Gauged Cosmic Strings and Superconductor Vortices

Transcription

1 Gauged Cosmic Strings and Superconductor Vortices Hindmarsh&Rajantie, PRL2000 Rajantie, IJMPA2001 Kibble&Rajantie, PRB2003 Donaire,Kibble&Rajantie, cond-mat/ Donaire&Rajantie, hep-ph/ September 2005 Smolenice, Slovakia

2 Outline Global vs Gauge symmetry Kibble-Zurek scenario Flux trapping Simulations Experiments Page 2

3 Global Symmetry Same transformation everywhere Condender matter: Superfluids, Liquid crystals, Ferromagnets,... Particle physics: Chiral symmetry (approx) Theoretically simple: Effectively just a scalar field φ (=order parameter) F [φ] = d 3 x [ φ 2 λ ( φ 2 v 2) 2 ] Transformation φ(x) e iα φ(x) Symmetry breaking Goldstone modes Infinite correlation length Page 3

4 Global Symmetry Breaking Breaking of global U(1) symmetry F [φ] = d 3 x [ φ 2 λ ( φ 2 v 2) 2 ] Classical ground state: Potential φ = v Gradient term θ = 0 (where θ = arg φ) Ground states characterised by θ ( π, π] (represented by different colours) Page 4

5 Global Symmetry Breaking Breaking of global U(1) symmetry F [φ] = d 3 x [ φ 2 λ ( φ 2 v 2) 2 ] Classical ground state: Potential φ = v Gradient term θ = 0 (where θ = arg φ) Ground states characterised by θ ( π, π] (represented by different colours) Vortex solution: φ(r, ϕ) = f(r) exp(iϕ) f(r) v φ 2 v 2 /r 2 Logarithmically diverging energy Confining interaction potential Page 5

6 Gauge Symmetry Symmetry under position-dependent transformations Condender matter: Superconductors [U(1) symmetry of electrodynamics] Particle physics: Electroweak, GUT symmetries Effective theory: Scalar field φ (order parameter) Gauge field A (vector potential) F [φ] = ( ) 2+ d x[ 3 A ( +ie A)φ 2 λ ( φ 2 v 2) ] 2 Page 6 Transformation φ(x) e iα(x) φ(x) A A (1/e) α To compare θ = arg φ between two points, one must specify how! (=Gauge fixing)

7 Gauge Symmetry Breaking F [φ] = ( ) 2+ d x[ 3 A ( +ie A)φ 2 λ ( φ 2 v 2) ] 2 Again φ = v Higgs mechanism: m γ = ev ( Meissner effect) No massless Goldstone Covariant derivative term θ + e A = 0 Φ = d x A = (1/e) d x θ = (2π/e)N W (2π/e)Z Page 7 Flux quantization Also important in vortex formation! Vortex solution: φ(r, ϕ) = f(r) exp(iϕ) A(r, ϕ) ˆϕ/er ( +ie A)φ 2 exp( 2m γ r) Exponentially decaying short-range interaction potential

8 Kibble-Zurek Mechanism Assume global symmetry (e = 0) Gradual cooling from T > T c (symmetric) to T < T c (broken) Second-order transition at T = T c Critical phenomena: Correlation length ξ diverges Causality/Critical slowing down: Cannot grow arbitrarily fast ξ will still have a finite value ˆξ at T = T c (Zurek 1985) System must choose a direction for symmetry breaking = value of θ U(1) symmetry: Any value just as likely Choice uncorrelated at distances > ˆξ Page 8

9 Kibble-Zurek Mechanism System must choose a direction for symmetry breaking = value of θ U(1) symmetry: Any value equally likely Choice uncorrelated at distances > ˆξ Consider a circle of radius R ˆξ Three points on different sides: Random θ Interpolates smoothly between these values (= Geodesic rule ) Finite probability that θ = ±2π: Vortex (Kibble 1976) Number density per unit area n ˆξ 2 Independent on the number of dimensions Page 9

10 Kibble-Zurek Mechanism System must choose a direction for symmetry breaking = value of θ U(1) symmetry: Any value equally likely Choice uncorrelated at distances > ˆξ Consider a circle of radius R ˆξ Three points on different sides: Random θ Interpolates smoothly between these values (= Geodesic rule ) Finite probability that θ = ±2π: Vortex (Kibble 1976) Number density per unit area n ˆξ 2 Independent on the number of dimensions First order transitions Replace domains by bubbles, ˆξ by bubble size Negative vortex correlation N W R/ˆξ (A. Srivastava s talk) Page 10

11 Kibble-Zurek and Gauge Symmetry Global theory: Ground states satisfy θ = 0 Characterised by θ ( π, π] Gauge symmetry: Can rotate θ arbirarily at any point Page 11

12 Kibble-Zurek and Gauge Symmetry Gauge symmetry: Can rotate θ arbirarily at any point Is θ completely random at all points? Yes, unless you fix the gauge! Page 12

13 Kibble-Zurek and Gauge Symmetry Gauge fixing: Many possibilities, but does not affect physics Let us choose the Coulomb gauge A = 0 Note: Constant A is still allowed Page 13

14 Kibble-Zurek and Gauge Symmetry Constant A allowed Ground states satisfy θ + e A = 0 and A = 0 Characterised by constant A and integration constant θ 0 Page 14

15 Kibble-Zurek and Gauge Symmetry Constant A allowed Ground states satisfy θ + e A = 0 Characterised by constant A and integration constant θ 0 Page 15

16 Kibble-Zurek and Gauge Symmetry Constant A allowed Ground states satisfy θ + e A = 0 Characterised by constant A and integration constant θ 0 Page 16

17 Kibble-Zurek and Gauge Symmetry Bubbles touch: Trapped flux Φ = d r A Cannot escape: Must turn into a thick vortex Alternatively, use geodesic rule same result N W = Φ/Φ 0 Page 17

18 Kibble-Zurek and Gauge Symmetry What is Φ? Choice of ground state ( A) cannot be completely random Non-compact space: No uniform measure Long-range interactions Page 18

19 Kibble-Zurek and Gauge Symmetry Magnetic field carries energy: Why should it be non-zero? Thermal fluctuations! p exp( βh) Page 19

20 Typical Flux R Saddle point method: Φ given by E min (Φ) = T E min minimum energy for a given flux Dimensional analysis E min Φ 2 R D 4 Φ T R 4 D (in D space dimensions) Page 20

21 Flux Trapping in Bubble Collisions N W e T R 4 D cf. N W 1 from KZ (M. Donaire s talk) Page 21

22 Simulations Discretize theory on lattice λ/e 2 = 0.05 Type I, 1st order phase transition Thermal initial conditions p exp( βh) Start in the symmetric phase T < T c metastable, but with a very long lifetime Generated using standard Monte Carlo methods Evolve using classical eqs of motion Add a small damping term Temperature decreases Nucleation rate increases Bubbles of broken phase start to nucleate Visualization: Page 22 Walls: Phase of φ in Coulomb gauge White regions: Broken phase (early stages) Symmetric phase (late stages) V(φ) φ

23 Simulations σ = σ = Slower cooling: Lower nucleation rate, bigger bubbles, fewer strings Strings with N W = 2 Page 23

24 Simulations Phase angle Magnetic field 2D simulations: Stronger effect Page 24

25 Second Order Transition Type II λ/e 2 > 1: Continuous transition No bubbles, no flux trapping?? Not true: Causality argument (Hindmarsh&Rajantie 2000) Consider a circular loop of radius R Typical flux before transition Φ T R 4 D After transition, the flux should disappear Causality: Takes at least time t = R If R larger than some critical value λ c, not enough time λ c depends on the cooling rate and dynamics B B Small region of size λ c : Sees a uniform background field B Φ/λ 2 c Abrikosov lattice of n B/Φ 0 e T/R D vortices/unit area Clusters of e T R 4 D vortices Page 25

26 Simulations 2 dimensions: Complex phase in global theory Complex phase in local theory Magnetic field in local theory 2-dimensional film in three-dimensional space: (Donaire,Kibble&Rajantie 2004) Magnetic field in local theory Page 26

27 Quantitative Tests τ Q =500, t=500 τ Q =10, t=240 random n C (r) r (Hindmarsh& Rajantie 2000) Winding number around a circle of radius r around a vortex Above 1: Sign of positive correlation Page 27

28 Quantitative Tests a) 24 b) ρ 8 ρ e T (Stephens et al. 2001) 2D simulations: n e T as predicted Page 28

29 Superconductor Experiments 1999 Carmi et al: 1cm 2 YBCO superconductor film Relatively slow cooling 20K/s Measured total net flux Detected no signal N W 20 Predictions from KZ and FT 10 Page 29

30 Superconductor Experiments 1999 Carmi et al: 1cm 2 YBCO superconductor film Relatively slow cooling 20K/s measured total net flux Detected no signal N W 20 Predictions from KZ and FT Carmi et al: 1cm 2 loop of 214 Josephson junctions Produced 7.4 ± 0.7 flux quanta KZ predicts 15, flux trapping 4 (Both independent of the cooling rate) Page 30

31 Superconductor Experiments 1999 Carmi et al: 1cm 2 YBCO superconductor film Relatively slow cooling 20K/s measured total net flux Detected no signal N W 20 Predictions from KZ and FT Carmi et al: 1cm 2 loop of 214 Josephson junctions Produced 7.4 ± 0.7 flux quanta KZ predicts 15, flux trapping 4 (Both independent of the cooling rate) 2003 Maniv et al: YBCO film again Faster cooling 10 8 K/s Page 31

32 Superconductor Experiments 2003 Maniv et al: YBCO film again Faster cooling 10 8 K/s Find 30 vortices Consistent with (Kibble-Zurek) prediction No conflict with FT either Page 32

33 Superconductor Experiments 2003 Maniv et al: YBCO film again Faster cooling 10 8 K/s Find 30 vortices Consistent with (Kibble-Zurek) prediction No conflict with FT either 2002 Monaco et al: Annular Josephson junction (R.Rivers s talk) Two superposed superconductor rings Measure difference in magnetic flux Relatively long boundary, but negligible area No flux trapping expected Results agree with KZ (?) Page 33

34 Superconductor Experiments 2003 Kirtley et al: Array of Mo 3 Si rings Diameter 40µm Dependence on cooling time supports thermal activation Simpler dynamics Page 34

35 What We Need? Theories exist Predictions still only orders of magnitude (rather than precise number) More robust: Vortex distribution Film experiment (rather than rings) High temperature T c 100K Larger area Locations of individual vortices Page 35

36 What We Need? Theories exist Predictions still only orders of magnitude (rather than precise number) More robust: Vortex distribution Film experiment (rather than rings) High temperature T c 100K Larger area Locations of individual vortices No reason why this would not be possible soon!? Page 36