Journal of High Energy Physics

, 2014:33 | Cite as

Volume law for the entanglement entropy in non-local QFTs

Open Access
Article

Abstract

In this paper, we present a simple class of non-local field theories whose ground state entanglement entropy follows a volume law as long as the size of subsystem is smaller than a certain scale. We will confirm this volume law both from numerical calculations and from analytical estimation. This behavior fits nicely with holographic results for spacetimes whose curvatures are much smaller than AdS spaces such as those in the flat spacetime.

Keywords

AdS-CFT Correspondence Holography and condensed matter physics (AdS/CMT) Renormalization Group Lattice Quantum Field Theory 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    J. Eisert, M. Cramer and M. Plenio, Area laws for the entanglement entropya review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    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].ADSMathSciNetGoogle Scholar
  3. [3]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  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].
  5. [5]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
  9. [9]
    L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    D. Bigatti and L. Susskind, TASI lectures on the holographic principle, hep-th/0002044 [INSPIRE].
  11. [11]
    W. Li and T. Takayanagi, Holography and Entanglement in Flat Spacetime, Phys. Rev. Lett. 106 (2011) 141301 [arXiv:1010.3700] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J.L. Barbon and C.A. Fuertes, A note on the extensivity of the holographic entanglement entropy, JHEP 05 (2008) 053 [arXiv:0801.2153] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    J.L. Barbon and C.A. Fuertes, Holographic entanglement entropy probes (non)locality, JHEP 04 (2008) 096 [arXiv:0803.1928] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    W. Fischler, A. Kundu and S. Kundu, Holographic Entanglement in a Noncommutative Gauge Theory, JHEP 01 (2014) 137 [arXiv:1307.2932] [INSPIRE].CrossRefGoogle Scholar
  15. [15]
    J.L. Karczmarek and C. Rabideau, Holographic entanglement entropy in nonlocal theories, arXiv:1307.3517 [INSPIRE].
  16. [16]
    D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    G. Vitagliano, A. Riera and J. Latorre, Violation of area-law scaling for the entanglement entropy in spin 1/2 chains, New J. Phys. 12 (2010) 113049 [arXiv:1003.1292] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of Entanglement Renormalization in Quantum Field Theories, JHEP 10 (2012) 193 [arXiv:1208.3469] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    M. Ghasemi Nezhadhaghighi and M. Rajabpour, Quantum entanglement entropy and classical mutual information in long-range harmonic oscillators, Phys. Rev. B 88 (2013) 045426 [arXiv:1306.0982] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    J.L. Karczmarek and P. Sabella-Garnier, Entanglement entropy on the fuzzy sphere, arXiv:1310.8345 [INSPIRE].
  21. [21]
    D. Dou and B. Ydri, Entanglement entropy on fuzzy spaces, Phys. Rev. D 74 (2006) 044014 [gr-qc/0605003] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    D. Dou, Comments on the Entanglement Entropy on Fuzzy Spaces, Mod. Phys. Lett. A 24 (2009) 2467 [arXiv:0903.3731] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    I. Peschel, Letter to the editor: Calculation of reduced density matrices from correlation functions, J. Phys. A: Math. Gen. 36 (2003) L205 [cond-mat/0212631].ADSCrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    M.-C. Chung and I. Peschel, Density-matrix spectra for two-dimensional quantum systems, Phys. Rev. B 62 (2000) 4191 [cond-mat/0004222].ADSCrossRefGoogle Scholar
  25. [25]
    H. Casini and M. Huerta, Entanglement and alpha entropies for a massive scalar field in two dimensions, J. Stat. Mech. 0512 (2005) P12012 [cond-mat/0511014] [INSPIRE].CrossRefGoogle Scholar
  26. [26]
    H. Casini, C. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 0507 (2005) P07007 [cond-mat/0505563] [INSPIRE].Google Scholar
  27. [27]
    T. Azeyanagi, T. Nishioka and T. Takayanagi, Near Extremal Black Hole Entropy as Entanglement Entropy via AdS 2 /CF T 1, Phys. Rev. D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].ADSMathSciNetGoogle Scholar
  28. [28]
    H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].MathSciNetGoogle Scholar
  29. [29]
    C.P. Herzog and M. Spillane, Tracing Through Scalar Entanglement, Phys. Rev. D 87 (2013) 025012 [arXiv:1209.6368] [INSPIRE].ADSGoogle Scholar
  30. [30]
    C.P. Herzog and T. Nishioka, Entanglement Entropy of a Massive Fermion on a Torus, JHEP 03 (2013) 077 [arXiv:1301.0336] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  31. [31]
    N. Shiba, Entanglement Entropy of Two Black Holes and Entanglement Entropic Force, Phys. Rev. D 83 (2011) 065002 [arXiv:1011.3760] [INSPIRE].ADSGoogle Scholar
  32. [32]
    N. Shiba, Entanglement Entropy of Two Spheres, JHEP 07 (2012) 100 [arXiv:1201.4865] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    M. Creutz, Quarks, gluons and lattices, Cambridge University Press, (1985).Google Scholar
  34. [34]
    B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].ADSGoogle Scholar
  35. [35]
    G. Vidal, Entanglement renormalization: an introduction, arXiv:0912.1651.
  36. [36]
    G. Evenbly and G. Vidal, Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz, arXiv:1109.5334.
  37. [37]
    J. Haegeman, T.J. Osborne, H. Verschelde and F. Verstraete, Entanglement Renormalization for Quantum Fields in Real Space, Phys. Rev. Lett. 110 (2013) 100402 [arXiv:1102.5524] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    T. Nishioka and T. Takayanagi, AdS Bubbles, Entropy and Closed String Tachyons, JHEP 01 (2007) 090 [hep-th/0611035] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    I.R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796 (2008) 274 [arXiv:0709.2140] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Yukawa Institute for Theoretical Physics (YITP)Kyoto UniversityKyotoJapan
  2. 2.Kavli Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan

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