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Doktorarbeit Supersymmetry in Random Matrix Theory Universität Duisburg-Essen Fakultät für Physik Doktorarbeit zur Erlangung des Doktorgrades „Dr. rer. nat.“ Supersymmetry in Random Matrix Theory von Mario Kieburg geboren am 23.06.1981 in Zossen Tag der Einreichung: 22.01.2010 Jahrelang in einer Wüste lebst du nun auf off’nem Meer, erlebst ein paar neu’ Gelüste and die Angst fährt hinterher. Jahrelang auf flachem Land steigst du auf den höchsten Berg. Die Sicht war einst dir unbekannt, nun fühlst du dich wie ein Zwerg. Jahrelang mit was Vertrautem, jahrelang hast du nichts gesehen, nach Jahren mit was Verbautem hast du es schwer, neu zu verstehen. Mario Kieburg Tag der Disputation: 04.05.2010 Erstgutachter: Prof. Dr. Thomas Guhr Zweitgutachter: Prof. Dr. Tilo Wettig Drittgutachter: Prof. Dr. Hans Werner Diehl Prüfungsvorsitzender: Prof. Dr. Rolf Möller weiterer Prüfer: Prof. Dr. Hans-Jürgen Sommers Teile dieser Arbeit sind in den wissenschaftlichen Artikeln: [1] M. Kieburg, H. Kohler, and T. Guhr. J. Math. Phys., 50:013528, 2009. preprint: arXiv:0809.2674. [2] M. Kieburg, J. Grönqvist, and T. Guhr. J. Phys., A 42:275205, 2009. preprint: arXiv:0905.3253. [3] M. Kieburg, H.-J. Sommers, and T. Guhr. J. Phys., A 42:275206, 2009. preprint: arXiv:0905.3256. [4] M. Kieburg and T. Guhr. J. Phys., A 43:075201, 2010. preprint: arXiv:0912.0654. [5] M. Kieburg and T. Guhr. J. Phys., A 43:135204, 2010. preprint: arXiv:0912.0658. schon veröffentlicht worden. Acknowledgements First of all, I thank my wife who gave me strength and perseverance to finish this thesis. I am glad that she has so much patience with me for working not only at the university but also at home. I am deeply grateful to Thomas Guhr for being such an open minded and helpful supervisor. I have not only learned expertise from him but also to write scientific works and to give presentations at international conferences. He gave me helpful instructions to improve my English and introduced me to experts in random matrix theory and related areas. I thank J. Grönqvist, H. Kohler and H.-J. Sommers for fruitful coopera- tions which led to the articles [1, 2, 3]. I am also thankful for the mathemat- ical advice of P. Heinzner and E. Vishnyakova which clarified some technical problems. The discussions with G. Akemann, A. Huckleberry, S. Mandt, M.J. Phillips and M.R. Zirnbauer were also quite helpful to understand where our new methods might be applicable and where it fits in the framework of random matrix theory. I am thankful to them for pointing out so many important references. I am indebted to all members of the research group of Thomas Guhr. To this group the following members belonged or are belonging: B. Gutkin, J. Hämmerling, H. Kohler, E. Ljudko, M. Münnix, V. Osipov, C. Recher, R. Schäfer and H.-J. Sommers. I am grateful to them for discussions, help- ful advice and pleasant small talks which often evolved beyond physics and mathematics. In particular, I thank R. Schäfer for proof-reading this thesis. I thank the Universität Duisburg–Essen for the support. Furthermore, I acknowledge financial support from and workshops organized by the Sonder- forschungsbereich Transregio 12 “Symmetries and Universality in Mesoscopic Systems” of the Deutsche Forschungsgemeinschaft. Contents 1 Introduction 9 1.1 Randommatrixtheoryanditsapplications . 9 1.1.1 Quantumchaos ...................... 12 1.1.2 Quantumchromodynamics . 17 1.2 Supersymmetry inrandommatrixtheory . 19 1.2.1 Supersymmetry in other branches of physics . 19 1.2.2 The supersymmetry method in the theory of disordered systems andinrandommatrixtheory . 21 1.3 Outline............................... 24 I Harmonicalanalysisinsuperspaces 27 2 Definitions and notations 29 2.1 TheGrassmannalgebra . 29 2.2 Supervectors and supermatrices and their symmetries . ... 30 2.3 Integrationinsuperspaces . 34 3 Integrations in superspaces and Cauchy–like theorems 39 3.1 Integration of Grassmann variables over invariant functions . 40 3.2 Integral theorems for invariant functions on supervectors ... 42 3.3 Integral theorems for invariant functions on supermatrices . 47 3.4 On certain integrals for functions invariant under UOSp (1/2) 55 4 Applications of theorem 3.1.1 59 4.1 One–point correlation functions and supermatrix Bessel func- tions................................ 60 4.2 Supermatrix Bessel functions for U (2)(p/q)) .......... 64 4.3 Supermatrix Bessel-functions for arbitrary Dyson index β ... 66 5 Summary of part I 69 1 2 CONTENTS II The supersymmetry method 71 6 k-point correlation functions and the supersymmetry method 73 6.1 Posingtheproblem ........................ 73 6.2 Sketchofourapproach . 75 7 From ordinary matrix space to superspace 79 7.1 Average over the ensemble and the characteristic function... 79 7.2 Duality between ordinary and superspace . 81 7.3 Analysis of the duality between ordinary and superspace ... 82 7.4 Problems when diagonalizing K ................. 84 7.5 Ambiguity of the characteristic function in the supersymmet- ricextension............................b 85 7.6 Symmetriesofthesupermatrices. 86 8 The generalized Hubbard–Stratonovich transformation 89 8.1 Transformation to supermatrices by a Dirac distribution ... 90 8.2 The supersymmetric Ingham–Siegel integral . 91 8.3 The generalized Hubbard–Stratonovich transformation for su- persymmetricWishartmatrices . 96 9 Discussion and results of the integrals in superspace 99 9.1 Eigenvalue integral representation . 99 9.2 Theunitarycaserevisited . .100 9.3 Independence statement . 101 9.4 One–point and higher order correlation functions . 103 10 Comparison of the generalized Hubbard–Stratonovich trans- formation and the superbosonization formula 107 10.1 Thesuperbosonizationformula . 107 10.2 Equivalence of and connections between the two approaches . 109 10.3 The general case for arbitrary positive integers a, b, c, d and arbitrary Dyson–index β 1, 2, 4 ...............111 ∈{ } 11 Explicit formulas for the superfunctions Φ and Φ 115 F 11.1 The case c a, b = d =0 and arbitrary Dyson index . 116 ≤ 11.2 The case c = d =1, b =0 and β =2 ..............117 11.3 Extension to arbitrary dimensions . 119 12 Summary of part II 121 CONTENTS 3 III An algebraical method to derive determinantal and Pfaffian structures 123 13 Sketch of the idea: “Supersymmetry without supersymme- try” 125 13.1 Determinantalstructures . 125 13.2 Pfaffianstructures . .128 14 Determinantal structure of Berezinians 131 14.1 Berezinians related to the supergroup U (2)(p/q) ........131 14.2 Berezinians related to the supergroup UOSp (p/q) .......133 15 Main result 135 15.1 Integrals of square root–Berezinian type . 135 15.2 Integrals of squared–Vandermonde type . 138 15.3 Integrals of coupled square–root Vandermonde type . 142 16 Applications to random matrix theory 147 16.1 Applications for integrals of squared–Vandermonde type . .147 16.1.1 Hermitian matrix ensemble . 148 16.1.2 Listofothermatrixensembles . 149 16.2 Applications for integrals of square root–Berezinian type . 152 16.2.1 The Hermitian matrices revisited . 152 16.2.2 The Hermitian matrix ensemble in an external field . 153 16.2.3 Determinantal and Pfaffian structures for intermediate ensemble..........................156 16.2.4 Laguerre ensembles in an external field . 160 16.3 Applications for integrals of coupled square–root Vandermonde type ................................162 16.3.1 Rotation invariant ensembles of real symmetric matri- ces and Hermitian self-dual matrices . 162 16.3.2 A list of other matrix ensembles . 164 17 Summary of part III 169 18 Conclusions and outlook 171 A Derivations for part I 173 A.1 Proofoftheorem3.1.1 . .173 A.2 Proofoflemma3.1.2 . .175 A.3 Proofoftheorem3.2.1 . .176 A.4 Proofoftheorem3.3.1 . .177 4 CONTENTS A.5 Proofoftheorem3.3.2 . .179 B Derivations for part II 181 B.1 Circularity of the supertrace for rectangular supermatrices . 181 B.2 Proofofproposition7.4.1 . .182 B.3 A matrix Bessel version of the Sekiguchi differential operator . 183 B.4 Calculation of the supersymmetric Ingham–Siegel integral. 186 B.4.1 Decomposition of the boson–boson and fermion–fermion block integration . 186 B.4.2 Proofoflemma8.2.1 . .188 B.4.3 Proofoftheorem8.2.2 . .190 B.5 Proofoftheorem8.3.1 . .192 B.6 Proofoftheorem10.1.1. .194 B.7 Proofoftheorem10.2.1. .201 C Derivations for part III 205 C.1 DerivationoftheBerezinians. 205 C.1.1 The case β =2 ......................205 C.1.2 The case β 1, 4 with p 2q .............206 ∈{ } ≤ C.1.3 The case β 1, 4 with p 2q .............207 C.2 Calculating integrals∈{ of square} root–Berezinian≥ type . .207 C.2.1 The case k1 = k2 = k ...................207 C.2.2 The case k1 k2 = k ...................208 C.2.3 The case k =≤k k ...................209 1 ≥ 2 C.3 Extension of integration theorems for determinantal kernels . 209 C.3.1 Extension of Andréief’s integral theorem . 209 C.3.2 Extension of de Bruijn’s integral theorem . 210 C.4 Calculating integrals of squared–Vandermonde type . 212 C.4.1 The case (k k )=(l l ) N ...........212 1 − 2 1 − 2 ≤ C.4.2 The case (k1 k2), (l1 l2) N .............213 C.5 Calculating integrals− of coupled− square–root≥ Vandermonde type214 C.5.1 The case k +2N +1 k ................214 2 ≥ 1 C.5.2 The case k2 +2N +1 k1 ................215 C.6 Calculation of the flat Fourier≤ transform in Eq. (16.11) . 216 CONTENTS 5 Zusammenfassung Die Untersuchung des Verhältnisses zwischen der Supersymmetrie und der Zufallsmatrixtheorie steht im Mittelpunkt dieser Arbeit. Es wird die Su- persymmetriemethode verallgemeinert. Weiterhin werden drei neue Berech- nungsmethoden von Eigenwertkorrelationsfunktionen entwickelt. Diese Kor- relationsfunktionen sind Mittelwerte von Quotienten, welche aus charakte- ristischen Polynomen aufgebaut sind. Im ersten Teil dieser Arbeit wird ein Zusammenhang zwischen Integralen über antikommutierenden Variablen (Grassmann–Variablen) und Differen- tialoperatoren hergeleitet. Die Differentialoperatoren wirken nur auf den kommutierenden Anteil der Variablen. Mittels dieses Zusammenhangs wer- den Cauchy–ähnliche Integraltheoreme verifiziert. Außerdem werden die Supermatrix–Bessel–Funktionen auf ein Produkt von zwei gewöhnlichen Ma- trix–Bessel–Funktionen zurückgeführt.
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