Journal of High Energy Physics

, 2012:193 | Cite as

Holographic geometry of entanglement renormalization in quantum field theories

Article

Abstract

We study a conjectured connection between AdS/CFT and a real-space quantum renormalization group scheme, the multi-scale entanglement renormalization ansatz (MERA). By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction. The metric is formulated purely in terms of quantum field theoretical data. Using the continuum version of the MERA (cMERA), we calculate this emergent holographic metric explicitly for free scalar boson and free fermions theories, and check that the metric so computed has the properties expected from AdS/CFT. We also discuss the cMERA in a time-dependent background induced by quantum quench and estimate its corresponding metric.

Keywords

AdS-CFT Correspondence Holography and condensed matter physics (AdS/CMT) Renormalization Group 

References

  1. [1]
    G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
  2. [2]
    L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  3. [3]
    D. Bigatti and L. Susskind, TASI lectures on the holographic principle, hep-th/0002044 [INSPIRE].
  4. [4]
    J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113 ] [hep-th/9711200] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  5. [5]
    S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].MathSciNetADSGoogle Scholar
  6. [6]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].MathSciNetADSMATHGoogle Scholar
  7. [7]
    G. Vidal, Entanglement Renormalization, Phys. Rev. Lett. 99 (2007) 220405 [cond-mat/0512165].ADSCrossRefGoogle Scholar
  8. [8]
    G. Vidal, Entanglement Renormalization: an introduction, arXiv:0912.1651.
  9. [9]
    G. Evenbly and G. Vidal, Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz, arXiv:1109.5334.
  10. [10]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  11. [11]
    M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
  12. [12]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [Int. J. Mod. Phys. D 19 (2010) 2429] [arXiv:1005.3035] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  13. [13]
    B. Swingle, Mutual information and the structure of entanglement in quantum field theory, arXiv:1010.4038 [INSPIRE].
  14. [14]
    J. Molina-Vilaplana and P. Sodano, Holographic View on Quantum Correlations and Mutual Information between Disjoint Blocks of a Quantum Critical System, JHEP 10 (2011) 011 [arXiv:1108.1277] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    J. Molina-Vilaplana, Connecting Entanglement Renormalization and Gauge/Gravity dualities, arXiv:1109.5592 [INSPIRE].
  16. [16]
    V. Balasubramanian, M.B. McDermott and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory, Phys. Rev. D 86 (2012) 045014 [arXiv:1108.3568] [INSPIRE].ADSGoogle Scholar
  17. [17]
    H. Matsueda, Scaling of entanglement entropy and hyperbolic geometry, arXiv:1112.5566 [INSPIRE].
  18. [18]
    M. Ishihara, F.-L. Lin and B. Ning, Refined Holographic Entanglement Entropy for the AdS Solitons and AdS black Holes, arXiv:1203.6153 [INSPIRE].
  19. [19]
    H. Matsueda, M. Ishihara and Y. Hashizume, Tensor Network and Black Hole, arXiv:1208.0206 [INSPIRE].
  20. [20]
    K. Okunishi, Wilsons numerical renormalization group and AdS 3 geometry, arXiv:1208.1645 [INSPIRE].
  21. [21]
    H. Matsueda, Multiscale Entanglement Renormalization Ansatz for Kondo Problem, arXiv:1208.2872 [INSPIRE].
  22. [22]
    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 [INSPIRE].MathSciNetADSGoogle Scholar
  23. [23]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  24. [24]
    J. Eisert, M. Cramer and M. Plenio, Area laws for the entanglement entropy - a review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  25. [25]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].MathSciNetGoogle Scholar
  26. [26]
    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].MathSciNetGoogle Scholar
  27. [27]
    J.I. Latorre and A. Riera, A short review on entanglement in quantum spin systems, J. Phys. A 42 (2009) 4002 [arXiv:0906.1499].MathSciNetGoogle Scholar
  28. [28]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  29. [29]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].MathSciNetGoogle Scholar
  31. [31]
    T. Takayanagi, Entanglement Entropy from a Holographic Viewpoint, Class. Quant. Grav. 29 (2012) 153001 [arXiv:1204.2450] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  32. [32]
    M. Headrick and T. Takayanagi, A Holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  34. [34]
    S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].MathSciNetADSGoogle Scholar
  35. [35]
    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  37. [37]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    J. Hammersley, Extracting the bulk metric from boundary information in asymptotically AdS spacetimes, JHEP 12 (2006) 047 [hep-th/0609202] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    J. Hammersley, Numerical metric extraction in AdS/CFT, Gen. Rel. Grav. 40 (2008) 1619 [arXiv:0705.0159] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  40. [40]
    V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].CrossRefGoogle Scholar
  42. [42]
    I. Heemskerk and J. Polchinski, Holographic and Wilsonian Renormalization Groups, JHEP 06 (2011) 031 [arXiv:1010.1264] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  43. [43]
    T. Faulkner, H. Liu and M. Rangamani, Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm, JHEP 08 (2011) 051 [arXiv:1010.4036] [INSPIRE].MathSciNetADSGoogle Scholar
  44. [44]
    J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement renormalization for quantum fields, arXiv:1102.5524 [INSPIRE].
  45. [45]
    S.-S. Lee, Holographic description of quantum field theory, Nucl. Phys. B 832 (2010) 567 [arXiv:0912.5223] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    S.-S. Lee, Holographic description of large-N gauge theory, Nucl. Phys. B 851 (2011) 143 [arXiv:1011.1474] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    S.-S. Lee, Background independent holographic description : From matrix field theory to quantum gravity, arXiv:1204.1780 [INSPIRE].
  48. [48]
    J.I. Cirac and F. Verstraete, Renormalization and tensor product states in spin chains and lattices, J. Phys. A 42 (2009) 504004 [INSPIRE].MathSciNetGoogle Scholar
  49. [49]
    S.R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69 (1992) 2863 [INSPIRE].ADSCrossRefGoogle Scholar
  50. [50]
    F. Verstraete and J. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, cond-mat/0407066 [INSPIRE].
  51. [51]
    T. Nishino and K. Okunishi, A Density Matrix Algorithm for 3D Classical Models, J. Phys. Soc. Japan 67 (1998) 3066 [cond-mat/9804134].ADSCrossRefGoogle Scholar
  52. [52]
    C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  53. [53]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].CrossRefGoogle Scholar
  54. [54]
    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  55. [55]
    G. Evenbly and G. Vidal, Tensor Network States and Geometry, J. Stat. Phys. 145 (2011) 891 [arXiv:1106.1082].MathSciNetADSMATHCrossRefGoogle Scholar
  56. [56]
    L.-Y. Hung, R.C. Myers and M. Smolkin, On Holographic Entanglement Entropy and Higher Curvature Gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Entanglement Entropy in Lovelock Gravities, JHEP 07 (2011) 109 [arXiv:1101.5781] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    B. Swingle, Constructing holographic spacetimes using entanglement renormalization, arXiv:1209.3304 [INSPIRE].
  59. [59]
    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
  60. [60]
    J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic Evolution of Entanglement Entropy, JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    T. Albash and C.V. Johnson, Evolution of Holographic Entanglement Entropy after Thermal and Electromagnetic Quenches, New J. Phys. 13 (2011) 045017 [arXiv:1008.3027] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    V. Balasubramanian et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    W. Li and T. Takayanagi, Holography and Entanglement in Flat Spacetime, Phys. Rev. Lett. 106 (2011) 141301 [arXiv:1010.3700] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    D. Lynden-Bell, Negative specific heat in astronomy, physics and chemistry, Physica A 263 (1999) 293 [cond-mat/9812172] [INSPIRE].ADSGoogle Scholar
  65. [65]
    F. Bouchet, S. Gupta and D. Mukamel, Thermodynamics and dynamics of systems with long-range interactions, Physica A 389 (2010) 4389 [arXiv:1001.1479].MathSciNetADSGoogle Scholar
  66. [66]
    M.A. Vasiliev, Higher spin gauge theories: Star product and AdS space, hep-th/9910096 [INSPIRE].
  67. [67]
    M.A. Vasiliev, Holography, Unfolding and Higher-Spin Theory, arXiv:1203.5554 [INSPIRE].
  68. [68]
    I. Klebanov and A. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].MathSciNetADSGoogle Scholar
  69. [69]
    S. Giombi and X. Yin, Higher Spin Gauge Theory and Holography: The Three-Point Functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  70. [70]
    R. de Mello Koch, A. Jevicki, K. Jin and J.P. Rodrigues, AdS 4 /CF T 3 Construction from Collective Fields, Phys. Rev. D 83 (2011) 025006 [arXiv:1008.0633] [INSPIRE].ADSGoogle Scholar
  71. [71]
    M.R. Douglas, L. Mazzucato and S.S. Razamat, Holographic dual of free field theory, Phys. Rev. D 83 (2011) 071701 [arXiv:1011.4926] [INSPIRE].ADSGoogle Scholar
  72. [72]
    J. Maldacena and A. Zhiboedov, Constraining Conformal Field Theories with A Higher Spin Symmetry, arXiv:1112.1016 [INSPIRE].
  73. [73]
    J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a slightly broken higher spin symmetry, arXiv:1204.3882 [INSPIRE].
  74. [74]
    X-G. Wen, Quantum Field Theory of Many-Body Systems, Oxford University Press, Oxford, U.K. (2004).Google Scholar
  75. [75]
    M. Aguado and G. Vidal, Entanglement Renormalization and Topological Order, Phys. Rev. Lett. 100 (2008) 070404 [arXiv:0712.0348].ADSCrossRefGoogle Scholar
  76. [76]
    M. Fujita, W. Li, S. Ryu and T. Takayanagi, Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States and Hierarchy, JHEP 06 (2009) 066 [arXiv:0901.0924] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • Masahiro Nozaki
    • 1
  • Shinsei Ryu
    • 2
  • Tadashi Takayanagi
    • 1
    • 3
  1. 1.Yukawa Institute for Theoretical PhysicsKyoto UniversityKyotoJapan
  2. 2.Department of PhysicsUniversity of Illinois at Urbana-ChampaignUrbanaU.S.A.
  3. 3.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

Personalised recommendations