kleinert. collective quantum fields

Collective Quantum Fields in Plasmas, Superconductors, and Superfluid 3 He, Collective Quantum Fields in Plasmas, Superconductors, and Superfluid 3 He Hagen Kleinert Professor of Physics Freie Universit¨at Berlin Preface Under certain circumstances, many-body systems behave approximatly like a gas of weakly interacting collective excitations. Once this happens it is desirable to replace the original action involving the fundamental fields (electrons, nucleons, 3 He, 4 He atoms, quarks etc.) by another one in which all these excitations appear as explicit independent quantum fields. It will turn out that such replacements can be performed in many different ways without changing the physical content of the theory. Sometimes, there esists a choice of fields associated with dominant collective excitations displaying weak residual interactions which can be treated perturbatively. Then the collective field language greatly simplifies the description of the physical system. It is the purpose of this book to discuss a simple technique via Feynman path integral formulas in which the transformation to collective fields amounts to mere changes of integration variables in functional integrals. After the transformation, the path formulation will again be discarded. The resulting field theory is quantized in the standard fashion and the fundamental quanta directly describe the collective excitations. For systems showing plasma type of excitations, a real field depending on one space and time variable is most convenient to describe all physics. For the opposite situation in which dominant bound states are formed, a complex field depending on two space and one or two time coordinates will render th emore ecnonomic description. Such fields will be called bilocal. If the potential becomes extremely short range, the bilocal field degenerates into a local field. In the latter case a classical approximation to the action of a superconducting electron system has been known for some time: the Ginzburg-Landau equation. The complete bilocal theory has been studied in elementary-particle physics where it plays a role in the transition from inobservable quark to observable hadron fields. The change of integration variables in path integrals will be shown to correspond to an exact resummation of the perturbation series thereby accounting for phenom- ena which cannot be described perturbatively. The path formulation has the great advantage of translating all quantum effects among the fundamental particles com- pletely into the field language of collective excitations. All radiative corrections may be computed using only propagators and interaction vertices of the collective fields. The method presented here is particularly powerful when a system is in a region where several collective effects becomes simultaneously important. An example is the electron gas at lower density where ladder graphs gain increasing importance vii viii with respect to ring graphs thus mixing plasma and pair effects. 3 He pair effects are dominant but plasma effects provide strong corrections. In Part I of the book we shall illustrate the functional approach by discussing first conventional systems such as electron gas and superconductors. Also, we investigate a simple soluble model to understand precisely the mechanism of the functional field transformations as well as the relation between the Hilbert spaces generated once from fundamental and once from collective quantum fields. In Part II we apply the same techniques to superfluid 3 He. In Part III, finally, we illustrate the working of the functional techniques by applying it to some simple solvabel models. Berlin, January 1990 H. Kleinert H. Kleinert, COLLECTIVE QUNATUM FIELDS Contents I Functional Integral Techniques 1 1 Nonrelativistic Fields 4 1.1 Free Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Functional Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Equivalence of Functional and Operator Methods . . . . . . . . . . . 12 1.5 Grand-Canonical Ensembles at Zero Temperature . . . . . . . . . . 13 2 Relativistic Fields 19 2.1 Lorentz and Poincar´e Invariance . . . . . . . . . . . . . . . . . . . . 19 2.2 Relativistic Free Scalar Fields . . . . . . . . . . . . . . . . . . . . . 24 2.2.1 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Relativistic Free Fermi Fields . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Perturbation Theory of Relativistic Fields . . . . . . . . . . . . . . . 33 II Plasmas and Superconductors 39 1 Introduction 41 2 Plasmas 42 3 Superconductors 48 3.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Local Interaction and Ginzburg-Landau Equations . . . . . . . . . . 55 3.3 Inclusion of Electromagnetic Fields into the Pair Field Theory . . . 63 3.4 Far below the Critical Temperature . . . . . . . . . . . . . . . . . . 66 3.4.1 The Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.2 The Free Pair field . . . . . . . . . . . . . . . . . . . . . . . 70 3.5 Ground State Properties . . . . . . . . . . . . . . . . . . . . . . . . 83 3.5.1 Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.5.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.5.3 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.6 Plasmons versus Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Appendix 3A Propagator of the Bilocal Pair Field . . . . . . . . . . . . . 92 ix x Appendix 3B Fluctuations around the Composite Field . . . . . . . . . . 94 III Superfluid 3 He 101 1 Introduction 103 2 Preparation of Functional Integral 107 2.1 The Action of the System . . . . . . . . . . . . . . . . . . . . . . . . 107 2.2 From Particles to Quasiparticles . . . . . . . . . . . . . . . . . . . . 109 2.3 The Approximate Quasiparticle Action . . . . . . . . . . . . . . . . 110 2.4 The Effective Interaction . . . . . . . . . . . . . . . . . . . . . . . . 112 2.5 Pairing Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.6 Dipole Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3 Transformation from Fundamental to Collective Fields 117 4 General Properties of Collective Action 123 Appendix 4A Comparison with O(3)-Symmetric Linearσ-Model . . . . . 127 Appendix 4B Other Possible Phases . . . . . . . . . . . . . . . . . . . . 128 5 Hydrodynamic Limit Close to T c 141 Appendix 5A Hydrodynamic Coefficients for T ≈ T c . . . . . . . . . . . . 149 6 Bending the Superfluid 3 He-A 153 6.1 Monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.2 Line Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.3 Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.4 Localized Lumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.5 Use of Topology in the A-Phase . . . . . . . . . . . . . . . . . . . . 163 6.6 Topology in the B-Phase . . . . . . . . . . . . . . . . . . . . . . . . 165 7 Hydrodynamic Limit at any Temperature T < T c 169 7.1 Fermi Liquid Corrections . . . . . . . . . . . . . . . . . . . . . . . . 185 7.2 Ground State Properties . . . . . . . . . . . . . . . . . . . . . . . . 192 7.2.1 Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.2.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.2.3 Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . 200 Appendix 7A Hydrodynamic Coefficients for all T ≤ T c . . . . . . . . . . 201 8 Large Currents and Magnetic Fields in the Ginzburg-Landau Regime 206 8.1 B-Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.1.1 Neglecting Gap Distortion . . . . . . . . . . . . . . . . . . . 207 8.1.2 Including a Magnetic Field . . . . . . . . . . . . . . . . . . 210 H. Kleinert, COLLECTIVE QUNATUM FIELDS xi 8.1.3 Allowing for a Gap Distortion . . . . . . . . . . . . . . . . . 212 8.2 A-phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.3 Critical Current in Other Phases for T ∼ T c . . . . . . . . . . . . . . 218 9 Is 3 He-A a Superfluid? 227 9.1 Magnetic Field and Transition between A- and B-Phases . . . . . . 252 Appendix 9A Generalized Ginzburg-Landau Energy . . . . . . . . . . . . 254 10 Currents at Any Temperature T ≤ T c 255 10.1 Energy at Nonzero Velocities . . . . . . . . . . . . . . . . . . . . . . 255 10.2 The Gap Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 10.3 Superfluid Densities and Currents . . . . . . . . . . . . . . . . . . . 264 10.4 Critical Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 10.5 Ground State Energy at Large Velocities . . . . . . . . . . . . . . . 270 10.6 Fermi Liquid Corrections . . . . . . . . . . . . . . . . . . . . . . . . 270 11 Collective Modes in Presence of Current . . . 274 11.1 Quadratic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 274 11.2 Time-Dependent Fluctuations at Infinite Wavelength . . . . . . . . . 277 11.3 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 11.4 Simple Limiting Results at Zero Gap Deformation . . . . . . . . . . 283 11.5 Static Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 12 Fluctuation Coefficients 286 12.1 Stability of Super-Flow in the B-phase under Small Fluctuations for T ∼ T c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 IV Hadronization of Quark Theories 303 1 Introduction 306 2 Abelian Quark Gluon Theory 308 3 The Limit of Heavy Gluons 325 Appendix 3A Remarks on the Bethe-Salpeter Equation . . . . . . . . . . 341 Appendix 3B TVertices for Heavy Gluons . . . . . . . . . . . . . . . . . 348 Appendix 3C Some Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 350 IV Exactly Solvable Field Theoretic Models via Collec- tive Quantum Fields 303 xii 1 Low-Dimensional Models 305 1.1 The Pet Model in One Time Dimension . . . . . . . . . . . . . . . . 305 1.2 The Generalized BCS Model in a Degenerate Shell . . . . . . . . . . 313 1.3 The Hilbert Space of Generalized BCS Model . . . . . . . . . . . . . 323 2 Massive Thirring Model in 1+1 Dimensions 327 3 O(N)-Symmetric Four-Fermi Interaction in 2 +  Dimensions 330 3.0.1 Relation between Pairing and Gross-Neveu Model . . . . . . 355 3.0.2 Comparison with O(N)-Symmetric φ 4 Model . . . . . . . . 357 3.1 Finite-Temperature Properties . . . . . . . . . . . . . . . . . . . . . 361 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Notes and References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 H. Kleinert, COLLECTIVE QUNATUM FIELDS List of Figures 1.1 Contour C in the complex z-plane . . . . . . . . . . . . . . . . . . . 16 2.1 The pure current piece of the collective action . . . . . . . . . . . . 44 2.2 The non-polynomial self-interaction terms of the plasmons . . . . . 44 2.3 The free plasmon propagator . . . . . . . . . . . . . . . . . . . . . . 45 3.1 The fundamental particles entering and any diagram only via the external currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 The free pair field following the Bethe-Salpeter equation . . . . . . . 52 3.3 Graphical content of free pair propagator . . . . . . . . . . . . . . . 54 3.4 The self-interaction terms of the non-polynomial pair Lagrangian . 55 3.5 The free part of the pair field ∆ Lagrangian . . . . . . . . . . . . . 57 3.6 Energy gap of superconductor as a function of temperature . . . . . 69 3.7 Temperature behaviour of superfluid density ρ s /ρ (Yoshida function) and the gap function ¯ρ s /ρ . . . . . . . . . . . . . . . . . . . . . . . 80 3.8 Temperature behaviour of the inverse square coherence length ξ −2 (T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.9 The gap function ˜ρ s appearing in the condensation energy of a su- perconductor as a function of temperature . . . . . . . . . . . . . . . 87 3.10 Condensation energy of a superconductor . . . . . . . . . . . . . . . 87 3.11 The temperature behaviour of the condensation entropy . . . . . . . 90 3.12 Total specific heat of a superconductor as a function of temperature 91 1.1 Interatomic potential between 3 He atoms as a function of the dis- tance r. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 1.2 Imaginary part of the susceptibility caused by repeated exchange of spin fluctuations,as a function of energy ω. . . . . . . . . . . . . . . 104 1.3 Phase diagram of 3 He plotted against temperature, pressure, and magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1 Three fundamental planar textures, splay, bend, and twist of a di- rector field in liquid crystals . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Sphere with no, one, or two handles and Euler characteristics E = 2, E = 0, E = −2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.3 Local tangential coordinate system n, t, i for an arbitrary curve on the surface of a sphere. . . . . . . . . . . . . . . . . . . . . . . . . . 147 xiii xiv 6.1 The l  d field lines in a spherical container. . . . . . . . . . . . . . 154 6.2 Two possible parametrization of a sphere, either with two singular- ities, as the standard geographic coordinate frame on the globe, or with one singularity, as shown in the lower figure. . . . . . . . . . . 155 6.3 Spectra of Goldstone bosons versus gauge bosons. . . . . . . . . . . 156 6.4 Cylindrical container with the l  d field lines spreading outwards when moving upwards. . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.5 Field vectors in a composite soliton . . . . . . . . . . . . . . . . . . 160 6.6 Nuclear magnetic resonance frequencies in a superfluid 3 He-A sample in an external magnetic field . . . . . . . . . . . . . . . . . . . . . . 162 6.7 Vectors of orbital and spin orientation in The A-phase of superfluid 3 He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.8 Parameter space of 3 He-B containing that of the rotation group . . 166 6.9 Possible path followed by the order parameter in a planar texture (soliton) when going from z = −∞ to z = +∞ . . . . . . . . . . . . 167 6.10 Another possible class of solitons . . . . . . . . . . . . . . . . . . . 168 7.1 The fundamental quantities of superfluid 3 He-B and A are shown as a function of temperature. The superscript FL denotes the Fermi liquid corrected values. . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.2 Temperature behavior of the superfluid densities in the A- and B- phase of superfluid 4 He. . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.3 The superfluid stiffness functions K t , K b , K s of the A-phase as a function of temperature . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.4 The remaining hydrodynamic parameters of superfluid 3 He-A shown as a function of temperature together with their Fermi liquid cor- rected values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 7.5 Condensation energies of A- and B-phases as functions of temperature197 7.6 The temperature behavior of the condensation entropies in B- and A-phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.7 Specific heat of A- and B-phases as a function of temperature . . . . 201 8.1 Shape of potential determining stability of superflow . . . . . . . . 209 9.1 Superflow in a torus which can relax by vortex rings forming, in- creasing, and meeting their death at the surface . . . . . . . . . . . 227 9.2 In the presence of a superflow in 3 He-A, the l-vector is attracted to the direction of flow . . . . . . . . . . . . . . . . . . . . . . . . . . 233 9.3 Doubly connected parameter space of the rotation group correspond- ing to integer and half-integer spin representations . . . . . . . . . . 234 9.4 Helical texture in the presence of a supercurrent . . . . . . . . . . . 235 9.5 Three different regions in which there are equilibrium configurations of the texture at H = 0 (schematically) . . . . . . . . . . . . . . . . 237 H. Kleinert, COLLECTIVE QUNATUM FIELDS [...]... energy-momentum, the thermodynamic formulation specifies the way to specifies how to avoid singularities H Kleinert, COLLECTIVE QUNATUM FIELDS H Kleinert, COLLECTIVE QUNATUM FIELDS /amd/apollo/0/home/ag-kleinert/kleinert/kleinert/books/cqf/techniqs.tex June 21, 2001 2 Relativistic Fields We shall also study collective pheomena in relativistic fermion systems For this we shall need fields describing relativistic... entire ladder of fermion pairs Such a formulation can also be given to quantum electrodynamics of electrons and positrons, where the bare mesons are positronium atoms [9] 3 H Kleinert, COLLECTIVE QUNATUM FIELDS /amd/apollo/0/home/ag-kleinert/kleinert/kleinert/books/cqf/techniqs.tex June 21, 2001 1 Nonrelativistic Fields 1.1 Free Fields Consider free nonrelativistic particles, whose energy ε depends... presented in what follows As soon as bound states or collective excitations are formed, it is very suggestive to use them as new quantum fields rather than the original fundamental particles ψ The goal would then to be rewrite the expression (1.39) for Z[η ∗ , η] in terms of new fields whose unperturbed propagator has the free energy spectrum of the bound states or collective excitations and whose Aint describes... conceive The ideal theoretical framework for describing a system in terms of the new quantum fields Z[η ∗ , η] is offered by the above-intoduced functional integral techniques [1, 2, 3] In these, changes of fields amount to changes of integration variables, as we shall see in the sequel H Kleinert, COLLECTIVE QUNATUM FIELDS 11 1.3 Functional Formulation 1.3 Functional Formulation In the functional integral... 2πi dξ √ ξ = 1, 2πi dξ n √ ξ = 0, 2πi n>1 (1.17) for real ξ, from which we derive dξ dξ ∗ dξ √ √ √ = 0, iξ ∗ ξ = 1, 2πi 2πi 2πi dξ ∗ dξ √ √ (ξ ∗ ξ)n = 0, n > 1, 2πi 2πi H Kleinert, COLLECTIVE QUNATUM FIELDS (1.18) 7 1.1 Free Fields for complex ξ ∗ , ξ Note that these integration rules imply that under a linear change of a Grassmann integration variable, the integral multiplies by the inverse of the usual... 3.1 Solution of the temperature dependent gap equation, showing the decrease of the fermion mass M (T ) = Σ(T ) with increasing temperature T /Tc 364 H Kleinert, COLLECTIVE QUNATUM FIELDS List of Tables 1.1 There is a factor of roughly 1000 between the characteristic quantities of superconductors and 3 He 106 2.1 a s s Pressure dependence of Landau... F0 , and F1 in 3 He together with the molar volume and the effective mass ratio m∗ /m 111 8.1 Parameters of the critical currents in all theoretically known phases 221 xvii xviii H Kleinert, COLLECTIVE QUNATUM FIELDS Part I Functional Integral Techniques 1 Introduction In this book we shall study certain classes of phenomena which occur in systems of many fermions interacting with each other via two-body... sum of normal products with all possible contractions taken via Feynman propagators The formula for an arbitrary functional of free fields ψ, ψ ∗ is T F [ψ ∗ , ψ] = e d3 xdtd3 x dt H Kleinert, COLLECTIVE QUNATUM FIELDS δ G (x,t;x δψ(x,t) 0 δ ,t ) δψ∗ (x,t ) : F [ψ ∗ , ψ] : (1.54) 13 1.5 Grand-Canonical Ensembles at Zero Temperature Applying this to 0|T F [ψ ∗ , ψ]|0 = 0|T exp i dxdt(ψ ∗ η + η ∗ ψ) |0... Matsubara frequencies accounts for the density of these frequencies yielding the correct T → 0-limit 1 Throughout these lectures we shall use natural units so that kB = 1, ¯ = 1 h H Kleinert, COLLECTIVE QUNATUM FIELDS 15 1.5 Grand-Canonical Ensembles at Zero Temperature Thus we obtain for the imaginary-time Green function of a free nonrelativistic field at finite temperature (the so-called free thermal... called the euclidean four-momentum Correspondingly, we define the euclidean spacetime coordinate xE ≡ (−τ, x) (1.77) The the exponential in (1.68) can be written as pE xE = −ωτ + px H Kleinert, COLLECTIVE QUNATUM FIELDS (1.78) 17 1.5 Grand-Canonical Ensembles at Zero Temperature Collecting integral and sum in a single four-summation symbol, we shall write (1.68) as T G0 (xE − x ) ≡ − T V pE exp [−ipE . Collective Quantum Fields in Plasmas, Superconductors, and Superfluid 3 He, Collective Quantum Fields in Plasmas, Superconductors, and Superfluid 3 He Hagen Kleinert Professor. given to quantum electrodynamics of electrons and positrons, where the bare mesons are positronium atoms [9]. 3 H. Kleinert, COLLECTIVE QUNATUM FIELDS /amd/apollo/0/home/ag -kleinert/ kleinert /kleinert/ books/cqf/techniqs.tex June. models. Berlin, January 1990 H. Kleinert H. Kleinert, COLLECTIVE QUNATUM FIELDS Contents I Functional Integral Techniques 1 1 Nonrelativistic Fields 4 1.1 Free Fields . . . . . . . . . . . . .