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Symmetry decomposition of relative entropies in conformal field theory

Journal of High Energy Physics volume 2021, Article number: 84 (2021) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We consider the symmetry resolution of relative entropies in the 1+1 dimensional free massless compact boson conformal field theory (CFT) which presents an internal U(1) symmetry. We calculate various symmetry resolved Rényi relative entropies between one interval reduced density matrices of CFT primary states using the replica method. By taking the replica limit, the symmetry resolved relative entropy can be obtained. We also take the XX spin chain model as a concrete lattice realization of this CFT to perform numerical computation. The CFT predictions are tested against exact numerical calculations finding perfect agreement.

References

  1. L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80 (2008) 517 [quant-ph/0703044] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  2. P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  3. J. Eisert, M. Cramer and M. B. Plenio, Area laws for the entanglement entropy — a review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  4. N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rept. 646 (2016) 1 [arXiv:1512.03388] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  5. S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].

  6. S. W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  7. S. D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  8. J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  9. A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, The entropy of Hawking radiation, arXiv:2006.06872 [INSPIRE].

  10. V. Vedral, The role of relative entropy in quantum information theory, Rev. Mod. Phys. 74 (2002) 197 [quant-ph/0102094] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  11. D. D. Blanco, H. Casini, L.-Y. Hung and R. C. Myers, Relative Entropy and Holography, JHEP 08 (2013) 060 [arXiv:1305.3182] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  12. N. Lashkari, Relative Entropies in Conformal Field Theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].

    ADS  Article  Google Scholar 

  13. N. Lashkari, Modular Hamiltonian for Excited States in Conformal Field Theory, Phys. Rev. Lett. 117 (2016) 041601 [arXiv:1508.03506] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  14. G. Sárosi and T. Ugajin, Relative entropy of excited states in two dimensional conformal field theories, JHEP 07 (2016) 114 [arXiv:1603.03057] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. P. Ruggiero and P. Calabrese, Relative Entanglement Entropies in 1 + 1-dimensional conformal field theories, JHEP 02 (2017) 039 [arXiv:1612.00659] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  16. D. L. Jafferis, A. Lewkowycz, J. Maldacena and S. J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  17. H. Casini, E. Teste and G. Torroba, Relative entropy and the RG flow, JHEP 03 (2017) 089 [arXiv:1611.00016] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  18. J. Bhattacharya, M. Nozaki, T. Takayanagi and T. Ugajin, Thermodynamical Property of Entanglement Entropy for Excited States, Phys. Rev. Lett. 110 (2013) 091602 [arXiv:1212.1164] [INSPIRE].

    ADS  Article  Google Scholar 

  19. S. Balakrishnan, T. Faulkner, Z. U. Khandker and H. Wang, A General Proof of the Quantum Null Energy Condition, JHEP 09 (2019) 020 [arXiv:1706.09432] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  20. H. Casini, I. Salazar Landea and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140 [arXiv:1607.00390] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  21. H.-Q. Zhou, R. Orus and G. Vidal, Ground State Fidelity from Tensor Network Representations, Phys. Rev. Lett. 100 (2008) 080601 [arXiv:0709.4596] [INSPIRE].

    ADS  Article  Google Scholar 

  22. J. Zhang, P. Ruggiero and P. Calabrese, Subsystem Trace Distance in Quantum Field Theory, Phys. Rev. Lett. 122 (2019) 141602 [arXiv:1901.10993] [INSPIRE].

    ADS  Article  Google Scholar 

  23. J. Zhang, P. Ruggiero and P. Calabrese, Subsystem trace distance in low-lying states of (1 + 1)-dimensional conformal field theories, JHEP 10 (2019) 181 [arXiv:1907.04332] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  24. S. Kullback and R. A. Leibler, On Information and Sufficiency, Annals Math. Statist. 22 (1951) 79 [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  25. H. Araki, Relative Entropy of States of Von Neumann Algebras, Publ. Res. Inst. Math. Sci. Kyoto 1976 (1976) 809 [INSPIRE].

    MATH  Google Scholar 

  26. M. Goldstein and E. Sela, Symmetry-resolved entanglement in many-body systems, Phys. Rev. Lett. 120 (2018) 200602 [arXiv:1711.09418] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. R. Bonsignori, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement in free fermionic systems, J. Phys. A 52 (2019) 475302 [arXiv:1907.02084] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. S. Murciano, G. Di Giulio and P. Calabrese, Entanglement and symmetry resolution in two dimensional free quantum field theories, JHEP 08 (2020) 073 [arXiv:2006.09069] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  29. R. Bonsignori and P. Calabrese, Boundary effects on symmetry resolved entanglement, J. Phys. A 54 (2021) 015005 [arXiv:2009.08508] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  30. S. Fraenkel and M. Goldstein, Symmetry resolved entanglement: Exact results in 1D and beyond, J. Stat. Mech. 2003 (2020) 033106 [arXiv:1910.08459] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  31. B. Estienne, Y. Ikhlef and A. Morin-Duchesne, Finite-size corrections in critical symmetry-resolved entanglement, SciPost Phys. 10 (2021) 054 [arXiv:2010.10515] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. D. Azses and E. Sela, Symmetry-resolved entanglement in symmetry-protected topological phases, Phys. Rev. B 102 (2020) 235157 [arXiv:2008.09332] [INSPIRE].

    ADS  Article  Google Scholar 

  33. V. Vitale et al., Symmetry-resolved dynamical purification in synthetic quantum matter, arXiv:2101.07814 [INSPIRE].

  34. E. Cornfeld, M. Goldstein and E. Sela, Imbalance entanglement: Symmetry decomposition of negativity, Phys. Rev. A 98 (2018) 032302 [arXiv:1804.00632] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  35. S. Murciano, R. Bonsignori and P. Calabrese, Symmetry decomposition of negativity of massless free fermions, SciPost Phys. 10 (2021) 111 [arXiv:2102.10054] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  36. P. Caputa, G. Mandal and R. Sinha, Dynamical entanglement entropy with angular momentum and U(1) charge, JHEP 11 (2013) 052 [arXiv:1306.4974] [INSPIRE].

    ADS  Article  Google Scholar 

  37. A. Belin, L.-Y. Hung, A. Maloney, S. Matsuura, R. C. Myers and T. Sierens, Holographic Charged Renyi Entropies, JHEP 12 (2013) 059 [arXiv:1310.4180] [INSPIRE].

    ADS  Article  Google Scholar 

  38. A. Belin, L.-Y. Hung, A. Maloney and S. Matsuura, Charged Renyi entropies and holographic superconductors, JHEP 01 (2015) 059 [arXiv:1407.5630] [INSPIRE].

    ADS  MATH  Article  Google Scholar 

  39. P. Caputa, M. Nozaki and T. Numasawa, Charged Entanglement Entropy of Local Operators, Phys. Rev. D 93 (2016) 105032 [arXiv:1512.08132] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  40. S. Zhao, C. Northe and R. Meyer, Symmetry-Resolved Entanglement in AdS3/CFT2 coupled to U(1) Chern-Simons Theory, arXiv:2012.11274 [INSPIRE].

  41. P. Caputa and A. Veliz-Osorio, Entanglement constant for conformal families, Phys. Rev. D 92 (2015) 065010 [arXiv:1507.00582] [INSPIRE].

    ADS  Article  Google Scholar 

  42. J. S. Dowker, Charged Renyi entropies for free scalar fields, J. Phys. A 50 (2017) 165401 [arXiv:1512.01135] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  43. J. S. Dowker, Conformal weights of charged Rényi entropy twist operators for free scalar fields in arbitrary dimensions, J. Phys. A 49 (2016) 145401 [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  44. P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].

    MATH  Google Scholar 

  45. M. I. Berganza, F. C. Alcaraz and G. Sierra, Entanglement of excited states in critical spin chians, J. Stat. Mech. 1201 (2012) P01016 [arXiv:1109.5673] [INSPIRE].

    Google Scholar 

  46. P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997) [DOI].

  47. F. Essler, A. M. Laeuchli and P. Calabrese, Shell-filling effect in the entanglement entropies of spinful fermions, Phys. Rev. Lett. 110 (2013) 115701 [arXiv:1211.2474].

    ADS  Article  Google Scholar 

  48. P. Calabrese, F. Essler and A. M. Läuchli, Entanglement entropies of the quarter filled hubbard model, J. Stat. Mech. 1409 (2014) P09025 [arXiv:1406.7477].

    MathSciNet  MATH  Article  Google Scholar 

  49. L. Capizzi, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement entropy of excited states in a CFT, J. Stat. Mech. 2007 (2020) 073101 [arXiv:2003.04670] [INSPIRE].

    MATH  Article  Google Scholar 

  50. J. L. Cardy, O. A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. D. X. Horváth and P. Calabrese, Symmetry resolved entanglement in integrable field theories via form factor bootstrap, JHEP 11 (2020) 131 [arXiv:2008.08553] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  52. D. X. Horvath, L. Capizzi and P. Calabrese, U(1) symmetry resolved entanglement in free 1 + 1 dimensional field theories via form factor bootstrap, JHEP 05 (2021) 197 [arXiv:2103.03197] [INSPIRE].

    ADS  MathSciNet  MATH  Article  Google Scholar 

  53. L. Capizzi and P. Calabrese, Symmetry resolved relative entropies and distances in conformal field theory, arXiv:2105.08596 [INSPIRE].

  54. M. Fagotti and P. Calabrese, Entanglement entropy of two disjoint blocks in XY chains, J. Stat. Mech. 1004 (2010) P04016 [arXiv:1003.1110] [INSPIRE].

    Google Scholar 

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  1. College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, 330022, China

    Hui-Huang Chen

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Chen, HH. Symmetry decomposition of relative entropies in conformal field theory. J. High Energ. Phys. 2021, 84 (2021). https://doi.org/10.1007/JHEP07(2021)084

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  • DOI: https://doi.org/10.1007/JHEP07(2021)084

Keywords

  • Conformal Field Theory
  • Field Theories in Lower Dimensions