirrored path. 47 2.11 Path sum rule. 48 3.1 Corner transfer matrix. 64 3.2 RSOS path. 67 3.3 Local energy functions for 1 1 RSOS. 69 × 3.4 Local energy functions for 2 2 RSOS, 0 < λ < π . 73 × 2 3.5 Local energy functions for 2 2 RSOS, π < λ < π. 74 × 2 3.6 Local energy functions for 3 3 RSOS, 0 < λ < π . 76 × 3 3.7 2 2 fused groundstate paths. 78 × 3.8 Adjecancy graphs for unfused and 2 2 fused models. 81 × 3.9 Jacob-Mathieu – RSOS path bijection. 83 4.1 L = 16 free-fermionic Bethe ansatz solution with Cooper pair. 114 4.2 L = 10 non free-fermionic Bethe ansatz solution. 114 4.3 L = 16 non free-fermionic Bethe ansatz solution. 116 4.4 Action of symmetries between domain wall sectors . 120 xi A.1 Site percolation – dilute loop mapping. 130 A.2 Computing the T -matrix normalization. 132 A.3 Construction of a new K-matrix via insertion of a line. 137 A.4 KRR block forming a new K-matrix. 138 Chapter 1 Introduction Statistical physics is the branch of physics which aims to describe the collective behavior of a large number of constituting pieces, typically particles. The precise, or near precise description of such a system (e.g. the trajectories of all the particles in an interacting classical system) is a hopelessly complicated task, and statistical physics does not aim for that. Its goal is to describe the system through average macroscopic quantities, such as temperature, density, magnetization, etc. To describe the intrinsic behavior of systems, correlations between different parts of the system are probed in the context of statistical physics. An example is how the correlation of local magnetization decays with the distance. One important focus of statistical physical research is the description of phase transition: the phenomenon when some macroscopic quantity of the system undergoes a sudden change. It is observed that while usually the thermodynamic properties of the materials change smoothly, at certain values of the parameters, they undergo abrupt, often discontinuous changes.
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