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A Thermal Quench Induces Spatial Inhomogeneities in a Holographic Superconductor. Hai -Qing Zhang ( Instituto Superior Técnico , Lisboa ) INI, Cambridge, 17/09/2013 Coauthor: Antonio M. García-García & Hua -Bi Zeng , arXiv:1308.5398 . Introduction.

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a thermal quench induces spatial inhomogeneities in a holographic superconductor

A Thermal Quench Induces Spatial Inhomogeneities in a Holographic Superconductor

Hai-Qing Zhang

(Instituto Superior Técnico, Lisboa)

INI, Cambridge, 17/09/2013

Coauthor: Antonio M. García-García & Hua-Bi Zeng, arXiv:1308.5398

introduction
Introduction
  • Static homogeneous holographic superconductor (HSC)
  • Dynamical homogeneous HSC
  • Static inhomogeneous HSC
slide3

Gubser, Phys. Rev. D 78 (2008) 065034

  • Static homogeneous HSC:
  • Einstein-Maxwell-charged scalar action with negative cosmological constant
  • T>Tc (or μ< μc, since T~1/ μ): ψ=0;

T<Tc : two available solutions

ψ=0 ψ≠0

Hartnoll, Herzog, & Horowtiz, Phys.Rev.Lett. 101 (2008) 031601

superconducting solution

free energy

slide4

Murata, Kinoshita & Tanahashi, JHEP 1007 (2010) 050 

Gaussian quench

  • Dynamical homogeneous HSC after a quench:

T<Tc, RN-AdS static hairy black hole;

normal phase superconducting phase

slide5

Bhaseen, Gauntlett, Simons, Sonner & Wiseman, Phys.Rev.Lett. 110 (2013) 015301

three kinds of

decays are found,

consistent with

QNMs

slide6

Gao, Garcia-Garcia, Zeng & Zhang:arXiv:1212.1049

AdSsoliton background, undamped oscillation modes are found; lack of thermalization in CFT

slide7

Polkovnikov, Sengupta, Silva, Vengalattore, Rev.Mod.Phys. 83 (2011) 863 

CMT: lack of thermalization comes from integrability.

Question: what is the relation between a gravity without horizon and the integrable field theory on boundary?

slide8

Flauger, Pajer & Papanikolaou, Phys.Rev. D83 (2011) 064009 

  • Static inhomogeneous HSC:

Einstein- Maxwell-scalar action, modulated chemical potential, striped superconductor, explicitly break translational symmetry, free energy is lower than the homogeneous case

slide9

Donos & Gauntlett, JHEP 1108, 140 (2011)

Rozali, Smyth, Sorkin & Stang, Phys.Rev.Lett. 110, (2013) 201603  

Einstein-Maxwell-axion action, spontaneously break the translational symmetry, striped order parameter

slide10

Liu, Ooguri, Stoica & Yunes, Phys. Rev. Lett. 110, (2013) 211601

Chern-Simons term , spontaneously generate angular momentum

slide11

Other references:

  • dynamical studies :
  • inhomogeneous studies:

I. Amado, M. Kaminski & K. Landsteiner, JHEP, 0905021 (2009)

S. Bhattacharyya, and S. Minwalla, JHEP 0909:034, (2009)

P. Bizon, and A. Rostworowski, Phys. Rev. Lett. 107, 031102 (2011)

O. Dias, G. Horowitz, D. Marolf and J. Santos, Class. Quant. Grav. 29, 235019 (2012)

P. Basu, D. Das, S. Das and T. Nishioka, arXiv:1211.7076

S. Nakamura, H. Ooguri and C. -S. Park, Phys. Rev. D 81, 044018 (2010)

G. T. Horowitz, J. E. Santos and D. Tong, JHEP 1211, 102 (2012)

motivation to dynamical inhomogeneous hsc
Motivation to Dynamical Inhomogeneous HSC
  • Kibble-Zurek mechanisms:

topological defects generated from a sudden quench

Zel’dovich, Kobzarev & Okun, Zh. eksp. teor. Fiz. 67, 3 (1974);

Soviet Phys. JETP 67, 401 (1975)

Kibble, J. Phys. A 9, 1387 (1976)

Zurek, Nature 317, 505 (1985)

slide13

Dzero, Yuzbashyan & Altshuler, Eur. Phys. Lett. 85 (2009) 20004

system size larger than superconducting coherence length, quench can excite finite momentum states, results in spatial inhomogeneities

slide14

Donos & Gauntlett, JHEP 1108, 140 (2011)

Rozali, Smyth, Sorkin & Stang, Phys.Rev.Lett. 110, (2013) 201603  

  • From holography:
  • axion fields, spontaneously break the translational symmetry.
  • angular momentum generation from Chern-Simons terms.
  • How about a quench depending on time?

Liu, Ooguri, Stoica & Yunes, Phys. Rev. Lett. 110, (2013) 211601

dynamical inhomogeneous hsc
Dynamical inhomogeneous HSC
  • Einstein-Maxwell-complex scalar action:

in which

  • AdS-Schwarzschild black hole:

with

slide16

Fields depend on t,r & x.

  • Ansatz:
  • 4 independent EoMs, 1 constraint equation
  • Boundary conditions:
  • At horizon: Mt(t, x)|rh=0;
  • Other fields are finite
slide17

For m^2=-2, expansions near boundary:

-charge density;

-chemical potential;

source ;

order parameter

- superfluid velocity;

supercurrent

slide18

Thermal quench, since T~1/, it is equal to quench chemical potential.

  • Quench A:

i=4.5, f=4.8, corresponding to

Ti=0.903Tc, Tf=0.846Tc

since for homogeneous case c=4.063 (m^2=-2)

Hartnoll, Herzog, & Horowtiz, Phys.Rev.Lett. 101 (2008) 031601

slide19

chemical potential

expectation value

slide20

t=30

t=9.7

t=1.2

t=0

oscillations in time

inhomogeneous order parameter

Physical meaning: quench will excite finite

momentum

slide21

Quench B:

i=5.5, f=5.8, corresponding to

Ti=0.738 Tc, Tf=0.700 Tc.

chemical potential

expectation value

slide22

Free energy, grand canonical ensemble,

F=-TSos

where z=1/r, h=1-z^3

slide23

quench A

  • at late times

quench B

summaries
Summaries
  • Developments in HSC: static homogeneous, dynamical homogeneous, static inhomogeneous.
  • Motivation: spontaneous symmetry breaking
  • Dynamical inhomogeneous HSC, a quench induces translational symmetry breaking