Entanglement entropy and differential entropy for massive flavors

  • Peter A. R. Jones
  • Marika TaylorEmail author
Open Access
Regular Article - Theoretical Physics


In this paper we compute the holographic entanglement entropy for massive flavors in the D3-D7 system, for arbitrary mass and various entangling region geometries. We show that the universal terms in the entanglement entropy exactly match those computed in the dual theory using conformal perturbation theory. We derive holographically the universal terms in the entanglement entropy for a CFT perturbed by a relevant operator, up to second order in the coupling; our results are valid for any entangling region geometry. We present a new method for computing the entanglement entropy of any top-down brane probe system using Kaluza-Klein holography and illustrate our results with massive flavors at finite density. Finally we discuss the differential entropy for brane probe systems, emphasising that the differential entropy captures only the effective lower-dimensional Einstein metric rather than the ten-dimensional geometry.


Gauge-gravity correspondence AdS-CFT Correspondence 


  1. [1]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    T. Takayanagi, Entanglement Entropy from a Holographic Viewpoint, Class. Quant. Grav. 29 (2012) 153001 [arXiv:1204.2450] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, Phys. Rev. B 85 (2012) 035121 [arXiv:1112.0573] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    K. Skenderis and M. Taylor, Kaluza-Klein holography, JHEP 05 (2006) 057 [hep-th/0603016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    K. Jensen and A. O’Bannon, Holography, Entanglement Entropy and Conformal Field Theories with Boundaries or Defects, Phys. Rev. D 88 (2013) 106006 [arXiv:1309.4523] [INSPIRE].ADSGoogle Scholar
  9. [9]
    H.-C. Chang and A. Karch, Entanglement Entropy for Probe Branes, JHEP 01 (2014) 180 [arXiv:1307.5325] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    A. Karch and C.F. Uhlemann, Generalized gravitational entropy of probe branes: flavor entanglement holographically, JHEP 05 (2014) 017 [arXiv:1402.4497] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    J. Estes, K. Jensen, A. O’Bannon, E. Tsatis and T. Wrase, On Holographic Defect Entropy, JHEP 05 (2014) 084 [arXiv:1403.6475] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    H.-C. Chang, A. Karch and C.F. Uhlemann, Flavored \( \mathcal{N}=4 \) SYMa highly entangled quantum liquid, JHEP 09 (2014) 110 [arXiv:1406.2705] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    I.R. Klebanov, T. Nishioka, S.S. Pufu and B.R. Safdi, On Shape Dependence and RG Flow of Entanglement Entropy, JHEP 07 (2012) 001 [arXiv:1204.4160] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    T. Nishioka, Relevant Perturbation of Entanglement Entropy and Stationarity, Phys. Rev. D 90 (2014) 045006 [arXiv:1405.3650] [INSPIRE].ADSGoogle Scholar
  15. [15]
    J. Lee, A. Lewkowycz, E. Perlmutter and B.R. Safdi, Rényi entropy, stationarity and entanglement of the conformal scalar, JHEP 03 (2015) 075 [arXiv:1407.7816] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Allais and M. Mezei, Some results on the shape dependence of entanglement and Rényi entropies, Phys. Rev. D 91 (2015) 046002 [arXiv:1407.7249] [INSPIRE].ADSGoogle Scholar
  17. [17]
    M. Mezei, Entanglement entropy across a deformed sphere, Phys. Rev. D 91 (2015) 045038 [arXiv:1411.7011] [INSPIRE].ADSGoogle Scholar
  18. [18]
    O. Ben-Ami, D. Carmi and M. Smolkin, Renormalization group flow of entanglement entropy on spheres, arXiv:1504.00913 [INSPIRE].
  19. [19]
    A. Schwimmer and S. Theisen, Entanglement Entropy, Trace Anomalies and Holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    V. Rosenhaus and M. Smolkin, Entanglement Entropy Flow and the Ward Identity, Phys. Rev. Lett. 113 (2014) 261602 [arXiv:1406.2716] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    V. Rosenhaus and M. Smolkin, Entanglement entropy, planar surfaces and spectral functions, JHEP 09 (2014) 119 [arXiv:1407.2891] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    V. Rosenhaus and M. Smolkin, Entanglement Entropy for Relevant and Geometric Perturbations, JHEP 02 (2015) 015 [arXiv:1410.6530] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S. He, J.-R. Sun and H.-Q. Zhang, On Holographic Entanglement Entropy with Second Order Excitations, arXiv:1411.6213 [INSPIRE].
  25. [25]
    H. Casini, F.D. Mazzitelli and E. Testé, Area terms in entanglement entropy, Phys. Rev. D 91 (2015) 104035 [arXiv:1412.6522] [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    C. Park, Logarithmic Corrections in the Entanglement Entropy, arXiv:1505.03951 [INSPIRE].
  27. [27]
    A. Karch, A. O’Bannon and K. Skenderis, Holographic renormalization of probe D-branes in AdS/CFT, JHEP 04 (2006) 015 [hep-th/0512125] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Swingle and M. Van Raamsdonk, Universality of Gravity from Entanglement, arXiv:1405.2933 [INSPIRE].
  30. [30]
    E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, Class. Quant. Grav. 31 (2014) 214002 [arXiv:1212.5183] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    V. Balasubramanian, B. Czech, B.D. Chowdhury and J. de Boer, The entropy of a hole in spacetime, JHEP 10 (2013) 220 [arXiv:1305.0856] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    V. Balasubramanian, B.D. Chowdhury, B. Czech, J. de Boer and M.P. Heller, Bulk curves from boundary data in holography, Phys. Rev. D 89 (2014) 086004 [arXiv:1310.4204] [INSPIRE].ADSGoogle Scholar
  33. [33]
    R.C. Myers, J. Rao and S. Sugishita, Holographic Holes in Higher Dimensions, JHEP 06 (2014) 044 [arXiv:1403.3416] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    B. Czech, X. Dong and J. Sully, Holographic Reconstruction of General Bulk Surfaces, JHEP 11 (2014) 015 [arXiv:1406.4889] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    M. Headrick, R.C. Myers and J. Wien, Holographic Holes and Differential Entropy, JHEP 10 (2014) 149 [arXiv:1408.4770] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    A. Fayyazuddin and M. Spalinski, Large-N superconformal gauge theories and supergravity orientifolds, Nucl. Phys. B 535 (1998) 219 [hep-th/9805096] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    O. Aharony, A. Fayyazuddin and J.M. Maldacena, The large-N limit of N = 2, N = 1 field theories from three-branes in F-theory, JHEP 07 (1998) 013 [hep-th/9806159] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  39. [39]
    A. Karch and E. Katz, Adding flavor to AdS/CFT, JHEP 06 (2002) 043 [hep-th/0205236] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    J. Erdmenger, N. Evans, I. Kirsch and E. Threlfall, Mesons in Gauge/Gravity DualsA Review, Eur. Phys. J. A 35 (2008) 81 [arXiv:0711.4467] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    F. Bigazzi, A.L. Cotrone, J. Mas, A. Paredes, A.V. Ramallo and J. Tarrio, D3-D7 quark-gluon Plasmas, JHEP 11 (2009) 117 [arXiv:0909.2865] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    F. Bigazzi, A.L. Cotrone, J. Mas, D. Mayerson and J. Tarrio, D3-D7 quark-gluon Plasmas at Finite Baryon Density, JHEP 04 (2011) 060 [arXiv:1101.3560] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  43. [43]
    F. Bigazzi, A.L. Cotrone, J. Mas, D. Mayerson and J. Tarrio, Holographic Duals of Quark Gluon Plasmas with Unquenched Flavors, Commun. Theor. Phys. 57 (2012) 364 [arXiv:1110.1744] [INSPIRE].CrossRefzbMATHGoogle Scholar
  44. [44]
    D. Mateos, R.C. Myers and R.M. Thomson, Holographic phase transitions with fundamental matter, Phys. Rev. Lett. 97 (2006) 091601 [hep-th/0605046] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    S. Kobayashi, D. Mateos, S. Matsuura, R.C. Myers and R.M. Thomson, Holographic phase transitions at finite baryon density, JHEP 02 (2007) 016 [hep-th/0611099] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    K. Kontoudi and G. Policastro, Flavor corrections to the entanglement entropy, JHEP 01 (2014) 043 [arXiv:1310.4549] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    M.P. Hertzberg and F. Wilczek, Some Calculable Contributions to Entanglement Entropy, Phys. Rev. Lett. 106 (2011) 050404 [arXiv:1007.0993] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    M.P. Hertzberg, Entanglement Entropy in Scalar Field Theory, J. Phys. A 46 (2013) 015402 [arXiv:1209.4646] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  49. [49]
    H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    H. Casini, C.D. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. (2005) P07007 [cond-mat/0505563] [INSPIRE].
  51. [51]
    H. Casini and M. Huerta, Entanglement and alpha entropies for a massive scalar field in two dimensions, J. Stat. Mech. (2005) P12012 [cond-mat/0511014] [INSPIRE].
  52. [52]
    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  53. [53]
    H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    H. Liu and M. Mezei, Probing renormalization group flows using entanglement entropy, JHEP 01 (2014) 098 [arXiv:1309.6935] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, The Mass Spectrum of Chiral N = 2 D = 10 Supergravity on S 5, Phys. Rev. D 32 (1985) 389 [INSPIRE].ADSGoogle Scholar
  56. [56]
    M.J. Duff, R.R. Khuri and J.X. Lu, String solitons, Phys. Rept. 259 (1995) 213 [hep-th/9412184] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    K. Skenderis and M. Taylor, Branes in AdS and p p wave space-times, JHEP 06 (2002) 025 [hep-th/0204054] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    M.M. Wolf, Violation of the entropic area law for Fermions, Phys. Rev. Lett. 96 (2006) 010404 [quant-ph/0503219] [INSPIRE].ADSCrossRefGoogle Scholar
  59. [59]
    D. Gioev and I. Klich, Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture, Phys. Rev. Lett. 96 (2006) 100503 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    M. Cramer, J. Eisert and M.B. Plenio, Statistics dependence of the entanglement entropy, Phys. Rev. Lett. 98 (2007) 220603 [quant-ph/0611264] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    N. Evans, A. Gebauer, K.-Y. Kim and M. Magou, Holographic Description of the Phase Diagram of a Chiral Symmetry Breaking Gauge Theory, JHEP 03 (2010) 132 [arXiv:1002.1885] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  62. [62]
    K. Jensen, A. Karch and E.G. Thompson, A Holographic Quantum Critical Point at Finite Magnetic Field and Finite Density, JHEP 05 (2010) 015 [arXiv:1002.2447] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  63. [63]
    K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    S. Banerjee, Trace Anomaly Matching and Exact Results For Entanglement Entropy, arXiv:1405.4876 [INSPIRE].
  65. [65]
    S. Banerjee, Note On The Dilaton Effective Action And Entanglement Entropy, arXiv:1406.3038 [INSPIRE].
  66. [66]
    L.-Y. Hung, R.C. Myers and M. Smolkin, Some Calculable Contributions to Holographic Entanglement Entropy, JHEP 08 (2011) 039 [arXiv:1105.6055] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  67. [67]
    D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT(d)/AdS(d+1) correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Continuous distributions of D3-branes and gauged supergravity, JHEP 07 (2000) 038 [hep-th/9906194] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    A. Mollabashi, N. Shiba and T. Takayanagi, Entanglement between Two Interacting CFTs and Generalized Holographic Entanglement Entropy, JHEP 04 (2014) 185 [arXiv:1403.1393] [INSPIRE].ADSCrossRefGoogle Scholar
  71. [71]
    F. Aprile and V. Niarchos, Large-N transitions of the connectivity index, JHEP 02 (2015) 083 [arXiv:1410.7773] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    K. Skenderis and M. Taylor, Holographic Coulomb branch vevs, JHEP 08 (2006) 001 [hep-th/0604169] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    A. Karch and C.F. Uhlemann, Holographic entanglement entropy and the internal space, Phys. Rev. D 91 (2015) 086005 [arXiv:1501.00003] [INSPIRE].ADSGoogle Scholar
  74. [74]
    O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Quantum Hall Effect in a Holographic Model, JHEP 10 (2010) 063 [arXiv:1003.4965] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  75. [75]
    C. Kristjansen and G.W. Semenoff, Giant D5 Brane Holographic Hall State, JHEP 06 (2013) 048 [arXiv:1212.5609] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    J. Erdmenger, C. Hoyos, A. O’Bannon and J. Wu, A Holographic Model of the Kondo Effect, JHEP 12 (2013) 086 [arXiv:1310.3271] [INSPIRE].ADSCrossRefGoogle Scholar
  77. [77]
    Y. Bea, E. Conde, N. Jokela and A.V. Ramallo, Unquenched massive flavors and flows in Chern-Simons matter theories, JHEP 12 (2013) 033 [arXiv:1309.4453] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  78. [78]
    K.K. Kim, O.-K. Kwon, C. Park and H. Shin, Renormalized Entanglement Entropy Flow in Mass-deformed ABJM Theory, Phys. Rev. D 90 (2014) 046006 [arXiv:1404.1044] [INSPIRE].ADSGoogle Scholar
  79. [79]
    J. Cardy and C.P. Herzog, Universal Thermal Corrections to Single Interval Entanglement Entropy for Two Dimensional Conformal Field Theories, Phys. Rev. Lett. 112 (2014) 171603 [arXiv:1403.0578] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    C.P. Herzog, Universal Thermal Corrections to Entanglement Entropy for Conformal Field Theories on Spheres, JHEP 10 (2014) 28 [arXiv:1407.1358] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Higher spin entanglement entropy from CFT, JHEP 06 (2014) 096 [arXiv:1402.0007] [INSPIRE].ADSCrossRefGoogle Scholar
  82. [82]
    S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Universal correction to higher spin entanglement entropy, Phys. Rev. D 90 (2014) 041903 [arXiv:1405.0015] [INSPIRE].ADSGoogle Scholar
  83. [83]
    S. Datta, J.R. David and S.P. Kumar, Conformal perturbation theory and higher spin entanglement entropy on the torus, JHEP 04 (2015) 041 [arXiv:1412.3946] [INSPIRE].ADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2015

Authors and Affiliations

  1. 1.Physics and Astronomy and STAG Research CentreUniversity of SouthamptonSouthamptonUnited Kingdom
  2. 2.Mathematical Sciences and STAG Research CentreUniversity of SouthamptonSouthamptonUnited Kingdom

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