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Magnetic field induced lattice ground states from holography


We study the holographic field theory dual of a probe SU(2) Yang-Mills field in a background (4 + 1)-dimensional asymptotically Anti-de Sitter space. We find a new ground state when a magnetic component of the gauge field is larger than a critical value. The ground state forms a triangular Abrikosov lattice in the spatial directions perpendicular to the magnetic field. The lattice is composed of superconducting vortices induced by the condensation of a charged vector operator. We perform this calculation both at finite temperature and at zero temperature with a hard wall cutoff dual to a confining gauge theory. The study of this state may be of relevance to both holographic condensed matter models as well as to heavy ion physics. The results shown here provide support for the proposal that such a ground state may be found in the QCD vacuum when a large magnetic field is present.

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  1. [1]

    M. Ammon, J. Erdmenger, P. Kerner and M. Strydom, Black hole instability induced by a magnetic field, Phys. Lett. B 706 (2011) 94 [arXiv:1106.4551] [INSPIRE].

    ADS  Google Scholar 

  2. [2]

    A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5 (1957) 1174 [INSPIRE].

    Google Scholar 

  3. [3]

    G.T. Horowitz, J.E. Santos and D. Tong, Optical conductivity with holographic lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  4. [4]

    R. Flauger, E. Pajer and S. Papanikolaou, A striped holographic superconductor, Phys. Rev. D 83 (2011) 064009 [arXiv:1010.1775] [INSPIRE].

    ADS  Google Scholar 

  5. [5]

    M. Chernodub, Superconductivity of QCD vacuum in strong magnetic field, Phys. Rev. D 82 (2010)085011 [arXiv:1008.1055] [INSPIRE].

    ADS  Google Scholar 

  6. [6]

    M. Chernodub, Electromagnetically superconducting phase of QCD vacuum induced by strong magnetic field, AIP Conf. Proc. 1343 (2011) 149 [arXiv:1011.2658] [INSPIRE].

    ADS  Article  Google Scholar 

  7. [7]

    M. Chernodub, J. Van Doorsselaere and H. Verschelde, Electromagnetically superconducting phase of vacuum in strong magnetic field: structure of superconductor and superfluid vortex lattices in the ground state, Phys. Rev. D 85 (2012) 045002 [arXiv:1111.4401] [INSPIRE].

    ADS  Google Scholar 

  8. [8]

    N. Nielsen and P. Olesen, An unstable Yang-Mills field mode, Nucl. Phys. B 144 (1978) 376 [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  9. [9]

    J. Ambjørn and P. Olesen, On electroweak magnetism, Nucl. Phys. B 315 (1989) 606 [INSPIRE].

    ADS  Article  Google Scholar 

  10. [10]

    J. Ambjørn and P. Olesen, A condensate solution of the electroweak theory which interpolates between the broken and the symmetric phase, Nucl. Phys. B 330 (1990) 193 [INSPIRE].

    ADS  Article  Google Scholar 

  11. [11]

    J. Ambjørn and P. Olesen, Electroweak magnetism: theory and application, Int. J. Mod. Phys. A 5 (1990) 4525 [INSPIRE].

    ADS  Google Scholar 

  12. [12]

    S.K. Domokos and J.A. Harvey, Baryon number-induced Chern-Simons couplings of vector and axial-vector mesons in holographic QCD, Phys. Rev. Lett. 99 (2007) 141602 [arXiv:0704.1604] [INSPIRE].

    ADS  Article  Google Scholar 

  13. [13]

    S. Nakamura, H. Ooguri and C.-S. Park, Gravity dual of spatially modulated phase, Phys. Rev. D 81 (2010) 044018 [arXiv:0911.0679] [INSPIRE].

    ADS  Google Scholar 

  14. [14]

    W.-y. Chuang, S.-H. Dai, S. Kawamoto, F.-L. Lin and C.-P. Yeh, Dynamical instability of holographic QCD at finite density, Phys. Rev. D 83 (2011) 106003 [arXiv:1004.0162] [INSPIRE].

    ADS  Google Scholar 

  15. [15]

    O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Striped instability of a holographic Fermi-like liquid, JHEP 10 (2011) 034 [arXiv:1106.3883] [INSPIRE].

    ADS  Article  Google Scholar 

  16. [16]

    C.B. Bayona, K. Peeters and M. Zamaklar, A non-homogeneous ground state of the low-temperature Sakai-Sugimoto model, JHEP 06 (2011) 092 [arXiv:1104.2291] [INSPIRE].

    ADS  Article  Google Scholar 

  17. [17]

    S. Takeuchi, Modulated instability in five-dimensional U(1) charged AdS black hole with R 2 -term, JHEP 01 (2012) 160 [arXiv:1108.2064] [INSPIRE].

    ADS  Article  Google Scholar 

  18. [18]

    H. Ooguri and C.-S. Park, Holographic end-point of spatially modulated phase transition, Phys. Rev. D 82 (2010) 126001 [arXiv:1007.3737] [INSPIRE].

    ADS  Google Scholar 

  19. [19]

    A. Donos and J.P. Gauntlett, Helical superconducting black holes, Phys. Rev. Lett. 108 (2012)211601 [arXiv:1203.0533] [INSPIRE].

    ADS  Article  Google Scholar 

  20. [20]

    A. Donos and J.P. Gauntlett, Black holes dual to helical current phases, Phys. Rev. D 86 (2012)064010 [arXiv:1204.1734] [INSPIRE].

    ADS  Google Scholar 

  21. [21]

    M. Ammon et al., On stability and transport of cold holographic matter, JHEP 09 (2011) 030 [arXiv:1108.1798] [INSPIRE].

    ADS  Article  Google Scholar 

  22. [22]

    A. Donos, J.P. Gauntlett and C. Pantelidou, Spatially modulated instabilities of magnetic black branes, JHEP 01 (2012) 061 [arXiv:1109.0471] [INSPIRE].

    ADS  Article  Google Scholar 

  23. [23]

    A. Donos and J.P. Gauntlett, Holographic striped phases, JHEP 08 (2011) 140 [arXiv:1106.2004] [INSPIRE].

    ADS  Article  Google Scholar 

  24. [24]

    S. Bolognesi and D. Tong, Monopoles and holography, JHEP 01 (2011) 153 [arXiv:1010.4178] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  25. [25]

    P. Sutcliffe, Monopoles in AdS, JHEP 08 (2011) 032 [arXiv:1104.1888] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  26. [26]

    D. Allahbakhshi, On Holography of Julia-Zee Dyon, JHEP 09 (2011) 085 [arXiv:1105.3677] [INSPIRE].

    ADS  Article  Google Scholar 

  27. [27]

    K. Maeda, M. Natsuume and T. Okamura, Vortex lattice for a holographic superconductor, Phys. Rev. D 81 (2010) 026002 [arXiv:0910.4475] [INSPIRE].

    ADS  Google Scholar 

  28. [28]

    O. Domenech, M. Montull, A. Pomarol, A. Salvio and P.J. Silva, Emergent gauge fields in holographic superconductors, JHEP 08 (2010) 033 [arXiv:1005.1776] [INSPIRE].

    ADS  Article  Google Scholar 

  29. [29]

    J.M. Murray and Z. Tesanovic, Isolated vortex and vortex lattice in a holographic p-wave superconductor, Phys. Rev. D 83 (2011) 126011 [arXiv:1103.3232] [INSPIRE].

    ADS  Google Scholar 

  30. [30]

    S.S. Gubser and S.S. Pufu, The gravity dual of a p-wave superconductor, JHEP 11 (2008) 033 [arXiv:0805.2960] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  31. [31]

    M. Ammon, J. Erdmenger, M. Kaminski and P. Kerner, Superconductivity from gauge/gravity duality with flavor, Phys. Lett. B 680 (2009) 516 [arXiv:0810.2316] [INSPIRE].

    ADS  Google Scholar 

  32. [32]

    M. Ammon, J. Erdmenger, M. Kaminski and P. Kerner, Flavor superconductivity from gauge/gravity duality, JHEP 10 (2009) 067 [arXiv:0903.1864] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  33. [33]

    S. Chunlen, K. Peeters, P. Vanichchapongjaroen and M. Zamaklar, Instability of N = 2 gauge theory in compact space with an isospin chemical potential, JHEP 01 (2013) 035 [arXiv:1210.6188] [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    D. Djukanovic, M.R. Schindler, J. Gegelia and S. Scherer, Quantum electrodynamics for vector mesons, Phys. Rev. Lett. 95 (2005) 012001 [hep-ph/0505180] [INSPIRE].

    ADS  Article  Google Scholar 

  35. [35]

    M. Chernodub, Spontaneous electromagnetic superconductivity of vacuum in strong magnetic field: evidence from the Nambu-Jona-Lasinio model, Phys. Rev. Lett. 106 (2011) 142003 [arXiv:1101.0117] [INSPIRE].

    ADS  Article  Google Scholar 

  36. [36]

    V. Skokov, A.Y. Illarionov and V. Toneev, Estimate of the magnetic field strength in heavy-ion collisions, Int. J. Mod. Phys. A 24 (2009) 5925 [arXiv:0907.1396] [INSPIRE].

    ADS  Google Scholar 

  37. [37]

    A. Bzdak and V. Skokov, Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions, Phys. Lett. B 710 (2012) 171 [arXiv:1111.1949] [INSPIRE].

    ADS  Google Scholar 

  38. [38]

    N. Callebaut, D. Dudal and H. Verschelde, Holographic ρ mesons in an external magnetic field, JHEP 03 (2013) 033 [arXiv:1105.2217] [INSPIRE].

    ADS  Article  Google Scholar 

  39. [39]

    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

    MathSciNet  ADS  MATH  Google Scholar 

  40. [40]

    J. Erlich, E. Katz, D.T. Son and M.A. Stephanov, QCD and a holographic model of hadrons, Phys. Rev. Lett. 95 (2005) 261602 [hep-ph/0501128] [INSPIRE].

    ADS  Article  Google Scholar 

  41. [41]

    L. Da Rold and A. Pomarol, Chiral symmetry breaking from five dimensional spaces, Nucl. Phys. B 721 (2005) 79 [hep-ph/0501218] [INSPIRE].

    ADS  Article  Google Scholar 

  42. [42]

    A. Abrikosov, Fundamentals of the theory of metals, North-Holland, Amsterdam Netherlands (1988).

    Google Scholar 

  43. [43]

    B. Rosenstein and D. Li, Ginzburg-Landau theory of type II superconductors in magnetic field, Rev. Mod. Phys. 82 (2010) 109.

    MathSciNet  ADS  MATH  Article  Google Scholar 

  44. [44]

    W.H. Kleiner, L.M. Roth and S.H. Autler, Bulk solution of Ginzburg-Landau equations for type II superconductors: upper critical field region, Phys. Rev. 133 (1964) A1226.

    ADS  Article  Google Scholar 

  45. [45]

    M. Tinkham, Introduction to superconductivity, Robert E. Krieger Publishing Company, Malabar U.S.A. (1980).

    Google Scholar 

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Correspondence to Migael Strydom.

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ArXiv ePrint: 1210.6669

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Bu, YY., Erdmenger, J., Shock, J.P. et al. Magnetic field induced lattice ground states from holography. J. High Energ. Phys. 2013, 165 (2013).

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  • Gauge-gravity correspondence
  • Holography and condensed matter physics (AdS/CMT)