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β FUNCTION OF ANDERSON LOCALIZATION TRANSITION IN THREE DIMENSIONS AT UNITARY SYMMETRY By TOMOYUKI NAKAYAMA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011 (cid:176)c 2011 Tomoyuki Nakayama 2 To my family 3 ACKNOWLEDGMENTS First, I would like to thank my advisor, Prof. Khandker Muttalib, who has spent extended amount of time to help me on my work toward Ph.D. Discussion with him has given me plentiful of physical insights. I would also like to thank my collaborators, Prof. Peter Wo¨lﬂe and Pavel Ostrovskii. My reserch is based on their recent studies, hence it was impossible to do this work without them. They also gave me useful advice and directions. I really thank my supervisory committee members, Prof. Peter Hirschfeld, Prof. Dmitrii Maslov, Prof. Arthur Hebard and Prof. Murali Rao for spending their valuable time in reviewing my dissertation and hearing my qualifying exam and ﬁnal defense. My thank also goes to Hridis Pal and Chungwei Wang for useful discussion on many body theory and quantrum transport. Finally, I would like to thank the department of physics at Universtiy of Florida and Institute of Condensed Matter Theory at Karlsruhe Institute of Technology, Germany for providing me with a great environment to research. 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 CHAPTER 1 ANDERSON LOCALIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 Anderson’s Original Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Minimum Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Scaling Theory of Localization . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4.1 1-Parameter Scaling Theory . . . . . . . . . . . . . . . . . . . . . . 15 1.4.2 Scaling for Frequency-Dependent Conductivity . . . . . . . . . . . 20 1.5 Perturbation Theory of Localization . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Non-Linear Sigma Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 UNITARY ENSEMBLES IN 2 DIMENSIONS . . . . . . . . . . . . . . . . . . . . 27 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Diffuson Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3 Overall Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Calculation of Unitary Set of Diagrams . . . . . . . . . . . . . . . . . . . . 29 2.4.1 4-Vertex Hikami Box . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.2 6-Vertex Hikami Box . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Correction to Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6 Restoration of Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.1 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.6.2 Diagrams with Single Diffuson . . . . . . . . . . . . . . . . . . . . . 36 2.7 Second Order Correction to Conductivity . . . . . . . . . . . . . . . . . . . 39 2.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3 UNITARY ENSEMBLE IN 3 DIMENSIONS . . . . . . . . . . . . . . . . . . . . . 41 3.1 Overall Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Diffuson Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3 Calculation of Unitary Set of Diagrams . . . . . . . . . . . . . . . . . . . . 42 3.3.1 Coordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.2 4-Vertex Vector Hikami Box . . . . . . . . . . . . . . . . . . . . . . 44 3.3.3 6-Vertex Scalar Hikami Box . . . . . . . . . . . . . . . . . . . . . . 44 3.3.4 2-Vertex Hikami Box . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 3.4 Correction to the Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Higher Order Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4 FREQUENCY DEPENDENT CONDUCTIVITY . . . . . . . . . . . . . . . . . . 54 4.1 Overall Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.2 Diffuson Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3 Calculation of Unitary Set of Diagrams . . . . . . . . . . . . . . . . . . . . 56 4.3.1 4-Vertex Vector Hikami Box . . . . . . . . . . . . . . . . . . . . . . 56 4.3.2 6-Vertex Scalar Hikami Box . . . . . . . . . . . . . . . . . . . . . . 56 4.3.3 2-Vertex Hikami Box . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4 Reevaluation of h¯ and h . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6a 2a 4.5 Correction to the Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.6 Calculation of β-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 APPENDIX A ORTHOGONAL ENSEMBLE IN 2 DIMENSIONS . . . . . . . . . . . . . . . . . 66 A.1 Overall Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 A.2 Diffuson and Cooperon Propagators . . . . . . . . . . . . . . . . . . . . . 66 A.3 Calculaiton of Orthogonal Set of Diagmrams . . . . . . . . . . . . . . . . 67 A.3.1 Hikami A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 A.3.2 Hikami B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 A.3.3 Hikami C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 A.3.4 Hikami D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A.3.5 Hikimi E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.3.6 Hikami F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 A.4 Diagrams Restoring Gauge Symmetry . . . . . . . . . . . . . . . . . . . . 81 A.4.1 Additional Diagram G . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A.4.2 Additional Diagram H . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.5 Summation of All Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.5.1 Power Law Divergence . . . . . . . . . . . . . . . . . . . . . . . . . 82 A.5.2 Logarithmic Divergence . . . . . . . . . . . . . . . . . . . . . . . . 84 A.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 B UNITARY ENSEMBLE IN DIFFUSIVE REGIME . . . . . . . . . . . . . . . . . . 87 B.1 Overall Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 B.2 Calculation of Diagrams at Unitary Symmetry . . . . . . . . . . . . . . . . 88 B.2.1 Hikami Diagram A . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 B.2.2 Hikami Diagram D . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 B.2.3 Hikami Diagram G . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 B.3 Summation of Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6 BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7 LIST OF TABLES Table page 1-1 Three symmetry classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1-2 Values of critical exponent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8 LIST OF FIGURES Figure page 1-1 Graph of β(g) versus lng. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1-2 Maximally crossed diagrams. The dotted lines show impurity scattering and the wavy line is the cooperon propagator. . . . . . . . . . . . . . . . . . . . . . 21 1-3 Cooperon diagram. The wavy line is the cooperon propagator. . . . . . . . . . 22 1-4 Coherent backscattering process. . . . . . . . . . . . . . . . . . . . . . . . . . 23 2-1 Diffuson propagator. The dotted lines show impurity scattering and the solid line is diffuson propagator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2-2 3-diffuson diagram and 4-vertex vector Hikami box. The box on the right only corresponds to the numbered part of the diagram on the left. . . . . . . . . . . 29 2-3 2-diffuson diagram and 6-vertex scalar Hikami box. . . . . . . . . . . . . . . . . 30 2-4 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4a 2-5 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4b 2-6 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6a 2-7 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6b 2-8 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6c 2-9 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6d 2-10 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6e 2-11 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6f 2-12 Self energy diagram and generated three diagrams, h , h and h . . . . . 37 6,a 2,a 2,a(cid:48) 2-13 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2a 2-14 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2b 2-15 Diagram h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2c 3-1 Coordinate transformation. We align z-axis with Q. . . . . . . . . . . . . . . . . 42 3-2 Two diffuson diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3-3 Four-diffuson diagram. Only one diffuson carries small momentum. Others are ballsitic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4-1 Diagram h¯ . This diagram needs more accurate evaluation. . . . . . . . . . . 54 6a 9 4-2 Diagram h . This diagram needs more accurate evaluation. . . . . . . . . . . . 55 2a 4-3 Diagram h¯ , reevaluated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6a 4-4 Diagram h , reevaluated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2a 4-5 β versus lng. The solid line is the β-function without additional term A, and the dotted line is the β-function with A. Note adding A does not change the slope at g = g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 c A-1 Hikami Diagram A. This diagram is the same as h . It contains three diffusons. 67 4 A-2 Hikami Diagram B. This diagram consists of two vector Hikami Box, two cooperon and one diffuson. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A-3 Hikami Diagram C. This diagram contains two cooperons and one diffuson. . . 68 A-4 Hikami Diagram D. This diagram is the same as h . It contains two diffusons. . 68 6 A-5 Hikami Diagram E. This diagram contains two cooperons. . . . . . . . . . . . . 68 A-6 Hikami F. This diagram contains one cooperon and one diffuson. . . . . . . . . 68 A-7 Diagram D in standard notion. Solid lines are diffusons. . . . . . . . . . . . . 83 0 A-8 Diagram E in standard notaion. Wavy lines express cooperons. . . . . . . . . 83 0 A-9 Diagram F in standard notation. . . . . . . . . . . . . . . . . . . . . . . . . . . 84 0 A-10Diagram F(cid:48) in standard notation. . . . . . . . . . . . . . . . . . . . . . . . . . . 84 0 A-11Diagram G in standard notation. . . . . . . . . . . . . . . . . . . . . . . . . . . 84 1 A-12Diagram H in standard notation. . . . . . . . . . . . . . . . . . . . . . . . . . 84 1 10

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β FUNCTION OF ANDERSON LOCALIZATION TRANSITION IN THREE DIMENSIONS .. Electrons are scattered by this disorder to a state with a.

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β FUNCTION OF ANDERSON LOCALIZATION TRANSITION IN THREE DIMENSIONS .. Electrons are scattered by this disorder to a state with a.

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