Modern Aspects of Superconductivity Theory of Superconductivity Second Edition TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk Modern Aspects of Superconductivity Theory of Superconductivity Second Edition Sergei Kruchinin Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. MODERN ASPECTS OF SUPERCONDUCTIVITY Theory of Superconductivity Second Edition Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. 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ISBN 978-981-123-451-4 (hardcover) ISBN 978-981-123-452-1 (ebook for institutions) ISBN 978-981-123-453-8 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/12215#t=suppl Desk Editor: Rhaimie Wahap Typeset by Stallion Press Email: [email protected] Printed in Singapore RRhhaaiimmiiee -- 1122221155 -- MMooddeerrnn AAssppeeccttss ooff SSuuppeerrccoonndduuccttiivviittyy..iinndddd 11 2233//33//22002211 1111::3333::5533 aamm March24,2021 20:14 ModernAspects ofSuperconductivity (2ndEdn.)-9.61inx6.69in b4202-fm pagev “The single reason for our inability to treat the problems of superconductivity consists in the absence of a sufficient imagination.” Richard P. Feynman v TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk March24,2021 20:14 ModernAspectsofSuperconductivity (2ndEdn.)-9.61inx6.69in b4202-fm pagevii Preface The study in the field of superconductivity theory is one of the most bright, fruitful,andpromisingtrendsinthetheoretical physicsofcondensedmatter, since superconductivity remains to be one of the most interesting research areas in physics. The goal of the book is to give representation of certain modern aspects of superconductivity. We discuss such important aspects of the theory of superconductivity as the nature of high-Tc superconductivity, two- gap superconductivity, room-temperature superconductivity, mesoscopic superconductivity, the pairing state and the mechanism of cuprate high-Tc superconductivity. We will consider also the up-to-date state of the problem of real construction of a quantum computer on the base of superconducting units. InChapter1,weconsiderthefield-theoreticmethodofsuperconductivity and discuss the basic idea of superconductivity and the elaboration of the Ginzburg–Landau and BCS theories in the frame of many-particle quantum field theory. In Chapter 2, we consider such topics of high-Tc superconductivity as the structures of high-Tc superconductors, phase diagrams, and the problem of pseudogaps andanalyze the mechanisms of superconductivity.We present general arguments as for the pairing symmetry in cuprate superconductors andinvestigatetheirthermodynamicalpropertieswithinthespin-fluctuation mechanismofsuperconductivity,byusingthemethodoffunctionalintegrals. In Chapter 3, we consider the new class of high-temperature supercon- ductors such as Fe-based laminar compounds.We analyze thephysical prop- erties and electron models of Fe-based high-temperature superconductors. Theparticularinterest inthemisexplained beperspectivesoftheir practical application. In the chapter, we give the complete pattern of the formation of their physical properties within theoretical models on the basis of their electron structure. vii March24,2021 20:14 ModernAspectsofSuperconductivity (2ndEdn.)-9.61inx6.69in b4202-fm pageviii viii Modern Aspects of Superconductivity: Theory of Superconductivity Chapter 4 concentrates on the two-band and multiband superconductiv- ity. Weconsiderthephysicalpropertiesof superconductorMgB anduseour 2 two-band model to explain the two coupled superconductor’s gaps of MgB . 2 TostudytheeffectoftheincreasingTc inMgB2,weusedtherenormalization group approach and phase diagrams. In the field of superconductivity, we meet the problem-maximum — it consists in the creation of room- temperature superconductors. We consider this problem in our book, and give some recommendations on the search for these superconductors. We consider the problem of high-temperature superconductivity at high pressures in hydrides and present the E´liashberg–Migdal theory explaining the room-temperature superconductivity in this case. Chapter 5 deals with the problem of nanoscale superconductivity. We consider the two-band superconductivity in ultrasmall grains, by extending the Richardson exact solution to two-band systems, and develop the theory of interactions between nano-scale ferromagnetic particles and superconduc- tors. The properties of nano-sized two-gap superconductors and the Kondo effect in superconducting ultrasmall grains are investigated as well. At the present time, popular is the topic of the nano-engineering with superconducting states. In this connection, we deal with the physics of magnetic dots on the superconductors and the problem of superconducting wires. We analyze ideas of the quantum information and quantum calcula- tions with the use of superconducting nano-units. We describe the theory of the Josephson effect and its applications in quantum calculations, as well as the idea of qubits. We consider a quantum computer on the basis of superconducting qubits and give a short review of the modern state of the problem of physical realization of quantum computers. We present the idea of a topological quantum computer with qubits in the basis of Majorana fermion quasiparticles. This bookdeals with a widescopeof theoretical andexperimental topics in superconductivity and is written for advanced students and researchers in the field of superconductors. Kyiv, November 2020 S. Kruchinin March24,2021 20:14 ModernAspectsofSuperconductivity (2ndEdn.)-9.61inx6.69in b4202-fm pageix Contents Preface vii 1. Theory of Superconductivity 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Spinor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Noether theorem and Nambu–Goldstone theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Non-interacting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Interacting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5.1 Unrestricted HF . . . . . . . . . . . . . . . . . . . . . . 20 1.5.2 Gap equation for superconductivity. . . . . . . . . . 21 1.6 Illustrative Example, Critical Temperature . . . . . . . . . . 22 1.6.1 Bond alternation . . . . . . . . . . . . . . . . . . . . . . 23 1.6.2 Deformation energy . . . . . . . . . . . . . . . . . . . . 24 1.6.3 Polyacene, gap equation, critical temperature . . . 25 1.6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.7 Linear Response Magnetic Resonance in Normal and Superconducting Species: Spin-Lattice Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.7.2 T1 in NMR . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.7.3 Theory with Green’s function . . . . . . . . . . . . . 35 1.7.4 Non-interacting . . . . . . . . . . . . . . . . . . . . . . . 37 1.7.5 Interacting; normal . . . . . . . . . . . . . . . . . . . . 39 ix