Anomalous breaking of anisotropic scaling symmetry in the quantum lifshitz model

Article

Abstract

In this note we investigate the anomalous breaking of anisotropic scaling symmetry (t, x) → (λzt, λ x) in a non-relativistic field theory with dynamical exponent z = 2. On general grounds, one can show that there exist two possible “central charges” which characterize the breaking of scale invariance. Using heat kernel methods, we compute these two central charges in the quantum Lifshitz model, a free field theory which is second order in time and fourth order in spatial derivatives. We find that one of the two central charges vanishes. Interestingly, this is also true for strongly coupled non-relativistic field theories with a geometric dual described by a metric and a massive vector field.

Keywords

Gauge-gravity correspondence Holography and condensed matter physics (AdS/CMT) Anomalies in Field and String Theories 

References

  1. [1]
    D. Son, Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].MathSciNetADSGoogle Scholar
  2. [2]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    S. Kachru, X. Liu and M. Mulligan, Gravity Duals of Lifshitz-like Fixed Points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].MathSciNetADSGoogle Scholar
  4. [4]
    P. Koroteev and M. Libanov, On existence of self-tuning solutions in static braneworlds without singularities, JHEP 02 (2008) 104 [arXiv:0712.1136].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    M. Guica, K. Skenderis, M. Taylor and B.C. van Rees, Holography for Schrödinger backgrounds, JHEP 02 (2011) 056 [arXiv:1008.1991].ADSCrossRefGoogle Scholar
  6. [6]
    S.F. Ross, Holography for asymptotically locally Lifshitz spacetimes, Class. Quant. Grav. 28 (2011) 215019 [arXiv:1107.4451].ADSCrossRefGoogle Scholar
  7. [7]
    M. Baggio, J. de Boer and K. Holsheimer, Hamilton-Jacobi renormalization for Lifshitz spacetime, JHEP 01 (2012) 058 [arXiv:1107.5562].ADSCrossRefGoogle Scholar
  8. [8]
    R.B. Mann and R. McNees, Holographic renormalization for asymptotically Lifshitz spacetimes, JHEP 10 (2011) 129 [arXiv:1107.5792].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    I. Adam, I.V. Melnikov and S. Theisen, A non-relativistic Weyl anomaly, JHEP 09 (2009) 130 [arXiv:0907.2156].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    P.R.S. Gomes and M. Gomes, Ward identities in Lifshitz-like field theories, Phys. Rev. D 85 (2012) 065010 [arXiv:1112.3887].ADSGoogle Scholar
  12. [12]
    E. Ardonne, P. Fendley and E. Fradkin, Topological order and conformal quantum critical points, Ann. Phys. 310 (2004) 493 [cond-mat/0311466].MathSciNetADSMATHCrossRefGoogle Scholar
  13. [13]
    T. Griffin, P. Hořava and C.M. Melby-Thompson, Conformal Lifshitz gravity from holography, JHEP 05 (2012) 010 [arXiv:1112.5660].ADSCrossRefGoogle Scholar
  14. [14]
    G. Grinstein, Anisotropic sine-gordon model and infinite-order phase transitions in three dimensions, Phys. Rev. B 23 (1981) 4615.MathSciNetADSGoogle Scholar
  15. [15]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory (1997).Google Scholar
  16. [16]
    D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138].MathSciNetADSMATHCrossRefGoogle Scholar
  17. [17]
    A. Ceresole, P. Pizzochero and P. van Nieuwenhuizen, The curved space trace, chiral and einstein anomalies from path integrals, using flat space plane waves, Phys. Rev. D 39 (1989) 1567.ADSGoogle Scholar
  18. [18]
    J. de Boer, E. Verlinde and H. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012].CrossRefGoogle Scholar
  19. [19]
    D. Martelli and W. Mück, Holographic renormalization and Ward identities with the Hamilton-Jacobi method, Nucl. Phys. B 654 (2003) 248 [hep-th/0205061].ADSCrossRefGoogle Scholar
  20. [20]
    S.F. Ross and O. Saremi, Holographic stress tensor for non-relativistic theories, JHEP 09 (2009) 009 [arXiv:0907.1846].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    A. Adams, A. Maloney, A. Sinha and S.E. Vázquez, 1/N effects in non-relativistic gauge-gravity duality, JHEP 03 (2009) 097 [arXiv:0812.0166].ADSCrossRefGoogle Scholar
  22. [22]
    P.B. Gilkey, The spectral geometry of the higher order laplacian, Duke Math. J. 47 (1980) 511.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Marco Baggio
    • 1
  • Jan de Boer
    • 1
  • Kristian Holsheimer
    • 1
  1. 1.Institute for Theoretical PhysicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations