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Non Chiral Bosonization of a Luttinger Liquid

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Non Chiral Bosonization of a Luttinger Liquid NON CHIRAL BOSONIZATION OF A LUTTINGER LIQUID A thesis submitted for the degree of Doctor of Philosophy Joy Prakash Das Under the guidance of Prof. Girish. S. Setlur Department of Physics Indian Institute of Technology Guwahati Guwahati 781 039, Assam, India TH-2101_146121021 TH-2101_146121021 NON CHIRAL BOSONIZATION OF A LUTTINGER LIQUID A thesis submitted for the degree of Doctor of Philosophy Joy Prakash Das Roll No. 146121021 Under the guidance of Prof. Girish. S. Setlur Department of Physics Indian Institute of Technology Guwahati Guwahati 781 039, Assam, India TH-2101_146121021 January 2019 TH-2101_146121021 In matters of science, a thousand proclamations by so-called experts are outweighed by the humble reasoning of a single individual. - Galileo Galilei TH-2101_146121021 TH-2101_146121021 Dedicated to my family TH-2101_146121021 TH-2101_146121021 Declaration The work in this thesis entitled \Non chiral bosonization of a Luttinger liquid" has been carried out by me under the supervision of Prof. Girish S. Setlur, Department of Physics, Indian Institute of Technology Guwahati. No part of this thesis has been submitted elsewhere for award of any other degree or qualification. The research works have been carried out in the period from January, 2015 to December, 2018. In keeping with general practice of reporting scientific observations, due acknowledg- ments have been made wherever the work described is based on the findings of other investi- gations. Place: IIT Guwahati Joy Prakash Das Date: Roll No. 146121021 TH-2101_146121021 i TH-2101_146121021 ii Certificate This is to certify that the research work contained in this thesis entitled \Non chiral bosonization of a Luttinger liquid" by Mr. Joy Prakash Das, a PhD student of the Department of Physics, IIT Guwahati was carried out under my supervision. This work is original and has not been submitted elsewhere for award of any degree. Place: IIT Guwahati Prof. Girish S. Setlur Date: Department of Physics, IIT Guwahati. Email: [email protected] TH-2101_146121021 iii TH-2101_146121021 iv Acknowledgements I will forever be grateful to my supervisor Prof. Girish S. Setlur for introducing me to this challenging yet interesting field of research. His intelligent ideas and constant guidance have always helped me in moving forward with my thesis work. I have always admired his courage to challenge the conventional notion of a subject which has prevailed for a long time among a huge number of popular figures. If I am sent four years back in time to the day of choosing a supervisor, I will again approach him with the better version of me - more sincere and more hardworking. I extend my sincere gratitude to my doctoral committee members - Prof. P. Poulose, Prof. Amarendra Sarma and Dr. P.A.S. Sree Krishna for reviewing my progress every year and sharing their invaluable comments, suggestions and feedback that have enabled me to improve and present my work in the shape that it is today. I would like to express my heartfelt thanks to all the faculties of the Physics department of IIT Guwahati for building my foundation of Physics when I was an undergraduate student in the same institute, which has tremendously influenced me to build a career in Physics. I would also like to thank all the staff members of the department for their support, co- operation and friendly behavior. My special thanks to the H.O.D's of the Physics department Prof. Basu, Prof. Poulose and Prof. Ghosh for all the facilities in the department and the timely meetings with the research scholars, addressing their concerns. I would like to thank my friends of IIT Guwahati from the bottom of my heart, especially the 2014 batch. I am grateful to the Art of living and the Hare Krishna Movement for teaching me yoga and imparting me spiritual knowledge that have helped in transcending the good and not so good times of my Ph.D. life. I am also highly grateful to Dr. Uma Dutta of Cotton University for her moral support and words of encouragement. I am also thankful to my seniors Dr. Enamullah, Dr. Vipin Kumar, Dr. Upendra Kumar for their supportive and helping nature. It was a lovely group of us. My deepest gratitude to my mother for shaping me for life and taking a lot of pains for my career. I have always felt the blessings of my father from the unseen world. I am thankful to all the members of my family for constant moral support and help. No amount of acknowl- edgment will ever be enough to do justice to the kind of support, love and encouragement I have received from my family and this thesis is dedicated to them. I thank all the people who have been in direct or indirect association with me for my research work and have helped me in even the slightest possible manner. Above all I thank the Divine for everything - just everything. TH-2101_146121021 v TH-2101_146121021 vi Abstract In this work, the powerful Non-chiral bosonization technique (NCBT) is introduced, which is a non-trivial modification of the standard Fermi-Bose correspondence in one spatial dimension made in order to facilitate the study of strongly inhomogeneous Luttinger liquids (LL) where the properties of free fermions plus the source of inhomogeneities are reproduced exactly. The formalism is applied to obtain the correlation functions of translationally non-invariant systems like LL with a cluster of impurities (barriers/wells) around an origin, a one step fermionic ladder, slowly moving impurities in a Luttinger liquid, etc. The obtained correlation functions are used to study various physical phenomena like Friedel oscillations, resonant tunneling, dynamical density of states, conductance, mobility (in case of mobile impurities) and so on. The results are validated using the Schwinger Dyson equation and perturbative methods. The present method is superior to the conventional bosonization methods (g-ology methods) which requires additional tools like re-normalization, etc. to deal with impurities. TH-2101_146121021 vii TH-2101_146121021 viii Contents Declaration i Certificate iii Acknowledgementsv Abstract vii List of Figures xiii List of Tables xvii 1 Introduction1 1.1 One dimensional systems..............................2 1.1.1 Failure of the Fermi Liquid theory.....................3 1.1.2 Tomonaga model..............................4 1.1.3 Luttinger model...............................6 1.1.4 Physical realizations............................9 1.2 Bosonization..................................... 10 1.2.1 History of bosonization........................... 10 1.2.2 Formalism.................................. 15 1.2.2.1 Field theoretical bosonization.................. 15 1.2.2.2 Constructive bosonization.................... 16 1.3 Impurity in a Luttinger liquid........................... 17 1.3.1 Renormalization group........................... 17 1.3.2 Numerical methods............................. 18 1.4 Summary...................................... 19 2 Methodology 21 2.1 Critique to g-ology based chiral bosonization................... 21 2.2 Working procedure................................. 24 2.2.1 Single particle Green's functions...................... 24 2.2.2 Density density correlation functions................... 29 2.2.3 Field operator reconstruction (Bosonization)............... 30 2.2.4 Including interactions............................ 33 2.2.5 Many body Green functions........................ 34 2.3 Summary...................................... 35 3 The quantum steeplechase 37 3.1 System description................................. 38 TH-2101_146121021 ix 3.2 Green's functions of free fermions......................... 40 3.2.1 Density density correlation function.................... 42 3.3 Bosonized version of the N-point Green's functions............... 43 3.4 Full two-point Green's function.......................... 45 3.4.1 Anomalous exponents........................... 47 3.4.2 Limiting case checks............................ 49 3.4.3 Spinless case................................. 50 3.4.4 Technical clarification............................ 50 3.5 Four-point functions (Friedel oscillations)..................... 51 3.6 Dynamical density of states............................ 53 3.7 Summary...................................... 55 4 Transport properties 57 4.1 Conductance.................................... 58 4.1.1 Kubo conductance............................. 59 4.1.1.1 Limiting cases........................... 60 4.1.2 Tunneling conductance........................... 61 4.1.2.1 Derivation of RG equation for the tunneling conductance.. 63 4.2 Finite bandwidth conductance........................... 64 4.2.1 Numerical solution............................. 65 4.2.2 Comparison with the results of Matveev et al............... 68 4.2.3 Anomalous conductance.......................... 71 4.2.4 Analytical solution............................. 72 4.2.5 Comparison of analytical and numerical solution............. 73 4.2.6 Both forward and backward scattering.................. 73 4.2.7 Comparison with Monte Carlo results................... 75 4.3 Resonant tunneling across a double barrier.................... 76 4.4 Summary...................................... 77 5 The one step fermionic ladder 79 5.1 Problem overview.................................. 80 5.2 Green's functions of free fermions......................... 81 5.2.1 Density density correlation function................... 81 5.3 Bosonized version of the two point Green functions............... 82 5.4 Full two-point Green's function.......................... 85 5.4.1 Anomalous exponents..........................
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