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

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• Static homogeneous holographic superconductor (HSC)

• Dynamical homogeneous HSC• Static inhomogeneous HSC

Introduction

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• 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

free energy superconducting solution

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

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• Dynamical homogeneous HSC after a quench: T<Tc, RN-AdS static hairy black hole; normal phase superconducting phase

Gaussian quench

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

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

three kinds ofdecays are found,consistent withQNMs

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AdS soliton background, undamped oscillation

modes are found; lack of thermalization in CFT

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

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CMT: lack of thermalization comes from

integrability. Question: what is the relation between a

gravity without horizon and the integrable field theory on boundary?

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

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• Static inhomogeneous HSC:

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

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

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

Rozali, Smyth, Sorkin & Stang, Phys.Rev.Lett. 110, (2013) 201603 Donos & Gauntlett, JHEP 1108, 140 (2011)

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

Chern-Simons term , spontaneously generate angular momentum

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• Other references: dynamical studies :

inhomogeneous studies: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)

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

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)

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

…

…

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

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Motivation to Dynamical Inhomogeneous HSC

• Kibble-Zurek mechanisms:

topological defects generated from a sudden quench

Zurek, Nature 317, 505 (1985)

Zel’dovich, Kobzarev & Okun, Zh. eksp. teor. Fiz. 67, 3 (1974); Soviet Phys. JETP 67, 401 (1975)

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

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

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

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• From holography:

axion fields, spontaneously break the translational symmetry.

angular momentum generation from Chern-Simons terms.

• How about a quench depending on time?

Rozali, Smyth, Sorkin & Stang, Phys.Rev.Lett. 110, (2013) 201603 Donos & Gauntlett, JHEP 1108, 140 (2011)

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

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Dynamical inhomogeneous HSC

• Einstein-Maxwell-complex scalar action:

in which• AdS-Schwarzschild black hole:

with

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Fields depend on t,r & x. Ansatz:

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

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

-charge density; -chemical potential; source ; order parameter - superfluid velocity; supercurrent

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• 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

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chemical potential expectation value

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Physical meaning: quench will excite finite momentum

oscillations in time inhomogeneous order parameter

t=0t=1.2

t=9.7

t=30

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• Quench B: i=5.5, f=5.8, corresponding to Ti=0.738 Tc, Tf=0.700 Tc.

chemical potential expectation value

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• Free energy, grand canonical ensemble, F=-TSos

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

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• at late timesquench A

quench B

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Summaries• Developments in HSC: static homogeneous,

dynamical homogeneous, static inhomogeneous.

• Motivation: spontaneous symmetry breaking • Dynamical inhomogeneous HSC, a quench

induces translational symmetry breaking

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Thanks for your attention!