On the Higgs mechanism in the theory of

Transcription

1 On the Higgs mechanism in the theory of superconductivity* ty Dietrich Einzel Walther-Meißner-Institut für Tieftemperaturforschung Bayerische Akademie der Wissenschaften D Garching Outline Phenomenological theories (overview) The London s theory in a nutshell The Ginzburg-Landau theory in a nutshell London theory, Ginzburg-Landau theory and Higgs masses Goldstone bosons and the Anderson-Higgs mechanism The Higgs mechanism in the standard d model * Graduiertenkolleg Theorie der Supraleitung für Experimentalphysiker, WMI,

2 Phenomenological theories (overview) London-BCS: (electro-) dynamics of homogeneous pair condensates [F. & H. London, 1935, 1950; J. Bardeen, L.N. Cooper & R. Schrieffer, 1957] Ginzburg-Landau-BCS (GLAG): thermodynamics & magnetostatics of non-homogeneous pair condensates [V.L. Ginzburg, L.D. Landau, 1950; A.A. Abrikosov,1957; L.P. Gorkov, 1958] Anderson-Higgs: electroweak interaction within the standard model: coupling of gauge (A ) and Higgs ( ) field [P.W. Anderson, 1962; P. Higgs, 1964, ] 2

3 Electrodynamics: potentials and fields scalar potential vector potential (gauge field) magnetic field electric field gauge invariance 3

4 The London s theory in a nutshell Schrödinger equation macroscopic wave function conservation law for n s supercurrent density j s Josephson equation Bohm potential 4

5 The London s theory in a nutshell quasiclassical i l limit quasiclassical approxi- mation Bohm potential Euler s accereration eq. Ampère s law screening length London penetration depth 5

6 The Ginzburg-Landau theory in a nutshell Schrödinger equation macroscopic wave function GL conservation law for a 2 GL supercurrent density j s Josephson equation GL Bohm potential 6

7 The Ginzburg-Landau theory in a nutshell Josephson equation Bohm potential divide by a 0 Ginzburg Landau coherence length dimensionless GL order parameter 7

8 The Ginzburg-Landau theory in a nutshell Ginzburg-Landau equation for f(r) 1 coupling to A(r) ( ) Ampère s law Ginzburg-Landau equation for A(r) 2 coupling to f(r) London limit: f(r)=1 only 2 remains! 8

9 The BCS energy gap and the superfluid density n s pair binding energy and order parameter e energy spectrum aquires gap superfluid density n s and order parameter, e 4 Ginzburg-Landau regime 9

10 The Ginzburg-Landau functional and the Higgs masses Ginzburg-Landau energy density Madelung representation linearization w.r.t. a Higgs field (from V GL ) 10

11 The Ginzburg-Landau free energy (equilibrium) f a 2 a 0 =- / 10a

12 The Ginzburg-Landau free energy (dynamics) f 2 a 0 =- / a amplitude mode (Higgs boson) Phase mode (Goldstone boson) a 10b

13 The Ginzburg-Landau functional and the Higgs masses photon mass m A (de Broglie): plasma frequency Higgs mass (de Broglie): energy gap mechanisms of mass formation mass m A of the gauge field A (photon mass) order parameter n s mass of the Higgs field a ( Higgs boson ) order parameter type-i vs. type-ii universe 11

14 Condensate plasma frequency and Higgs mass BCS result Ginzburg-Landau regime T/T c 12

15 Two fundamental theorems Noether theorem: Goldstone theorem: (Emmy Noether, 1918) (Jeffrey Goldstone, 1961) Every continuous symmetry of a system can be associated with a conserved quantity The spontaneous breaking of a continuous symmetry can be associated with a massless and spinless particle, the so- called Nambu-Goldstone boson symmetry operation conservation law broken symmetry Goldstone boson translation in time energy translation in space momentum rotation in space angular momentum phase charge liquids Galilean longitudinal phonon solids Galilean longit.+transv. phonon spin rotation magnon gauge phonon 13

16 Plasmons, gauge-invariance and mass perturbation potential amplitude phase order parameter response condensate response (Tsuneto, 1962; Anderson 1963) 14

17 Plasmons, gauge-invariance and mass gauge-invariant g density response phase fluct s (Goldstone boson) amplitude fluct s ( Higgs boson ) generalized Josephson relation gauge mode or Goldstone boson (massless) result for the density response 15

18 Plasmons, gauge-invariance and mass charge (quasi-) conservation conservation law violated (broken gauge symmetry) conservation law restored (Nambu-Goldstone boson) + = 0 density response (neutral systems) 16

19 Plasmons, gauge-invariance and mass density response charged systems renormalized density response Condensate dielectric function gauge mode gets shifted to plasma frequency referred to as Anderson-Higgs mechanism! 17

20 Plasmons, gauge-invariance and mass electromagnetic response dielectric function London-BCS Drude Equivalence of a superconductor and a charged Bose-Einstein Condensate! 18

21 London s and GL magnetic field penetration =0.05, GL (0)/d=0.5 z(x)/ /H 0 H z T/T c = field penetration field screening x/d 19

22 On the Higgs mechanism in the standard model* Lagrange density for Higgs field Higgs potential photon (gauge field) finite vacuum expectation value below T c local gauge symmetry broken * Y. Nambu, 1960; J. Goldstone, 1961; P.W. Anderson, 1962; P. Higgs, 1964; F. Englert & R. Brout, 1964; T.W.B. Kibble, C.R. Hagen & G. Guralnik, 1964; S. Glashow, S. Weinberg & A. Salam,

23 On the Higgs mechanism in the standard model Madelung representation Nambu- Goldstone boson condensate density current density charge conservation law violated (EWSSB) conservation law restored (Nambu-Goldstone boson) + = 0 21

24 On the Higgs mechanism in the standard model Lagrange density gauge boson disappears in A mechanisms of mass formation mass m A of the gauge field A (photon mass) mass of the Higgs field (Higgs boson) 22

25 Summary and conclusions Theories of neutral and charged pair fields London s theory Ginzburg-Landau theory BCS theory Ginzburg-Landau theory of electroweak interaction London s theory, Ginzburg-Landau theory and Higgs masses Noether vs. Goldstone theorem Goldstone bosons and the Anderson-Higgs mechanism Electromagnetic response and condensate dielectric function Photon mass and Higgs mass in the theory of superconductivity Electroweak interaction and the Higgs field The Higgs mechanism in the standard model Ginzburg-Landau vs. 4 theories: type-i or type-ii universe? 23

26 Field theory prototypes Klein-Gordon theory (massive bosons) Dirac s theory (massive fermions) Maxwell s theory (massless bosons) Proca s theory (masssive bosons) A1

27 Phenomenological theories (overview) London-BCS: (electro-) dynamics of homogeneous pair condensates [F. & H. London, 1935, 1950; J. Bardeen, L.N. Cooper & R. Schrieffer, 1957] Ginzburg-Landau-BCS (GLAG): thermodynamics & magnetostatics of non-homogeneous pair condensates [V.L. Ginzburg, L.D. Landau, 1950; A.A. Abrikosov,1957; L.P. Gorkov, 1958] Gross-Pitaevskii: quantum hydrodynamics of interacting (contact-i. g) Bose-Einstein condensates [E.P. Gross, L.P. Pitaevskii, 1961] A2

28 Phenomenological theories (overview) London-BCS: linear, time-dependent Schrödinger equation for (elektro-) dynamics of homogeneous pair condensates [F. & H. London, 1935, 1950; J. Bardeen, L.N. Cooper & R. Schrieffer, 1957] can explain: broken gauge symmetry and gauge invariance persistent currents screening of magnetic fields, London penetration depth L fluxoid- and flux quantization Josephson effect (via Hamilton-Jacobi equation) can not pair charge 2e (original London s theory) explain: microscopic origin of n s (T)=2 2 distinction type-i/type-ii superconductors non-local effects (surfaces, vortices, vortex lattice) local response: C V (T), L (T), s (T): T-dependencies existence and properties of the normal component A3

29 Phenomenological theories (overview) Ginzburg-Landau-BCS (GLAG): stationary, non-linear Schrödinger-eq. thermodynamics & magnetostatics of non-homogeneous pair condensates [V.L. Ginzburg, L.D. Landau, 1950; A.A. Abrikosov,1957; L.P. Gorkov, 1958] can broken gauge symmetry and gauge invariance explain: persistent currents / fluxoid- and flux quantization screening of magnetic fields, London penetration depth L phase transitions (with and w/o) mag. field (2.und 1.o.) local response: C V (T), L (T), close to T c type-i/type-ii SC, supercooling, superheating) non-local effects (surfaces, vortices, vortex lattice) can not pair charge 2e (original GL theory) explain: microscopic origin of n s (T)=2 2 behavior at low temperatures T<<T c dynamics, for example Josephson effect, properties of the normal component, for example s (T) A4

30 Phenomenological theories (overview) Gross-Pitaevskii: time-dependent, non-linear Schrödinger equation quantum hydrodynamics d of interacting ti (contact-i. t g) ) Bose-Einstein condensates [E.P. Gross, L.P. Pitaevskii, 1961] can explain: broken gauge symmetry and gauge invariance dissipationless condensate dynamics (Euler s eq.) quantized vortices) Phonon form normal component (Bogoliubov) Landau criterion for superfluidity can not microscopic origin of n s (T)= 2 explain: elementary excitations: phonon-roton spectrum properties of the normal component temperature dependencies A5