Symmetry resolved entanglement in integrable field theories via form factor bootstrap

Abstract

We consider the form factor bootstrap approach of integrable field theories to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are determined in an intuitive way and their solution is presented for the massive Ising field theory and for the genuinely interacting sinh-Gordon model, both possessing a ℤ2 symmetry. The solutions are carefully cross-checked by performing various limits and by the application of the ∆-theorem. The issue of symmetry resolution for discrete symmetries is also discussed. We show that entanglement equipartition is generically expected and we identify the first subleading term (in the UV cutoff) breaking it. We also present the complete computation of the symmetry resolved von Neumann entropy for an interval in the ground state of the paramagnetic phase of the Ising model. In particular, we compute the universal functions entering in the charged and symmetry resolved entanglement.

A preprint version of the article is available at ArXiv.

References

  1. [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].

  2. [2]

    P. Calabrese, J. Cardy and B. Doyon, Entanglement entropy in extended quantum systems, J. Phys. A 42 (2009) 500301.

    MathSciNet  MATH  Google Scholar 

  3. [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  Google Scholar 

  4. [4]

    D.J.E. Marsh, Axion Cosmology, Phys. Rept. 643 (2016) 1 [arXiv:1510.07633] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  5. [5]

    N. Laflorencie and S. Rachel, Spin-resolved entanglement spectroscopy of critical spin chains and Luttinger liquids, J. Stat. Mech. 2014 (2014) P11013.

  6. [6]

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

    ADS  MathSciNet  Google Scholar 

  7. [7]

    A. Lukin et al., Probing entanglement in a many-body localized system, Science 364 (2019) 6437.

    Google Scholar 

  8. [8]

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

  9. [9]

    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, Cambridge University Press, Cambridge, U.K., 10th anniversary ed. (2010) [DOI].

  10. [10]

    H.M. Wiseman and J.A. Vaccaro, Entanglement of Indistinguishable Particles Shared between Two Parties, Phys. Rev. Lett. 91 (2003) 097902.

  11. [11]

    M. Kiefer-Emmanouilidis, R. Unanyan, J. Sirker and M. Fleischhauer, Bounds on the entanglement entropy by the number entropy in non-interacting fermionic systems, SciPost Phys. 8 (2020) 083.

    ADS  MathSciNet  Google Scholar 

  12. [12]

    M. Kiefer-Emmanouilidis, R. Unanyan, M. Fleischhauer and J. Sirker, Evidence for Unbounded Growth of the Number Entropy in Many-Body Localized Phases, Phys. Rev. Lett. 124 (2020) 243601 [arXiv:2003.04849] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  13. [13]

    C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  14. [14]

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 2004 (2004) P06002.

  15. [15]

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

    MathSciNet  MATH  Google Scholar 

  16. [16]

    J.C. Xavier, F.C. Alcaraz and G. Sierra, Equipartition of the entanglement entropy, Phys. Rev. B 98 (2018) 041106 [arXiv:1804.06357] [INSPIRE].

  17. [17]

    N. Feldman and M. Goldstein, Dynamics of Charge-Resolved Entanglement after a Local Quench, Phys. Rev. B 100 (2019) 235146 [arXiv:1905.10749] [INSPIRE].

    ADS  Google Scholar 

  18. [18]

    L. Capizzi, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement entropy of excited states in a CFT, J. Stat. Mech. (2020) 073101.

  19. [19]

    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  Google Scholar 

  20. [20]

    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  Google Scholar 

  21. [21]

    S. Fraenkel and M. Goldstein, Symmetry resolved entanglement: Exact results in 1d and beyond, J. Stat. Mech. 2020 (2020) 033106.

  22. [22]

    H. Barghathi, C.M. Herdman and A. Del Maestro, Rényi Generalization of the Accessible Entanglement Entropy, Phys. Rev. Lett. 121 (2018) 150501.

    ADS  MathSciNet  Google Scholar 

  23. [23]

    H. Barghathi, E. Casiano-Diaz and A. Del Maestro, Operationally accessible entanglement of one-dimensional spinless fermions, Phys. Rev. A 100 (2019) 022324 [arXiv:1905.03312] [INSPIRE].

  24. [24]

    S. Murciano, G. Di Giulio and P. Calabrese, Symmetry resolved entanglement in gapped integrable systems: a corner transfer matrix approach, SciPost Phys. 8 (2020) 046 [arXiv:1911.09588] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  25. [25]

    P. Calabrese, M. Collura, G. Di Giulio and S. Murciano, Ful l counting statistics in the gapped XXZ spin chain, Europhys. Lett. 129 (2020) 60007.

    ADS  Google Scholar 

  26. [26]

    M.T. Tan and S. Ryu, Particle number fluctuations, Rényi entropy, and symmetry-resolved entanglement entropy in a two-dimensional Fermi gas from multidimensional bosonization, Phys. Rev. B 101 (2020) 235169 [arXiv:1911.01451] [INSPIRE].

    ADS  Google Scholar 

  27. [27]

    S. Murciano, P. Ruggiero and P. Calabrese, Symmetry resolved entanglement in two-dimensional systems via dimensional reduction, J. Stat. Mech. 2008 (2020) 083102 [arXiv:2003.11453] [INSPIRE].

  28. [28]

    X. Turkeshi, P. Ruggiero, V. Alba and P. Calabrese, Entanglement equipartition in critical random spin chains, Phys. Rev. B 102 (2020) 014455 [arXiv:2005.03331] [INSPIRE].

  29. [29]

    K. Monkman and J. Sirker, Operational entanglement of symmetry-protected topological edge states, Phys. Rev. Res. 2 (2020) 043191 [arXiv:2005.13026] [INSPIRE].

  30. [30]

    E. Cornfeld, L.A. Landau, K. Shtengel and E. Sela, Entanglement spectroscopy of non-Abelian anyons: Reading off quantum dimensions of individual anyons, Phys. Rev. B 99 (2019) 115429 [arXiv:1810.01853] [INSPIRE].

    ADS  Google Scholar 

  31. [31]

    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  Google Scholar 

  32. [32]

    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  Google Scholar 

  33. [33]

    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  Google Scholar 

  34. [34]

    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  Google Scholar 

  35. [35]

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

    ADS  MathSciNet  MATH  Google Scholar 

  36. [36]

    H. Shapourian, K. Shiozaki and S. Ryu, Partial time-reversal transformation and entanglement negativity in fermionic systems, Phys. Rev. B 95 (2017) 165101 [arXiv:1611.07536] [INSPIRE].

    ADS  Google Scholar 

  37. [37]

    H. Shapourian, P. Ruggiero, S. Ryu and P. Calabrese, Twisted and untwisted negativity spectrum of free fermions, SciPost Phys. 7 (2019) 037 [arXiv:1906.04211] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  38. [38]

    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  Google Scholar 

  39. [39]

    V.G. Knizhnik, Analytic Fields on Riemann Surfaces. 2, Commun. Math. Phys. 112 (1987) 567 [INSPIRE].

  40. [40]

    L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The Conformal Field Theory of Orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  41. [41]

    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 2009 (2009) P11001.

  42. [42]

    M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].

    ADS  Google Scholar 

  43. [43]

    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 2011 (2011) P01021.

  44. [44]

    V. Alba, L. Tagliacozzo and P. Calabrese, Entanglement entropy of two disjoint intervals in c = 1 theories, J. Stat. Mech. 2011 (2011) P06012.

  45. [45]

    M.A. Rajabpour and F. Gliozzi, Entanglement entropy of two disjoint intervals from fusion algebra of twist fields, J. Stat. Mech. 2012 (2012) P02016.

    MathSciNet  MATH  Google Scholar 

  46. [46]

    P. Ruggiero, E. Tonni and P. Calabrese, Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks, J. Stat. Mech. 2018 (2018) 113101.

    MathSciNet  MATH  Google Scholar 

  47. [47]

    T. Dupic, B. Estienne and Y. Ikhlef, Entanglement entropies of minimal models from null-vectors, SciPost Phys. 4 (2018) 031 [arXiv:1709.09270] [INSPIRE].

    ADS  Google Scholar 

  48. [48]

    A. Coser, L. Tagliacozzo and E. Tonni, On Rényi entropies of disjoint intervals in conformal field theory, J. Stat. Mech. 2014 (2014) P01008.

  49. [49]

    O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in integrable models with backscattering, J. Phys. A 41 (2008) 275203 [arXiv:0802.4231] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  50. [50]

    O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive 1 + 1-dimensional quantum field theories, J. Phys. A 42 (2009) 504006 [arXiv:0906.2946] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  51. [51]

    O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive QFT with a boundary: The Ising model, J. Statist. Phys. 134 (2009) 105 [arXiv:0810.0219] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  52. [52]

    O.A. Castro-Alvaredo and E. Levi, Higher particle form factors of branch point twist fields in integrable quantum field theories, J. Phys. A 44 (2011) 255401 [arXiv:1103.2069] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  53. [53]

    O.A. Castro-Alvaredo, B. Doyon and E. Levi, Arguments towards a c-theorem from branch-point twist fields, J. Phys. A 44 (2011) 492003 [arXiv:1107.4280] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  54. [54]

    E. Levi, Composite branch-point twist fields in the Ising model and their expectation values, J. Phys. A 45 (2012) 275401 [arXiv:1204.1192] [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  55. [55]

    E. Levi, O.A. Castro-Alvaredo and B. Doyon, Universal corrections to the entanglement entropy in gapped quantum spin chains: a numerical study, Phys. Rev. B 88 (2013) 094439 [arXiv:1304.6874] [INSPIRE].

  56. [56]

    D. Bianchini, O. Castro-Alvaredo, B. Doyon, E. Levi and F. Ravanini, Entanglement Entropy of Non Unitary Conformal Field Theory, J. Phys. A 48 (2014) 04FT01.

  57. [57]

    D. Bianchini, O.A. Castro-Alvaredo and B. Doyon, Entanglement Entropy of Non-Unitary Integrable Quantum Field Theory, Nucl. Phys. B 896 (2015) 835 [arXiv:1502.03275] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  58. [58]

    O. Blondeau-Fournier, O.A. Castro-Alvaredo and B. Doyon, Universal scaling of the logarithmic negativity in massive quantum field theory, J. Phys. A 49 (2016) 125401 [arXiv:1508.04026] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  59. [59]

    D. Bianchini and O.A. Castro-Alvaredo, Branch Point Twist Field Correlators in the Massive Free Boson Theory, Nucl. Phys. B 913 (2016) 879 [arXiv:1607.05656] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  60. [60]

    O.A. Castro-Alvaredo, Massive Corrections to Entanglement in Minimal E8 Toda Field Theory, SciPost Phys. 2 (2017) 008 [arXiv:1610.07040] [INSPIRE].

    ADS  Google Scholar 

  61. [61]

    O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement Content of Quasiparticle Excitations, Phys. Rev. Lett. 121 (2018) 170602 [arXiv:1805.04948] [INSPIRE].

    ADS  MATH  Google Scholar 

  62. [62]

    O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement content of quantum particle excitations. Part I. Free field theory, JHEP 10 (2018) 039 [arXiv:1806.03247] [INSPIRE].

  63. [63]

    O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement content of quantum particle excitations. Part II. Disconnected regions and logarithmic negativity, JHEP 11 (2019) 058 [arXiv:1904.01035] [INSPIRE].

  64. [64]

    O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement Content of Quantum Particle Excitations III. Graph Partition Functions, J. Math. Phys. 60 (2019) 082301 [arXiv:1904.02615] [INSPIRE].

  65. [65]

    O.A. Castro-Alvaredo, M. Lencsés, I.M. Szécsényi and J. Viti, Entanglement Dynamics after a Quench in Ising Field Theory: A Branch Point Twist Field Approach, JHEP 19 (2020) 079 [arXiv:1907.11735] [INSPIRE].

    Google Scholar 

  66. [66]

    O.A. Castro-Alvaredo, M. Lencsés, I.M. Szécsényi and J. Viti, Entanglement Oscillations near a Quantum Critical Point, Phys. Rev. Lett. 124 (2020) 230601 [arXiv:2001.10007] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  67. [67]

    G. Delfino, P. Simonetti and J.L. Cardy, Asymptotic factorization of form-factors in two-dimensional quantum field theory, Phys. Lett. B 387 (1996) 327 [hep-th/9607046] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  68. [68]

    B. Berg, M. Karowski and P. Weisz, Construction of Green Functions from an Exact S Matrix, Phys. Rev. D 19 (1979) 2477 [INSPIRE].

    ADS  Google Scholar 

  69. [69]

    A.N. Kirillov and F.A. Smirnov, A Representation of the Current Algebra Connected With the SU(2) Invariant Thirring Model, Phys. Lett. B 198 (1987) 506 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  70. [70]

    F.A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, World Scientific, Singapore (1992) [DOI].

  71. [71]

    G. Mussardo, Statistical field theory: an introduction to exactly solved models in statistical physics, 2nd edition, Oxford University Press (2020).

  72. [72]

    A.E. Arinshtein, V.A. Fateev and A.B. Zamolodchikov, Quantum s Matrix of the (1+1)-Dimensional Todd Chain, Phys. Lett. B 87 (1979) 389 [INSPIRE].

    ADS  Google Scholar 

  73. [73]

    A. Fring, G. Mussardo and P. Simonetti, Form-factors for integrable Lagrangian field theories, the sinh-Gordon theory, Nucl. Phys. B 393 (1993) 413 [hep-th/9211053] [INSPIRE].

    ADS  MATH  Google Scholar 

  74. [74]

    A. Koubek and G. Mussardo, On the operator content of the sinh-Gordon model, Phys. Lett. B 311 (1993) 193 [hep-th/9306044] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  75. [75]

    C. Ahn, G. Delfino and G. Mussardo, Mapping between the sinh-Gordon and Ising models, Phys. Lett. B 317 (1993) 573 [hep-th/9306103] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  76. [76]

    S. Negro and F. Smirnov, On one-point functions for sinh-Gordon model at finite temperature, Nucl. Phys. B 875 (2013) 166 [arXiv:1306.1476] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  77. [77]

    S. Negro, On sinh-Gordon Thermodynamic Bethe Ansatz and fermionic basis, Int. J. Mod. Phys. A 29 (2014) 1450111 [arXiv:1404.0619] [INSPIRE].

    ADS  MATH  Google Scholar 

  78. [78]

    B. Bertini, L. Piroli and P. Calabrese, Quantum quenches in the sinh-Gordon model: steady state and one point correlation functions, J. Stat. Mech. 2016 (2016) 063102.

  79. [79]

    B. Doyon, Exact large-scale correlations in integrable systems out of equilibrium, SciPost Phys. 5 (2018) 054 [arXiv:1711.04568] [INSPIRE].

    ADS  Google Scholar 

  80. [80]

    R. Konik, M. Lájer and G. Mussardo, Approaching the Self-Dual Point of the Sinh-Gordon model, arXiv:2007.00154 [INSPIRE].

  81. [81]

    E.H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. 1. The General solution and the ground state, Phys. Rev. 130 (1963) 1605 [INSPIRE].

  82. [82]

    E.H. Lieb, Exact Analysis of an Interacting Bose Gas. 2. The Excitation Spectrum, Phys. Rev. 130 (1963) 1616 [INSPIRE].

  83. [83]

    M.A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac and M. Rigol, One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases, Rev. Mod. Phys. 83 (2011) 1405.

    ADS  Google Scholar 

  84. [84]

    M. Kormos, G. Mussardo and A. Trombettoni, Expectation Values in the Lieb-Liniger Bose Gas, Phys. Rev. Lett. 103 (2009) 210404 [arXiv:0909.1336] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  85. [85]

    M. Kormos, G. Mussardo and A. Trombettoni, 1D Lieb-Liniger Bose Gas as Non-Relativistic Limit of the Sinh-Gordon Model, Phys. Rev. A 81 (2010) 043606 [arXiv:0912.3502] [INSPIRE].

  86. [86]

    M. Kormos, G. Mussardo and B. Pozsgay, Bethe Ansatz Matrix Elements as Non-Relativistic Limits of Form Factors of Quantum Field Theory, J. Stat. Mech. 2010 (2010) P05014.

  87. [87]

    A. Bastianello, L. Piroli and P. Calabrese, Exact Local Correlations and Ful l Counting Statistics for Arbitrary States of the One-Dimensional Interacting Bose Gas, Phys. Rev. Lett. 120 (2018) 190601 [arXiv:1802.02115] [INSPIRE].

    ADS  Google Scholar 

  88. [88]

    A. Bastianello and L. Piroli, From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and ful l counting statistics, J. Stat. Mech. 2018 (2018) 113104.

    MATH  Google Scholar 

  89. [89]

    A. Bastianello, A. De Luca and G. Mussardo, Non relativistic limit of integrable QFT and Lieb-Liniger models, J. Stat. Mech. 2016 (2016) 123104.

    MathSciNet  MATH  Google Scholar 

  90. [90]

    A. Bastianello, A. De Luca and G. Mussardo, Non Relativistic Limit of Integrable QFT with fermionic excitations, J. Phys. A 50 (2017) 234002 [arXiv:1701.06542] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  91. [91]

    A.B. Zamolodchikov, Two point correlation function in scaling Lee-Yang model, Nucl. Phys. B 348 (1991) 619 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  92. [92]

    J. Bisognano and E.H. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys. 16 (1975) 985 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  93. [93]

    J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  94. [94]

    P.D. Hislop and R. Longo, Modular Structure of the Local Algebras Associated With the Free Massless Scalar Field Theory, Commun. Math. Phys. 84 (1982) 71 [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  95. [95]

    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  96. [96]

    J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 2016 (2016) 123103.

    MathSciNet  MATH  Google Scholar 

  97. [97]

    D.X. Horvath, P.E. Dorey and G. Takács, Roaming form factors for the tricritical to critical Ising flow, JHEP 07 (2016) 051 [arXiv:1604.05635] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  98. [98]

    P.H. Ginsparg, Applied conformal field theory, in Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena, pp. 1–168 (1988) [hep-th/9108028] [INSPIRE].

  99. [99]

    P. Calabrese, M. Campostrini, F. Essler and B. Nienhuis, Parity effects in the scaling of block entanglement in gapless spin chains, Phys. Rev. Lett. 104 (2010) 095701 [arXiv:0911.4660] [INSPIRE].

  100. [100]

    J. Cardy and P. Calabrese, Unusual Corrections to Scaling in Entanglement Entropy, J. Stat. Mech. 2010 (2010) P04023.

  101. [101]

    P. Calabrese, J. Cardy and I. Peschel, Corrections to scaling for block entanglement in massive spin-chains, J. Stat. Mech. 2010 (2010) P09003.

  102. [102]

    S.L. Lukyanov and A.B. Zamolodchikov, Exact expectation values of local fields in quantum sine-Gordon model, Nucl. Phys. B 493 (1997) 571 [hep-th/9611238] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  103. [103]

    V. Fateev, S.L. Lukyanov, A.B. Zamolodchikov and A.B. Zamolodchikov, Expectation values of local fields in Bul lough-Dodd model and integrable perturbed conformal field theories, Nucl. Phys. B 516 (1998) 652 [hep-th/9709034] [INSPIRE].

    ADS  MATH  Google Scholar 

  104. [104]

    V. Fateev, D. Fradkin, S.L. Lukyanov, A.B. Zamolodchikov and A.B. Zamolodchikov, Expectation values of descendent fields in the sine-Gordon model, Nucl. Phys. B 540 (1999) 587 [hep-th/9807236] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  105. [105]

    E. Cornfeld and E. Sela, Entanglement entropy and boundary renormalization group flow: Exact results in the Ising universality class, Phys. Rev. B 96 (2017) 075153 [arXiv:1705.04181] [INSPIRE].

  106. [106]

    H. Casini, C.D. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 2005 (2005) P07007.

  107. [107]

    T.T. Wu, B.M. McCoy, C.A. Tracy and E. Barouch, Spin spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region, Phys. Rev. B 13 (1976) 316 [INSPIRE].

    ADS  Google Scholar 

  108. [108]

    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].

    ADS  MathSciNet  MATH  Google Scholar 

  109. [109]

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

  110. [110]

    L.A. Rubel, Necessary and sufficient conditions for Carlson’s theorem on entire functions, Proc. Natl. Acad. Sci. 41 (1955) 601.

    ADS  MathSciNet  MATH  Google Scholar 

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Correspondence to Dávid X. Horváth.

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Horváth, D.X., Calabrese, P. Symmetry resolved entanglement in integrable field theories via form factor bootstrap. J. High Energ. Phys. 2020, 131 (2020). https://doi.org/10.1007/JHEP11(2020)131

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Keywords

  • Discrete Symmetries
  • Integrable Field Theories
  • Bethe Ansatz