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SUPERFLUID STATES of MATTER This Page Intentionally Left Blank SUPERFLUID STATES of MATTER SUPERFLUID STATES OF MATTER This page intentionally left blank SUPERFLUID STATES OF MATTER Boris Svistunov Egor Babaev Nikolay Prokof’ev Chapter 10 is written in coauthorship with Evgeny Kozik. Manuscript editing by Elizabeth Earl Phillips. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Boris Svistunov, Egor Babaev, and Nikolay Prokof’ev CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20141202 International Standard Book Number-13: 978-1-4398-0276-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. 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CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface xv Authors xix Abbreviations xxi I Superfluidity from a Classical-Field Perspective 1 1 Neutral Matter Field 3 1.1 Classical Hamiltonian Formalism . 4 1.1.1 Hamiltonian Mechanics . 4 1.1.2 Complex-Valued Scalar Field as a Canonical Variable . 6 1.1.3 Canonical Transformation . 8 1.1.4 Normal Modes by Bogoliubov Transformation . 9 1.1.5 Statistical and Dynamical Instabilities . 13 1.2BasicDynamicandStaticProperties.................. 14 1.2.1 Nonrelativistic Limit for a Complex-Valued ScalarField............................. 14 1.2.2 Gross–Pitaevskii Equation, aka Nonlinear SchrodingerEquation.......................¨ 19 1.2.3 Healing Length . 21 1.2.4 Vortices . 23 1.2.5 Topological Invariant, Persistent Current, and Role ofVortices............................. 28 1.2.6 Normal Modes, Landau Criterion, and CriticalVelocity.......................... 30 1.2.7 Flow-Induced Density Wave: Supersolid . 34 1.3MatterFieldunderRotation....................... 35 1.3.1 Rotating Frame . 36 1.3.2 Vector Potential, Gauge Equivalence . 37 1.3.3 Vortex Array . 38 1.3.4 Twisted Boundary Condition . 41 1.3.5 Josephson Junction, SQUID . 44 2 Superfluidity at Finite Temperatures and Hydrodynamics 47 2.1EquilibriumStatisticsofaSuperfluid.................. 47 2.1.1 Gibbs Distribution in the Presence of a Superflow . 47 v vi Contents 2.1.2 Two-Fluid Picture: Superfluid and NormalComponents....................... 49 2.1.3 Superfluid and Normal Densities as Static ResponseFunctions........................ 51 2.1.4 Fluctuations of the Superfluid Velocity Field, Topological versusGenuineLong-RangeOrder............... 54 2.1.5 Generic Off-Diagonal Correlations: Long-Range Asymptotics, Mean-Field and Non-Mean-Field Regimes, andElongatedSystems...................... 59 2.1.6 Statistics of Supercurrents and Mesoscopic Effects...... 63 2.2 Basics of Superfluid Hydrodynamics, Thermomechanical Effects, HydrodynamicHamiltonianandAction................ 66 2.2.1 Time Evolution of the Phase . 67 2.2.2 Super Thermal Conductivity, Counterflow, and the Fountain Effect.......................... 68 2.2.3 AC Josephson Effect........................ 72 2.2.4 Hydrodynamic Hamiltonian and Action . 72 References.................................... 76 3 Superfluid Phase Transition 77 3.1 XY UniversalityClass .......................... 77 3.2 High-Temperature Expansion for the XY Model ........... 82 3.3 J-CurrentModels............................. 87 3.4 Dual Descriptions of Normal and Superfluid Phases . 88 3.5 High-Temperature Expansion for the |ψ|4 Model ........... 91 References.................................... 93 4 Berezinskii–Kosterlitz–Thouless Phase Transition 95 4.1StatisticsofVorticesatLargeScales................... 96 4.1.1 EffectiveActionforVortices................... 96 4.1.2 Nelson–Kosterlitz Relation . 98 4.1.3 Dilute Gas of Vortex Pairs . 98 4.2 Kosterlitz–Thouless Renormalization-Group Theory . 101 4.2.1 Flow Equations . 101 4.2.2 Off-Diagonal Correlations in the Vicinity of the BKTTransition..........................105 4.2.3 Alternative Form of the Effective Energy for Vortices . 106 References....................................109 II Superconducting and Multicomponent Systems 111 5 Charged Matter Fields 113 5.1U(1)GaugeTheory............................114 5.1.1 Nonrelativistic |ψ|4 GaugeModel................114 Contents vii 5.1.2 Superfluid Velocity and Topological Invariant . 117 5.1.3 Magnetic Flux Quantization . 119 5.1.4 London Law . 120 5.1.5 Hydroelectrodynamic Action, Plasmons, and Anderson Effect.......................121 5.2 U(1) Mean-Field Gauge Theories. London and Ginzburg–LandauModels........................124 5.2.1 London Model. Hydromagnetostatic Regime . 125 5.2.2 Behavior of a Magnetic Field in a Superconductor. The Meissner Effect and London Penetration Length . 126 5.2.3 Ginzburg–Landau Model . 127 5.2.4 Ginzburg–Landau Equations . 129 5.3Type-1andType-2Superconductors..................130 5.3.1 Coherence Length . 130 5.3.2 Ginzburg–Landau Parameter and Surface Tension between theSuperconductingandNormalPhases............131 5.3.3 Intermediate Landau State . 135 5.4VorticesinaSuperconductor ......................135 5.4.1 Vortices and Their Interaction in the London Model . 137 5.4.2 Vortices and Their Interaction in the Ginzburg–LandauModel.....................140 5.5 Upper and Lower Critical Magnetic Fields (Hc1 and Hc2) ......144 5.5.1 Lower Critical Field Hc1 .....................145 5.5.2 Vortex-Lattice State . 146 5.5.3 Upper Critical Field Hc2 .....................147 5.5.4 Nonpairwise Intervortex Forces . 149 5.6 Critical Coupling κ = 1: Bogomolny Bound and Equations . 149 5.7 Summary of the Magnetic Response and Vortex Liquid . 151 5.8Magneto-RotationalIsomorphism ...................154 5.9 Superconducting Phase Transition . 156 References....................................161 6 Multicomponent Superconductors and Superfluids. Superconducting and Metallic Superfluids 163 6.1 Mixtures of Superfluids and Entrainment Effect ...........164 6.2MulticomponentSuperconductors...................165 6.2.1 Liquid Metallic Hydrogen–Type System . 165 6.2.2 Other Multicomponent Ginzburg–Landau Models . 166 6.3FractionalVortices ............................167 6.4 Superfluid Sector of Liquid Metallic Hydrogen–TypeSystem .........................170 6.4.1 Nonuniversal Quantization of Superfluid Velocity . 170 6.4.2 Rotational Response in the Neutral Sector . 171 viii Contents 6.5 Rotational Response of a Charged Sector in the Multicomponent Superconductor and Violation of the London Law . 174 6.6ViolationofLondonElectrodynamics .................175 6.6.1 Fractional Vortices in the Two-Component Ginzburg–Landau Model and Magnetic Flux Delocalization andFieldInversion........................176 6.6.2 Magnetic Field Behavior and Higher Order Derivative Terms intheFree-EnergyFunctional..................178 6.7 Skyrmion and Hopfion Topological Defects . 182 6.7.1 Skyrmions in Two-Component Superconductors . 182 − 6.7.2 CPN 1 Skyrmions.........................183 6.7.3 Hopfions . 184 6.8 Magnetic Responses and Type-1.5 Superconductivity . 185 6.8.1 Phase Diagrams and Magnetization Curves . 188 6.8.2 Vortex Clusters and Phase Separation in Vortex and MeissnerDomains.........................188 6.9 Effects of Intercomponent Interactions in MultibandSystems............................193 6.9.1 Intercomponent Josephson Coupling . 193 6.9.2 Hybridization of Coherence Lengths in MultibandSystems........................194 6.9.3 Passive-Band Superconductivity . 196 6.9.4 Spontaneous Breaking of Time-Reversal Symmetry . 199 6.9.5 Fractional Vortices and Domain Walls in the Presence of Symmetry-Breaking Interactions . 200 6.10 Composite U(1) Order and Coflow/Counterflow Superfluidity ...............................201 6.10.1 Metallic Superfluid . 203 6.10.2 Counterflow Superfluidity in N-Component ChargedSystems.........................204 6.10.3 Coflow/Counterflow Composite Superfluidity in NeutralMixtures.........................205 6.10.4 General Classification of N-Component Coflow/Counterflow CompositeSuperfluids......................206 6.10.5 Multicomponent j-Current Models. Universality Class of the Phase Transition to Composite Superfluid State
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