Holographic view on quantum correlations and mutual information between disjoint blocks of a quantum critical system

  • Javier Molina-Vilaplana
  • Pasquale Sodano


In (d + 1) dimensional Multiscale Entanglement Renormalization Ansatz (MERA) networks, tensors are connected so as to reproduce the discrete, (d + 2) holographic geometry of Anti de Sitter space (AdS d+2) with the original system lying at the boundary. We analyze the MERA renormalization flow that arises when computing the quantum correlations between two disjoint blocks of a quantum critical system, to show that the structure of the causal cones characteristic of MERA, requires a transition between two different regimes attainable by changing the ratio between the size and the separation of the two disjoint blocks. We argue that this transition in the MERA causal developments of the blocks may be easily accounted by an AdS d+2 black hole geometry when the mutual information is computed using the Ryu-Takayanagi formula. As an explicit example, we use a BTZ AdS3 black hole to compute the MI and the quantum correlations between two disjoint intervals of a one dimensional boundary critical system. Our results for this low dimensional system not only show the existence of a phase transition emerging when the conformal four point ratio reaches a critical value but also provide an intuitive entropic argument accounting for the source of this instability. We discuss the robustness of this transition when finite temperature and finite size effects are taken into account.


Holography and condensed matter physics (AdS/CMT) Renormalization Group Field Theories in Lower Dimensions Black Holes in String Theory 


  1. [1]
    C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    M. Caraglio and F. Gliozzi, Entanglement Entropy and Twist Fields, JHEP 11 (2008) 076 [arXiv:0808.4094] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [SPIRES].
  4. [4]
    S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [SPIRES].MathSciNetADSGoogle Scholar
  9. [9]
    O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    L. Susskind and E. Witten, The holographic bound in Anti-de Sitter space, hep-th/9805114 [SPIRES].
  11. [11]
    S. Furukawa, V. Pasquier and J. Shiraishi, Mutual Information and Compactification Radius in a c = 1 Critical Phase in One Dimension, arXiv:0809.5113 [SPIRES].
  12. [12]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. (2009) P11001 [arXiv:0905.2069] [SPIRES].
  13. [13]
    M.W. Wolf, F. Verstraete, M. Hastings and J. Cirac, Area laws in quantum systems: Mutual information and correlations, Phys. Rev. Lett. 100 (2008) 070502 [arXiv:0704.3906].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    G. Vidal and R. F. Werner, A computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117].ADSGoogle Scholar
  15. [15]
    H. Wichterich, J. Molina-Vilaplana and S. Bose, Scaling of entanglement between separated blocks in spin chains at criticality, Phys. Rev. A 80 (2009) 010304 [arXiv:0811.1285].ADSGoogle Scholar
  16. [16]
    S. Marcovitch, A. Retzker, M.B. Plenio and B. Reznik, Critical and noncritical long-range entanglement in Klein-Gordon fields, Phys. Rev. A 80 (2009) 012325 [arXiv:0811.1288].MathSciNetADSGoogle Scholar
  17. [17]
    H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    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] [SPIRES].
  19. [19]
    M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [SPIRES].ADSGoogle Scholar
  20. [20]
    B.-Q. Jin and V. Korepin, Entanglement entropy for disjoint subsystems in XX spin chain, arXiv:1104.1004 [SPIRES].
  21. [21]
    S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [SPIRES].
  22. [22]
    B. Swingle, Entanglement Renormalization and Holography, arXiv:0905.1317 [SPIRES].
  23. [23]
    B. Swingle, Mutual information and the structure of entanglement in quantum field theory, arXiv:1010.4038 [SPIRES].
  24. [24]
    G. Evenbly and G. Vidal, Tensor network states and geometry, arXiv:1106.1082 [SPIRES].
  25. [25]
    G. Vidal, Class of quantum many-body states that can be efficiently simulated, Phys. Rev. Lett. 101 (2008) 110501 [quant-ph/0610099].ADSCrossRefGoogle Scholar
  26. [26]
    G. Vidal, Entanglement renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165v2].ADSCrossRefGoogle Scholar
  27. [27]
    D.J. Gross and H. Ooguri, Aspects of large-N gauge theory dynamics as seen by string theory, Phys. Rev. D 58 (1998) 106002 [hep-th/9805129] [SPIRES].MathSciNetADSGoogle Scholar
  28. [28]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [SPIRES].
  29. [29]
    J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement renormalization for quantum fields, arXiv:1102.5524 [SPIRES].
  30. [30]
    M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  31. [31]
    P. Kraus, Lectures on black holes and the AdS 3 /CFT 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [SPIRES].MathSciNetADSGoogle Scholar
  32. [32]
    C. Misner, K. Thorne and J. Wheeler, Gravitation, W.H. Freeman, New York U.S.A. (1973).Google Scholar
  33. [33]
    V. Balasubramanian, I. García-Etxebarria, F. Larsen and J. Simón, Helical Luttinger Liquids and Three Dimensional Black Holes, arXiv:1012.4363 [SPIRES].
  34. [34]
    J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  35. [35]
    J. Louko, D. Marolf and S.F. Ross, On geodesic propagators and black hole holography, Phys. Rev. D 62 (2000) 044041 [hep-th/0002111] [SPIRES].MathSciNetADSGoogle Scholar
  36. [36]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].MathSciNetADSMATHGoogle Scholar
  37. [37]
    E. Tonni, Holographic entanglement entropy: near horizon geometry and disconnected regions, JHEP 05 (2011) 004 [arXiv:1011.0166] [SPIRES].ADSCrossRefGoogle Scholar
  38. [38]
    D. Birmingham, I. Sachs and S.N. Solodukhin, Relaxation in conformal field theory, Hawking-Page transition and quasinormal/normal modes, Phys. Rev. D 67 (2003) 104026 [hep-th/0212308] [SPIRES].MathSciNetADSGoogle Scholar
  39. [39]
    P.D. Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer, New York U.S.A. (1997).MATHCrossRefGoogle Scholar
  40. [40]
    D. Mumford, TATA Lectures on Theta, Birkhauser, Basel Switzerland (1982).Google Scholar
  41. [41]
    L. Álvarez-Gaumé, G.W. Moore and C. Vafa, Theta functions, modular invariance and strings, Commun. Math. Phys. 106 (1986) 1 [SPIRES].ADSMATHCrossRefGoogle Scholar
  42. [42]
    P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, arXiv:1107.2940 [SPIRES].
  43. [43]
    V. Coffman, J. Kundu and W.K. Wootters, Distributed Entanglement, Phys. Rev. A 61 (2000) 052306 [quant-ph/9907047] [SPIRES].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Department of Systems Engineering and AutomationTechnical University of CartagenaCartagenaSpain
  2. 2.Perimeter Institute of Theoretical PhysicsWaterlooCanada
  3. 3.Permanent Address: Dipartimento di FisicaUniversità di PerugiaPerugiaItaly

Personalised recommendations