Journal of High Energy Physics

, 2018:30 | Cite as

Holographic discommensurations

  • Alexander KrikunEmail author
Open Access
Regular Article - Theoretical Physics


When the system with internal tendency to a spontaneous formation of a spatially periodic state is brought in contact with the external explicit periodic potential, the interesting phenomenon of commensurate lock in can be observed. In case when the explicit potential is strong enough and its period is close to the period of the spontaneous structure, the latter is forced to assume the periodicity of the former and the commensurate state becomes a thermodynamically preferred one. If instead the two periods are significantly different, the incommensurate state is formed. It is characterized by a finite density of solitonic objects — discommensurations — on top of the commensurate state. In this note I study the properties of discommensurations in holographic model with inhomogeneous translational symmetry breaking and explain how one can understand the commensurate/incommensurate phase transition as a proliferation of these solitons. Some useful numerical techniques are discussed in the appendix.


Holography and condensed matter physics (AdS/CMT) Space-Time Symmetries Solitons Monopoles and Instantons 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    H. Ooguri and C.-S. Park, Holographic End-Point of Spatially Modulated Phase Transition, Phys. Rev. D 82 (2010) 126001 [arXiv:1007.3737] [INSPIRE].ADSGoogle Scholar
  2. [2]
    A. Donos and J.P. Gauntlett, Black holes dual to helical current phases, Phys. Rev. D 86 (2012) 064010 [arXiv:1204.1734] [INSPIRE].ADSGoogle Scholar
  3. [3]
    A. Donos, Striped phases from holography, JHEP 05 (2013) 059 [arXiv:1303.7211] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    A. Donos and J.P. Gauntlett, Holographic charge density waves, Phys. Rev. D 87 (2013) 126008 [arXiv:1303.4398] [INSPIRE].ADSGoogle Scholar
  5. [5]
    B. Withers, Black branes dual to striped phases, Class. Quant. Grav. 30 (2013) 155025 [arXiv:1304.0129] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    B. Withers, Holographic Checkerboards, JHEP 09 (2014) 102 [arXiv:1407.1085] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  7. [7]
    G.T. Horowitz, J.E. Santos and D. Tong, Further Evidence for Lattice-Induced Scaling, JHEP 11 (2012) 102 [arXiv:1209.1098] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    G.T. Horowitz, J.E. Santos and D. Tong, Optical Conductivity with Holographic Lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Donos and S.A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9 (2013) 649 [arXiv:1212.2998] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    L.V. Delacrétaz, B. Goutéraux, S.A. Hartnoll and A. Karlsson, Bad Metals from Fluctuating Density Waves, SciPost Phys. 3 (2017) 025 [arXiv:1612.04381] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    L.V. Delacrétaz, B. Goutéraux, S.A. Hartnoll and A. Karlsson, Theory of hydrodynamic transport in fluctuating electronic charge density wave states, Phys. Rev. B 96 (2017) 195128 [arXiv:1702.05104] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    T. Andrade, M. Baggioli, A. Krikun and N. Poovuttikul, Pinning of longitudinal phonons in holographic spontaneous helices, JHEP 02 (2018) 085 [arXiv:1708.08306] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    A. Amoretti, D. Areán, R. Argurio, D. Musso and L.A. Pando Zayas, A holographic perspective on phonons and pseudo-phonons, JHEP 05 (2017) 051 [arXiv:1611.09344] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    N. Jokela, M. Jarvinen and M. Lippert, Holographic sliding stripes, Phys. Rev. D 95 (2017) 086006 [arXiv:1612.07323] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    N. Jokela, M. Jarvinen and M. Lippert, Pinning of holographic sliding stripes, Phys. Rev. D 96 (2017) 106017 [arXiv:1708.07837] [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    L. Alberte, M. Ammon, M. Baggioli, A. Jiménez and O. Pujolàs, Black hole elasticity and gapped transverse phonons in holography, JHEP 01 (2018) 129 [arXiv:1708.08477] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    T. Andrade and A. Krikun, Commensurability effects in holographic homogeneous lattices, JHEP 05 (2016) 039 [arXiv:1512.02465] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    T. Andrade and A. Krikun, Commensurate lock-in in holographic non-homogeneous lattices, JHEP 03 (2017) 168 [arXiv:1701.04625] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    T. Andrade, A. Krikun, K. Schalm and J. Zaanen, Doping the holographic Mott insulator, Nature Phys. 14 (2018) 1049 [arXiv:1710.05791] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    P. Bak, Commensurate phases, incommensurate phases and the devils staircase, Rept. Prog. Phys. 45 (1982) 587.ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    W. McMillan, Theory of discommensurations and the commensurate-incommensurate charge-density-wave phase transition, Phys. Rev. B 14 (1976) 1496.ADSCrossRefGoogle Scholar
  24. [24]
    V. Pokrovsky and A. Talapov, Ground state, spectrum, and phase diagram of two-dimensional incommensurate crystals, Phys. Rev. Lett. 42 (1979) 65.ADSCrossRefGoogle Scholar
  25. [25]
    O. Braun and Y. Kivshar, The Frenkel-Kontorova Model: Concepts, Methods and Applications, Springer-Verlag Berlin Heidelberg (2004).CrossRefzbMATHGoogle Scholar
  26. [26]
    A. Donos and J.P. Gauntlett, Holographic striped phases, JHEP 08 (2011) 140 [arXiv:1106.2004] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  27. [27]
    B. Withers, The moduli space of striped black branes, arXiv:1304.2011 [INSPIRE].
  28. [28]
    A. Donos and J.P. Gauntlett, The thermoelectric properties of inhomogeneous holographic lattices, JHEP 01 (2015) 035 [arXiv:1409.6875] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M. Rangamani, M. Rozali and D. Smyth, Spatial Modulation and Conductivities in Effective Holographic Theories, JHEP 07 (2015) 024 [arXiv:1505.05171] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    M. Headrick, S. Kitchen and T. Wiseman, A New approach to static numerical relativity and its application to Kaluza-Klein black holes, Class. Quant. Grav. 27 (2010) 035002 [arXiv:0905.1822] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    T. Wiseman, Numerical construction of static and stationary black holes, arXiv:1107.5513.
  32. [32]
    E.M. Lifshitz and L.P. Pitaevskii, Statistical physics: theory of the condensed state, vol. 9, Elsevier (2013).Google Scholar
  33. [33]
    A. Arancibia and M.S. Plyushchay, Chiral asymmetry in propagation of soliton defects in crystalline backgrounds, Phys. Rev. D 92 (2015) 105009 [arXiv:1507.07060] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    A. Arancibia, F. Correa, V. Jakubský, J. Mateos Guilarte and M.S. Plyushchay, Soliton defects in one-gap periodic system and exotic supersymmetry, Phys. Rev. D 90 (2014) 125041 [arXiv:1410.3565] [INSPIRE].ADSGoogle Scholar
  35. [35]
    J.P. Boyd, Chebyshev and Fourier spectral methods, Courier Corporation (2001).Google Scholar
  36. [36]
    L.N. Trefethen, Spectral methods in MATLAB, vol. 10, Siam (2000).Google Scholar
  37. [37]
    I. Wolfram Research, Mathematica, Version 10.2, Champaign, Illinois (2015).Google Scholar
  38. [38]
    W.L. Briggs et al., A multigrid tutorial, vol. 72, Siam (2000).Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute-Lorentz for Theoretical Physics, ΔITPLeiden UniversityLeidenThe Netherlands

Personalised recommendations