Entanglement entropy in non-relativistic field theories

  • Sergey N. Solodukhin
Open Access


We calculate entanglement entropy in a non-relativistic field theory described by the Schrödinger operator. We demonstrate that the entropy is characterized by i) the area law and ii) UV divergences that are identical to those in the relativistic field theory. These observations are further supported by a holographic consideration. We use the non-relativistic symmetry and completely specify entanglement entropy in large class of non-relativistic theories described by the field operators polynomial in derivatives. The entropy of interacting fields is analyzed in some detail. We suggest that the area law of the entropy can be tested in experiments with condensed matter systems such as liquid helium.


AdS-CFT Correspondence Space-Time Symmetries 


  1. [1]
    L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [SPIRES].MathSciNetADSGoogle Scholar
  2. [2]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  3. [3]
    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [SPIRES].MathSciNetGoogle Scholar
  4. [4]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [SPIRES].MathSciNetGoogle Scholar
  5. [5]
    S.N. Solodukhin, Entanglement entropy of black holes and AdS/CFT correspondence, Phys. Rev. Lett. 97 (2006) 201601 [hep-th/0606205] [SPIRES].CrossRefADSGoogle Scholar
  6. [6]
    S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [SPIRES].MathSciNetADSGoogle Scholar
  7. [7]
    D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schroedinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [SPIRES].MathSciNetADSGoogle Scholar
  8. [8]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  9. [9]
    C. Duval, G.W. Gibbons and P. Horvathy, Celestial mechanics, conformal structures and gravitational waves, Phys. Rev. D 43 (1991) 3907 [hep-th/0512188] [SPIRES].MathSciNetADSGoogle Scholar
  10. [10]
    S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [SPIRES].MathSciNetADSGoogle Scholar
  11. [11]
    C.G. Callan, Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [SPIRES].MathSciNetADSGoogle Scholar
  12. [12]
    J.S. Dowker, Quantum field theory on a cone, J. Phys. A 10 (1977) 115 [SPIRES].MathSciNetADSGoogle Scholar
  13. [13]
    D.V. Fursaev, Spectral geometry and one loop divergences on manifolds with conical singularities, Phys. Lett. B 334 (1994) 53 [hep-th/9405143] [SPIRES].MathSciNetADSGoogle Scholar
  14. [14]
    A. Sommerfeld, Über verzweigte Potentiale im Raum,, Proc. Lond. Math. Soc. 28 (1897) 395.CrossRefGoogle Scholar
  15. [15]
    I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, Academic Press, U.S.A. (1980).MATHGoogle Scholar
  16. [16]
    T. Azeyanagi, W. Li and T. Takayanagi, On string theory duals of Lifshitz-like fixed points, JHEP 06 (2009) 084 [arXiv:0905.0688] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  17. [17]
    D.V. Fursaev, The Heat kernel expansion on a cone and quantum fields near cosmic strings, Class. Quant. Grav. 11 (1994) 1431 [hep-th/9309050] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    D.N. Kabat, S.H. Shenker and M.J. Strassler, Black hole entropy in the O(N) model, Phys. Rev. D 52 (1995) 7027 [hep-th/9506182] [SPIRES].MathSciNetADSGoogle Scholar
  19. [19]
    M.A. Metlitski, C.A. Fuertes and S. Sachdev, Entanglement entropy in the O(N) model, arXiv:0904.4477.
  20. [20]
    O. Lahav et al., A sonic black hole in a density-inverted Bose-Einstein condensate, arXiv:0906.1337.

Copyright information

© The Author(s) 2010

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

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique Théorique CNRS-UMR 6083Université de ToursToursFrance

Personalised recommendations