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Lecture Notes in Physics 874 M. Yu. Kagan Modern Trends in Superconductivity and Superfluidity Lecture Notes in Physics Volume 874 Founding Editors W. Beiglböck J. Ehlers K. Hepp H. Weidenmüller Editorial Board B.-G. Englert, Singapore, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg, Germany W. Hillebrandt, Garching, Germany M. Hjorth-Jensen, Oslo, Norway R. A. L. Jones, Sheffield, UK H. von Löhneysen, Karlsruhe, Germany M. S. Longair, Cambridge, UK M. L. Mangano, Geneva, Switzerland J.-F. Pinton, Lyon, France J.-M. Raimond, Paris, France A. Rubio, Donostia, San Sebastian, Spain M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Munchen, Germany J. D. 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Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. The series contentisindexed,abstractedandreferencedbymanyabstractingandinformation services, bibliographic networks, subscription agencies, library networks, and consortia. Proposals should be sent toa member ofthe Editorial Board,ordirectly to the managing editor at Springer: Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany [email protected] M. Yu. Kagan Modern Trends in Superconductivity and Superfluidity 123 M.Yu.Kagan Theoretical Department P.L.Kapitza Institutefor PhysicalProblems ofRussian Academyof Sciences and MoscowStateInstitute ofElectronics and Mathematics National Research University Higher SchoolofEconomics Moscow Russia ISSN 0075-8450 ISSN 1616-6361 (electronic) ISBN 978-94-007-6960-1 ISBN 978-94-007-6961-8 (eBook) DOI 10.1007/978-94-007-6961-8 SpringerDordrechtHeidelbergNewYorkLondon LibraryofCongressControlNumber:2013943720 (cid:2)SpringerScience+BusinessMediaDordrecht2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthe work. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the CopyrightClearanceCenter.ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The idea to write this book came to my mind after two advanced lecture courses which I have read in summer semesters of the year 2010 as Leverhulme visiting professor in Loughborough University UK and of the year 2011 as a visiting professor in LTPMS-Orsay (associated with the University Paris-Sud). The first Chapters of this book had the approbation during 16 years of my pedagogicalworkasaProfessoronGalitskiichairofTheoreticalNuclearPhysics inMoscowEngineeringPhysicalInstituteandduringautumnsemesteroftheyear 1994 in the University of Amsterdam. ThecourseisbasedontheoriginalresearchworkwhereIactivelyparticipated andcontributedasaprincipalscientistandagroupleaderinP.L.KapitzaInstitute for Physical Problems in Moscow during 30 years of my scientific career. It includesthesetofeightlecturesandeightseminarswhichcoverseveralimportant topics of the modern condensed matter physics, namely: • Quantum hydrodynamics offermionic and bosonic superfluids and supersolids; • BCS-BEC crossover in ultracold quantum gases; • Non-phonon mechanisms of superconductivity in high-T materials and other C unconventional superconductors; • Nanoscale phase separation in CMR-materials, heavy-fermions, and other strongly correlated electron systems; • Mesoscopic electron transport in multi-band and phase-separated metallic and oxide compounds. Ihopethebookwillbeusefulforundergraduatestudentsoftheseniorcourses, postgraduate students, and postdocs specializing in solid-state and low-tempera- ture physics. I am very grateful to my teachers, colleagues and pupils, first of all to A. S. Alexandrov, A. F. Andreev, A. G. Aronov, S. Balibar, M. A. Baranov, H. Beck, J. G. Bednorz, I. V. Brodsky, P. Brussard, H. W. Capel, H. Capellman, M. Capezzalli, R. Combescot, A. V. Chubukov, V. N. Devyatko, D. V. Efremov, M. A. Efremov, I. A. Fomin, R. Fresard, G. Frossati, P. Fulde, B. Halperin, Yu. A. Izumov, V. V. Kabanov, Yu. Kagan, L. V. Keldysh, A. V. Klaptsov, Yu. V.Kopaev, K.I.Kugel, D.I.Khomskii,Yu. A.Kosevich,F.V.Kusmartsev, A.V. Kuznetsov, A. I. Larkin, N. P. Laverov, Yu. E. Lozovik, I. M. Lifshitz, v vi Preface S.Maekawa,M.S.Mar0enko,B.E.Meierovich,A.P.Menushenkov,M.Mezard, P. Nozieres, S. L. Ogarkov, V.M. Osadchiev, A.V. Ozharovskii,A. Ya. Parshin, L. P. Pitaevskii, N. M. Plakida, F. Pobell, N. V. Prokof’ev, A. M. M. Pruisken, A. L. Rakhmanov, T. M. Rice, G. Sawatzky, A. O. Sboychakov, T. Schneider, G. V. Shlyapnikov, S. Stringari, V. V. Val0kov, C. M. Varma, D. Vollhardt, G.E.Volovik,J. T.M.Walraven,G.Wendin,Ch. vanWeert, P. Wölfle,and Ya. B.Zeldovich,whoencouragedmetostartwritingthisbookandgreatlyimproved its quality during our intensive scientific collaboration, numerous and sometimes very hot discussions both in Moscow and abroad. I am also very grateful to my family for their patience during the work on this book and acknowledge very important technical support from my assistants M. M. Markina and A. M. Padokhin. Moscow, November 2013 M. Yu. Kagan Corresponding Member of Russian Academy of Sciences Principal scientist in P.L. Kapitza Institute for Physical Problems and Professor of Physics in Moscow State Institute of Electronics and Mathematics, National Research University Higher School of Economics Contents Part I 1 Hydrodynamics of Rotating Superfluids with Quantized Vortices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 The Foundation of Landau Theory for Superfluid Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 The Essence of the Hydrodynamics. Description of the Goldstone Modes. . . . . . . . . . . . 5 1.1.2 Landau Scheme of the Conservation Laws. Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Sound Waves in Classical Liquid. Damping of Sound Waves. . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.4 Rotational Fluid. Vorticity Conservation. Inertial Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.5 Two-Velocity Hydrodynamics for Superfluid Helium.~v and~v , q and q . . . . . . . . . . . . . . . . . 13 n s n s 1.1.6 First and Second Sound Modes in Superfluid Liquid . . . . . . . . . . . . . . . . . . . . . . . 17 1.1.7 Gross–Pitaevskii Equation for Dilute Bose-Gas. Connection Between Superfluid Hydrodynamics and Microscopic Theory at T = 0 . . . . . . . . . . . . . . 19 1.2 Hydrodynamics of Rotating Superfluids . . . . . . . . . . . . . . . . 20 1.2.1 Andronikashvili Experiments in Rotating Helium. . . 20 1.2.2 Feynman-Onsager Quantized Vortices. Critical Angular Velocities X and X . . . . . . . . . . . . . . . 22 C1 C2 1.2.3 Vortex Lattice. Nonlinear Elasticity Theory. Vorticity Conservation Law. . . . . . . . . . . . . . . . . . 25 1.2.4 Hydrodynamics of Slow Rotations. Hall-Vinen Friction Coefficients b and b0. . . . . . . . . . . . . . . . . 28 1.2.5 Linearization of the Elasticity Theory. Connection Between~v and~u in Linearized Theory. . . . . . . . . . 31 s vii viii Contents 1.2.6 Collective Modes of the Lattice. Tkachenko Waves and Lord Kelvin Waves. Melting of the Vortex Lattice. . . . . . . . . . . . . . . . . . . . . . . 35 1.3 Hydrodynamics of Fast Rotations. . . . . . . . . . . . . . . . . . . . . 40 1.3.1 The Foundation of the Hydrodynamics of Fast Rotations. The Role of Umklapp Processes . . . . . . . 40 1.3.2 The System of the Nonlinear Equations for the Hydrodynamics of Fast Rotations. . . . . . . . . 42 1.3.3 Linearized System of Equations of Fast Rotations. The Spectrum and the Damping of the Second Sound Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 1.4 Opposite Case of a Single Bended Vortex Line for Extremely Slow Rotations (X * X ). . . . . . . . . . . . . . . 47 C1 1.4.1 Stabilization of the Bending Oscillations by Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.4.2 Visualization of the Vortex Lattice in Rotating Superfluid. Packard Experiments . . . . . . . . . . . . . . 50 1.4.3 Contribution to Normal Density and Specific Heat from Bended Vortex Lines. . . . . . . . . . . . . . . 51 1.5 Experimental Situation and Discussion. How to Achieve the Limit of the Fast Rotations at Not Very High Frequencies in He II–3He Mixtures and in Superfluid 3He-B . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2 Quantum Crystals. The Search for Supersolidity. . . . . . . . . . . . . 57 2.1 Quantum Crystals. Phase-Diagram. The Search for Supersolidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.1.1 Lindemann and de Boer Parameters . . . . . . . . . . . . 58 2.1.2 Flow of Zero Vacancies. Andreev–Lifshitz Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.1.3 Chan Experiments with Rotating Cryostat. The Search for Supersolidity in Solid 4He. . . . . . . . 65 2.2 The Surface Physics of Quantum Crystals. Atomically Smooth and Atomically Rough Surfaces . . . . . . . . . . . . . . . . 66 2.2.1 The Concept of the Mobile Rough Interface Between Solid 4He and Superfluid He-II. . . . . . . . . 67 2.2.2 Growth and Melting Shape of a Crystal. . . . . . . . . . 68 2.2.3 Melting-Crystallization Waves and Phase Equilibrium on the Mobile Rough Surface. . . . . . . . 68 2.2.4 Rayleigh Waves on Rough and Smooth Surfaces . . . 69 2.2.5 Roughening Transition. . . . . . . . . . . . . . . . . . . . . . 70 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Contents ix 3 Melting-Crystallization Waves on the Phase-Interface Between Quantum Crystal and Superfluid . . . . . . . . . . . . . . . . . . . . . . . . 79 3.1 The Surface Hydrodynamics for Rough Interface at Low Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.1.1 Linear Equations of Surface Hydrodynamics in the Absence of Stationary Surface Flows. . . . . . . 81 3.1.2 The Spectrum of Melting-Crystallization Waves. . . . 87 3.1.3 The Growth Coefficient: Damping of the Melting-Crystallization Waves . . . . . . . . . . . 89 3.1.4 The Instability of Superfluid Tangential Flows on the Mobile Phase-Interface . . . . . . . . . . . . . . . . 93 3.1.5 The Spectrum of the Rayleigh Waves on the Rough Surface . . . . . . . . . . . . . . . . . . . . . . 96 3.1.6 The Angles of the Total Internal Reflection: Excitation of the Surface Wave by the Bulk Second Sound Wave. . . . . . . . . . . . . . . . . . . . . . . 98 3.2 Surface Hydrodynamics on the Mobile Interface at T = 0 and in the Presence of 3He Impurities. . . . . . . . . . . . . 102 3.2.1 Equations of the Surface Hydrodynamics at T = 0 and in the Presence of the Impurities . . . . . . 103 3.2.2 The Surface Dissipative Function and Kapitza Thermal Resistance. . . . . . . . . . . . . . . 107 3.2.3 Damping of Melting-Crystallization Waves . . . . . . . 109 3.2.4 Impurity Contribution to the Kapitza Thermal Resistance at Low Temperatures. . . . . . . . . . . . . . . 112 3.2.5 Cherenkov Emission of the Second Sound Quanta by the Thermal Surface Waves. . . . . . . . . . . . . . . . 113 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4 Quantum Hydrodynamics of the p-Wave Superfluids with the Symmetry of 3He-A. . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.1 Orbital Hydrodynamics of Bosonic and Fermionic Superfluids with the Symmetry of A-phase of 3He . . . . . . . . . . . . . . . . . 118 4.1.1 Orbital Hydrodynamics and Collective Modes in Bosonic Regime . . . . . . . . . . . . . . . . . . . . . . . . 119 4.1.2 Orbital Waves: The Paradox of the Intrinsic Angular Momentum and Anomalous Current in Fermionic Superfluids . . . . . . . . . . . . . . . . . . . . 122 4.2 Two Approaches to a Complicated Problem of Anomalous Current in Fermionic (BCS) A-phase . . . . . . . . . . . . . . . . . . 126 4.2.1 Supersymmetric Hydrodynamics of the A-phase. . . . 126 4.2.2 A Different Approach Based on the Formal Analogy with Quantum Electrodynamics. . . . . . . . . 135