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Keldysh Field Theory for Dissipation-Induced States of Fermions

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Keldysh Field Theory for Dissipation-Induced States of Fermions CORE Metadata, citation and similar papers at core.ac.uk Provided by Electronic Thesis and Dissertation Archive - Università di Pisa Department of Physics Master Degree in Physics Curriculum in Theoretical Physics Keldysh Field Theory for dissipation-induced states of Fermions Master Thesis Federico Tonielli Candidate: Supervisor: Federico Tonielli Prof. Dr. Sebastian Diehl University of Koln¨ Graduation Session May 26th, 2016 Academic Year 2015/2016 UNIVERSITY OF PISA Abstract Department of Physics \E. Fermi" Keldysh Field Theory for dissipation-induced states of Fermions by Federico Tonielli The recent experimental progress in manipulation and control of quantum systems, together with the achievement of the many-body regime in some settings like cold atoms and trapped ions, has given access to new scenarios where many-body coherent and dissipative dynamics can occur on an equal footing and the generators of both can be tuned externally. Such control is often guaranteed by the toolbox of quantum optics, hence a description of dynamics in terms of a Markovian Quantum Master Equation with the corresponding Liouvillian generator is sufficient. This led to a new state preparation paradigm where the target quantum state is the unique steady state of the engineered Liouvillian (i.e. the system evolves towards it irrespective of initial conditions). Recent research addressed the possibility of preparing topological fermionic states by means of such dissipative protocol: on one hand, it could overcome a well-known difficulty in cooling systems of fermionic atoms, making easier to induce exotic (also paired) fermionic states; on the other hand dissipatively preparing a topological state allows us to discuss the concept and explore the phenomenology of topological order in the non-equilibrium context. The relevant example here presented is the dissipative counterpart of the ground state of the Kitaev chain, a 1D p-wave superfluid of spinless fermions, whose topological properties crucially rely on the pairing terms. The goal of the thesis is to provide a wider many-body picture for the theoretical description of the protocol, previously based on a mean field approach, in order to understand the pairing mechanism more deeply. To this end, it is structured as follows. It starts explaining the protocol for fermionic systems, pointing out why a description of dynamics in terms of a Master Equation is allowed, discussing the properties of steady states and presenting a microscopic implementation on which the mean field approach is applied. Keldysh formalism is then used to reformulate the Master Equation description in terms of a non-equilibrium field theory and its generating functionals. The needed tools of quantum field theory are then briefly reviewed. Finally, a many body analysis of the problem is offered, both from a variational and a Dyson-Schwinger based point of view. It is shown that certain pairing schemes elude a standard understanding as order parameters of a BCS-type theory; the mean field results are instead recovered as a 1-loop approximation to Dyson-Schwinger equations for the two-point function. Acknowledgements I am grateful to Prof. Sebastian Diehl for the opportunity he gave me to study in his group and for the time he spent with me on this project; during that time I have learned the value of listening carefully to other people's ideas, especially in order to improve your work and your working style. My sincere gratitude goes also to Jamir Marino, whose office had been for me a safe Italian exclave within Germany and whose precious advice helped me in the hard task of orienting myself in this new country. I am grateful to Prof. Rosario Fazio, because I could meet Prof. Diehl thanks to his help (as well as to his desire of getting rid of me quickly) and, most importantly, because he is a major culprit of fueling my insatiable scientific interest in Condensed Matter physics. I am grateful to my entire family for their support during the best and the worst moments; in particular, I thank my father, for he involuntarily taught me the art of avoiding unnecessary conflicts, and I owe heartfelt thanks to my mother, for she has never hesitated to stop me when she thought I was wrong, even though \ogni scarrafone `ebello a mamma soja" (even cockroaches are beautiful according to their moms). I am grateful to all the beautiful people I had met in Pisa, inside and outside Scuola Normale, thanks to whom I could take life as well as myself less seriously; the latter is one of the most important lessons I have ever learned. Above the others, I thank the members of the Spezia gang and the people who shared with me the five-year adventure in Scuola Normale: living and studying with them was the best I could hope for. The end of this experience was incredibly painful, yet this proves how invaluable these five years had been for me, and how beautiful the memories are. Besides, I thank in particular the colleagues who supported (and \sopported") me during the thesis period, especially Carmelo and Marcello for the numerous discussions had with them, and Matteo for he always gave me an opportunity to give vent to my feelings: they gave me the strength to overcome difficulties when I needed it most. Finally, I would like to thank Scuola Normale, for the scolarship it provided me, and the Physics department, for the Erasmus+ grant that supported me while I was working on this project. ii Contents Abstract i Acknowledgements ii Contents ii Abbreviations iv Symbols v 1 Introduction 1 1.1 Topological phases...................................1 1.2 Non-Equilibrium phases................................3 1.3 Non-equilibrium topological phases?.........................4 1.4 Outline of the work...................................6 1.4.1 Contents and structure of the thesis.....................6 1.4.2 Original contents................................8 2 Dissipative state preparation of paired fermionic states 10 2.1 Open quantum systems and their time evolution.................. 10 2.1.1 Density matrices................................ 11 2.1.2 Markovian Quantum Master Equations................... 12 2.1.3 Properties of the time evolution super-operator............... 14 2.2 Dark states of a QME................................. 15 2.3 The ground state of the Kitaev chain as a dark state................ 18 2.4 BCS-like mean field theory of the quartic QME................... 21 2.4.1 Equivalence of linear and quadratic Lindblad operators.......... 22 2.4.2 Dissipative Mean Field Theory at late times................. 23 2.5 Physical implementation of the protocol....................... 26 3 Keldysh functional integral for driven open fermionic systems 30 3.1 Towards a non-equilibrium Quantum Field Theory................. 30 3.2 Keldysh action for a quadratic QME......................... 32 3.2.1 Trotter formula for the time evolution of density matrices......... 32 3.2.2 Explicit derivation of the Keldysh functional integral............ 34 3.3 Keldysh contour ordering and Green's functions................... 36 3.3.1 Keldysh contour ordering........................... 36 3.3.2 Green's functions on the Keldysh basis.................... 38 3.4 Fundamental constraints on a non-equilibrium QFT................ 40 3.5 Keldysh action for an interacting QME........................ 44 3.6 Generating functionals in the Keldysh formalism.................. 46 ii Contents iii 4 Dyson-Schwinger formalism 50 4.1 Introduction to Dyson-Schwinger equations..................... 50 4.2 Generating Master Equation formulation....................... 51 4.3 Diagrammatic formulation............................... 53 4.3.1 From DSME to diagrammatics........................ 53 4.3.2 From diagrammatics to DSE......................... 54 5 Functional approaches to Dissipative Mean Field Theory 57 5.1 Keldysh action for the quartic model......................... 58 5.2 Hubbard-Stratonovich approach............................ 59 5.3 Dyson-Schwinger approach.............................. 63 5.3.1 Study of self-energy contributions....................... 64 5.3.2 Keldysh structure of DS solutions...................... 70 5.3.3 Mean field results from 1-loop truncation.................. 73 6 Summary and Conclusions 79 6.1 Summary........................................ 79 6.2 Interpretation and Outlook.............................. 80 A Formulas on Grassmann algebra and coeherent states 82 B Correlation functions for open quantum systems 84 C Notation and formulas for Keldysh-Nambu space 86 D DS Master Equation 89 D.1 General Derivation................................... 89 D.2 Explicit form for quartic actions........................... 91 E Exactness of Conjugation symmetry 92 F The Covariance Matrix and Keldysh-Majorana action 95 F.1 The Covariance Matrix and its equation of motion................. 95 F.2 Majorana doubling trick and Keldysh-Majorana action............... 98 F.3 Equation for the Covariance Matrix from Dyson-Schwinger............ 100 Bibliography 102 Abbreviations IQHE and FQHE Integer and Fractional Quantum Hall Effect QME (Markovian) Quantum Master Equation NESS Non-Equilibrium Steady State BEC Bose-Einstein Condensate RW Rotating Wave (approximation) BZ Brillouin Zone CPTP Completely Positive (and) Trace Preserving QFT Quantum Field Theory RAK Retarded-Advanced-Keldysh (basis) iv Symbols ρ^ density matrix of the system y y y y a^i; a^q (^ai = (^ai) ; a^q = (^aq) ) fermionic annihilation (creation) operators c^,γ ^ Majorana/Bogoliubov operators ^b (^by) bosonic annihilation (creation)
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