Rényi entropy of locally excited states with thermal and boundary effect in 2D CFTs

Open Access
Regular Article - Theoretical Physics

Abstract

We study Rényi entropy of locally excited states with considering the thermal and boundary effects respectively in two dimensional conformal field theories (CFTs). Firstly, we consider locally excited states obtained by acting primary operators on a thermal state in low temperature limit. The Rényi entropy is summation of contribution from thermal effect and local excitation. Secondly, we mainly study the Rényi entropy of locally excited states in 2D CFT with a boundary. We show that the time evolution of Rényi entropy is affected by the boundary, but does not depend on the boundary condition. Moreover, we show that the maximal value of Rényi entropy always coincides with the log of quantum dimension of the primary operator. In terms of quasi-particle interpretation, the boundary behaves as an infinite potential barrier which reflects any energy moving towards it.

Keywords

Field Theories in Lower Dimensions Nonperturbative Effects 

Notes

Open Access

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References

  1. [1]
    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    M. Levin and X.G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].ADSCrossRefGoogle Scholar
  3. [3]
    F.C. Alcaraz, M.I. Berganza and G. Sierra, Entanglement of low-energy excitations in conformal field theory, Phys. Rev. Lett. 106 (2011) 201601 [arXiv:1101.2881] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    M. Nozaki, T. Numasawa and T. Takayanagi, Quantum entanglement of local operators in conformal field theories, Phys. Rev. Lett. 112 (2014) 111602 [arXiv:1401.0539] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    S. He, T. Numasawa, T. Takayanagi and K. Watanabe, Quantum dimension as entanglement entropy in two dimensional conformal field theories, Phys. Rev. D 90 (2014) 041701 [arXiv:1403.0702] [INSPIRE].ADSGoogle Scholar
  6. [6]
    M. Nozaki, Notes on quantum entanglement of local operators, JHEP 10 (2014) 147 [arXiv:1405.5875] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    C.P. Herzog and M. Spillane, Tracing through scalar entanglement, Phys. Rev. D 87 (2013) 025012 [arXiv:1209.6368] [INSPIRE].ADSGoogle Scholar
  8. [8]
    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
  9. [9]
    C.P. Herzog, Universal thermal corrections to entanglement entropy for conformal field theories on spheres, JHEP 10 (2014) 028 [arXiv:1407.1358] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    C.P. Herzog and J. Nian, Thermal corrections to Rényi entropies for conformal field theories, arXiv:1411.6505 [INSPIRE].
  11. [11]
    J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    J.L. Cardy, Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 275 (1986) 200 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    J. Fuchs and C. Schweigert, Completeness of boundary conditions for the critical three state Potts model, Phys. Lett. B 441 (1998) 141 [hep-th/9806121] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    T. Takayanagi, Holographic dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    M. Fujita, T. Takayanagi and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043 [arXiv:1108.5152] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    P. Caputa, J. Simón, A. Štikonas and T. Takayanagi, Quantum entanglement of localized excited states at finite temperature, JHEP 01 (2015) 102 [arXiv:1410.2287] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    B. Chen and J.-Q. Wu, Single interval Rényi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].MathSciNetGoogle Scholar
  21. [21]
    J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer, New York U.S.A. (1998) [INSPIRE].MATHGoogle Scholar
  24. [24]
    V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in two-dimensional statistical models, Nucl. Phys. B 240 (1984) 312 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    V.S. Dotsenko and V.A. Fateev, Four point correlation functions and the operator algebra in the two-dimensional conformal invariant theories with the central charge c < 1, Nucl. Phys. B 251 (1985) 691 [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    A.M. Polyakov, Non-Hamiltonian approach to the quantum field theory at small distances, submitted to Zh. Eksp. Teor. Fiz. (1974) [INSPIRE].
  29. [29]
    G.W. Moore and N. Seiberg, Polynomial equations for rational conformal field theories, Phys. Lett. B 212 (1988) 451 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    G.W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].ADSCrossRefMATHGoogle Scholar
  32. [32]
    J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    R. Dijkgraaf and E.P. Verlinde, Modular invariance and the fusion algebra, Nucl. Phys. Proc. Suppl. 5B (1988) 87 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372 (1992) 654 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    G.W. Moore and N. Seiberg, Naturality in conformal field theory, Nucl. Phys. B 313 (1989) 16 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    B. Chen and J.-Q. Wu, Large interval limit of Rényi entropy at high temperature, arXiv:1412.0763 [INSPIRE].
  37. [37]
    S. Jackson, L. McGough and H. Verlinde, Conformal bootstrap, universality and gravitational scattering, arXiv:1412.5205 [INSPIRE].
  38. [38]
    J.-M. Stéphan and J. Dubail, Local quantum quenches in critical one-dimensional systems: entanglement, the Loschmidt echo, and light-cone effects, J. Stat. Mech. (2011) P08019 [arXiv:1105.4846].
  39. [39]
    P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, Prog. Theor. Exp. Phys. 2014 (2014) 093B06 [arXiv:1405.5946] [INSPIRE].CrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.State Key Laboratory of Theoretical PhysicsInstitute of Theoretical Physics, Chinese Academy of ScienceBeijingChina
  2. 2.Yukawa Institute for Theoretical PhysicsKyoto UniversitySakyo-kuJapan

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