Zoology of condensed matter: framids, ordinary stuff, extra-ordinary stuff

  • Alberto Nicolis
  • Riccardo PencoEmail author
  • Federico Piazza
  • Riccardo Rattazzi
Open Access
Regular Article - Theoretical Physics


We classify condensed matter systems in terms of the spacetime symmetries they spontaneously break. In particular, we characterize condensed matter itself as any state in a Poincaré-invariant theory that spontaneously breaks Lorentz boosts while preserving at large distances some form of spatial translations, time-translations, and possibly spatial rotations. Surprisingly, the simplest, most minimal system achieving this symmetry breaking pattern — the framid — does not seem to be realized in Nature. Instead, Nature usually adopts a more cumbersome strategy: that of introducing internal translational symmetries — and possibly rotational ones — and of spontaneously breaking them along with their space-time counterparts, while preserving unbroken diagonal subgroups. This symmetry breaking pattern describes the infrared dynamics of ordinary solids, fluids, superfluids, and — if they exist — supersolids. A third, “extra-ordinary”, possibility involves replacing these internal symmetries with other symmetries that do not commute with the Poincaré group, for instance the galileon symmetry, supersymmetry or gauge symmetries. Among these options, we pick the systems based on the galileon symmetry, the “galileids”, for a more detailed study. Despite some similarity, all different patterns produce truly distinct physical systems with different observable properties. For instance, the low-energy 2 → 2 scattering amplitudes for the Goldstone excitations in the cases of framids, solids and galileids scale respectively as E 2, E 4, and E 6. Similarly the energy momentum tensor in the ground state is “trivial” for framids (ρ + p = 0), normal for solids (ρ + p > 0) and even inhomogenous for galileids.


Effective field theories Space-Time Symmetries Global Symmetries Gauge Symmetry 


Open Access

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  1. [1]
    D.T. Son, Low-energy quantum effective action for relativistic superfluids, hep-ph/0204199 [INSPIRE].
  2. [2]
    M. Greiter, F. Wilczek and E. Witten, Hydrodynamic Relations in Superconductivity, Mod. Phys. Lett. B 3 (1989) 903 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    S. Dubovsky, T. Gregoire, A. Nicolis and R. Rattazzi, Null energy condition and superluminal propagation, JHEP 03 (2006) 025 [hep-th/0512260] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Nicolis, R. Penco and R.A. Rosen, Relativistic Fluids, Superfluids, Solids and Supersolids from a Coset Construction, Phys. Rev. D 89 (2014) 045002 [arXiv:1307.0517] [INSPIRE].ADSGoogle Scholar
  5. [5]
    A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    I. Low and A.V. Manohar, Spontaneously broken space-time symmetries and Goldstones theorem, Phys. Rev. Lett. 88 (2002) 101602 [hep-th/0110285] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    A. Nicolis and F. Piazza, Spontaneous Symmetry Probing, JHEP 06 (2012) 025 [arXiv:1112.5174] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    A. Nicolis and F. Piazza, Implications of Relativity on Nonrelativistic Goldstone Theorems: Gapped Excitations at Finite Charge Density, Phys. Rev. Lett. 110 (2013) 011602 [arXiv:1204.1570] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    A. Kapustin, Remarks on nonrelativistic Goldstone bosons, arXiv:1207.0457 [INSPIRE].
  10. [10]
    H. Watanabe, T. Brauner and H. Murayama, Massive Nambu-Goldstone Bosons, Phys. Rev. Lett. 111 (2013) 021601 [arXiv:1303.1527] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A. Nicolis, R. Penco, F. Piazza and R.A. Rosen, More on gapped Goldstones at finite density: More gapped Goldstones, JHEP 11 (2013) 055 [arXiv:1306.1240] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    E. Ivanov and V. Ogievetsky, The Inverse Higgs Phenomenon in Nonlinear Realizations, Teor. Mat. Fiz. 25 (1975) 164.MathSciNetCrossRefGoogle Scholar
  13. [13]
    I.N. McArthur, Nonlinear realizations of symmetries and unphysical Goldstone bosons, JHEP 11 (2010) 140 [arXiv:1009.3696] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  14. [14]
    S. Endlich, A. Nicolis and R. Penco, Ultraviolet completion without symmetry restoration, Phys. Rev. D 89 (2014) 065006 [arXiv:1311.6491] [INSPIRE].ADSGoogle Scholar
  15. [15]
    T. Jacobson and D. Mattingly, Gravity with a dynamical preferred frame, Phys. Rev. D 64 (2001) 024028 [gr-qc/0007031] [INSPIRE].ADSMathSciNetGoogle Scholar
  16. [16]
    H. Watanabe and T. Brauner, On the number of Nambu-Goldstone bosons and its relation to charge densities, Phys. Rev. D 84 (2011) 125013 [arXiv:1109.6327] [INSPIRE].ADSGoogle Scholar
  17. [17]
    P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 (2009) 084008 [arXiv:0901.3775] [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    D. Blas, O. Pujolàs and S. Sibiryakov, On the Extra Mode and Inconsistency of Hořava Gravity, JHEP 10 (2009) 029 [arXiv:0906.3046] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    P. Creminelli, J. Norena, M. Pena and M. Simonovic, Khronon inflation, JCAP 11 (2012) 032 [arXiv:1206.1083] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    D. Vollhardt and P. Wölfle, The superfluid phases of helium 3, CRC Press, (1990).Google Scholar
  21. [21]
    D.T. Son, Effective Lagrangian and topological interactions in supersolids, Phys. Rev. Lett. 94 (2005) 175301 [cond-mat/0501658] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    R. Jackiw, V.P. Nair, S.Y. Pi and A.P. Polychronakos, Perfect fluid theory and its extensions, J. Phys. A 37 (2004) R327 [hep-ph/0407101] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  23. [23]
    A. Nicolis, Low-energy effective field theory for finite-temperature relativistic superfluids, arXiv:1108.2513 [INSPIRE].
  24. [24]
    S. Dubovsky, L. Hui, A. Nicolis and D.T. Son, Effective field theory for hydrodynamics: thermodynamics and the derivative expansion, Phys. Rev. D 85 (2012) 085029 [arXiv:1107.0731] [INSPIRE].ADSGoogle Scholar
  25. [25]
    G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Galileons as Wess-Zumino Terms, JHEP 06 (2012) 004 [arXiv:1203.3191] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    S. Endlich, A. Nicolis and J. Wang, Solid Inflation, JCAP 10 (2013) 011 [arXiv:1210.0569] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    C.G. Callan Jr., S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247 [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    S.R. Coleman, J. Wess and B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239 [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    D.V. Volkov, Phenomenological Lagrangians, Fiz. Elem. Chast. Atom. Yadra 4 (1973) 3.MathSciNetGoogle Scholar
  30. [30]
    V.I. Ogievetsky, Nonlinear realizations of internal and space-time symmetries, in X-th winter school of theoretical physics in Karpacz, Poland, (1974).Google Scholar
  31. [31]
    N. Arkani-Hamed and J. Kaplan, On Tree Amplitudes in Gauge Theory and Gravity, JHEP 04 (2008) 076 [arXiv:0801.2385] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    S. Endlich, A. Nicolis, R. Rattazzi and J. Wang, The quantum mechanics of perfect fluids, JHEP 04 (2011) 102 [arXiv:1011.6396] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    L.V. Delacrétaz, S. Endlich, A. Monin, R. Penco and F. Riva, (Re-)Inventing the Relativistic Wheel: Gravity, Cosets and Spinning Objects, JHEP 11 (2014) 008 [arXiv:1405.7384] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    N. Arkani-Hamed, H.-C. Cheng, M.A. Luty and S. Mukohyama, Ghost condensation and a consistent infrared modification of gravity, JHEP 05 (2004) 074 [hep-th/0312099] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    L. Landau, E. Lifshitz, H. Schopf and P. Ziesche, Theory of Elasticity, Vol. 7: Course of Theoretical Physics.Google Scholar
  36. [36]
    N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys. 305 (2003) 96 [hep-th/0210184] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    P. Creminelli, M. Serone and E. Trincherini, Non-linear Representations of the Conformal Group and Mapping of Galileons, JHEP 10 (2013) 040 [arXiv:1306.2946] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    S.R. Coleman and J. Mandula, All Possible Symmetries of the S Matrix, Phys. Rev. 159 (1967) 1251 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  39. [39]
    K. Hinterbichler and A. Joyce, Goldstones with Extended Shift Symmetries, Int. J. Mod. Phys. D 23 (2014) 1443001 [arXiv:1404.4047] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    C. de Rham, M. Fasiello and A.J. Tolley, Galileon Duality, Phys. Lett. B 733 (2014) 46 [arXiv:1308.2702] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    G. Goon, K. Hinterbichler and M. Trodden, Symmetries for Galileons and DBI scalars on curved space, JCAP 07 (2011) 017 [arXiv:1103.5745] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    C. Burrage, C. de Rham and L. Heisenberg, de Sitter Galileon, JCAP 05 (2011) 025 [arXiv:1104.0155] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    G.R. Dvali, G. Gabadadze and M. Porrati, 4D gravity on a brane in 5D Minkowski space, Phys. Lett. B 485 (2000) 208 [hep-th/0005016] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    M.A. Luty, M. Porrati and R. Rattazzi, Strong interactions and stability in the DGP model, JHEP 09 (2003) 029 [hep-th/0303116] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    A. Nicolis and R. Rattazzi, Classical and quantum consistency of the DGP model, JHEP 06 (2004) 059 [hep-th/0404159] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].ADSGoogle Scholar
  48. [48]
    C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    C. de Rham, G. Gabadadze, L. Heisenberg and D. Pirtskhalava, Cosmic Acceleration and the Helicity-0 Graviton, Phys. Rev. D 83 (2011) 103516 [arXiv:1010.1780] [INSPIRE].ADSGoogle Scholar
  50. [50]
    K. Hinterbichler and R.A. Rosen, Interacting Spin-2 Fields, JHEP 07 (2012) 047 [arXiv:1203.5783] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    G. Ballesteros and B. Bellazzini, Effective perfect fluids in cosmology, JCAP 04 (2013) 001 [arXiv:1210.1561] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    L. Boubekeur, P. Creminelli, J. Norena and F. Vernizzi, Action approach to cosmological perturbations: the 2nd order metric in matter dominance, JCAP 08 (2008) 028 [arXiv:0806.1016] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    A. Gruzinov, Elastic inflation, Phys. Rev. D 70 (2004) 063518 [astro-ph/0404548] [INSPIRE].ADSGoogle Scholar
  54. [54]
    M. Sitwell and K. Sigurdson, Quantization of Perturbations in an Inflating Elastic Solid, Phys. Rev. D 89 (2014) 123509 [arXiv:1306.5762] [INSPIRE].ADSGoogle Scholar
  55. [55]
    J.A. Pearson, Material models of dark energy, Annalen Phys. 526 (2014) 318 [arXiv:1403.1213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    S. Endlich, W. Goldberger, R. Rattazzi and I. Rothstein, to appear.Google Scholar
  57. [57]
    L. Pitaevskii and E. Lifshitz, Statistical Physics, Part 2, Vol. 9: Course of Theoretical Physics.Google Scholar
  58. [58]
    C. Eling and T. Jacobson, Spherical solutions in Einstein-aether theory: Static aether and stars, Class. Quant. Grav. 23 (2006) 5625 [gr-qc/0603058] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    T. Jacobson and D. Mattingly, Einstein-Aether waves, Phys. Rev. D 70 (2004) 024003 [gr-qc/0402005] [INSPIRE].ADSGoogle Scholar
  60. [60]
    S.M. Carroll and E.A. Lim, Lorentz-violating vector fields slow the universe down, Phys. Rev. D 70 (2004) 123525 [hep-th/0407149] [INSPIRE].ADSGoogle Scholar
  61. [61]
    S. Weinberg, Gravitation and Cosmology.Google Scholar
  62. [62]
    K. Hinterbichler, M. Trodden and D. Wesley, Multi-field galileons and higher co-dimension branes, Phys. Rev. D 82 (2010) 124018 [arXiv:1008.1305] [INSPIRE].ADSGoogle Scholar
  63. [63]
    A. Padilla, P.M. Saffin and S.-Y. Zhou, Bi-galileon theory I: Motivation and formulation, JHEP 12 (2010) 031 [arXiv:1007.5424] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Alberto Nicolis
    • 1
  • Riccardo Penco
    • 1
    Email author
  • Federico Piazza
    • 1
    • 2
    • 3
  • Riccardo Rattazzi
    • 4
  1. 1.Physics Department and Institute for Strings, Cosmology, and Astroparticle PhysicsColumbia UniversityNew YorkUSA
  2. 2.Paris Center for Cosmological Physics and Laboratoire APCParisFrance
  3. 3.CPT, Aix Marseille UniversitéMarseilleFrance
  4. 4.Institut de Théorie des Phénomènes Physiques, EPFLLausanneSwitzerland

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