Journal of High Energy Physics

, 2019:177 | Cite as

Chiral entanglement in massive quantum field theories in 1+1 dimensions

  • M. LencsésEmail author
  • J. Viti
  • G. Takács
Open Access
Regular Article - Theoretical Physics


We determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement entropy is computed both analytically and numerically as a function of the volume. The excellent agreement between the analytic and numerical results provides further validation for Cardy’s Ansatz. The chiral entanglement entropy contains a universal O(1) term γ for which an exact analytic result is obtained, and which can distinguish energetically degenerate ground states of gapped systems in 1+1 dimensions.


Conformal Field Theory Field Theories in Lower Dimensions Integrable Field Theories 


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.


  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]
    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].zbMATHGoogle Scholar
  3. [3]
    T. Nishioka, Entanglement entropy: holography and renormalization group, Rev. Mod. Phys. 90 (2018) 035007 [arXiv:1801.10352] [INSPIRE].MathSciNetGoogle Scholar
  4. [4]
    N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rept. 646 (2016) 1 [arXiv:1512.03388] [INSPIRE].MathSciNetGoogle Scholar
  5. [5]
    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].MathSciNetzbMATHGoogle Scholar
  6. [6]
    G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (2003) 227902 [quant-ph/0211074] [INSPIRE].
  7. [7]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].zbMATHGoogle Scholar
  8. [8]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
  9. [9]
    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].MathSciNetGoogle Scholar
  10. [10]
    M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].
  11. [11]
    H. Li and F.D.M. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States, Phys. Rev. Lett. 101 (2008) 010504 [arXiv:0805.0332].Google Scholar
  12. [12]
    P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
  13. [13]
    A.M. Kaufman et al., Quantum thermalization through entanglement in an isolated many-body system, Science 353 (2016) 794 [arXiv:1603.04409].Google Scholar
  14. [14]
    V. Alba and P. Calabrese, Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Nat. Acad. Sci. 114 (2017) 7947 [arXiv:1608.00614].MathSciNetzbMATHGoogle Scholar
  15. [15]
    M. Kormos, M. Collura, G. Takács and P. Calabrese, Real-time confinement following a quantum quench to a non-integrable model, Nature Phys. 13 (2017) 246 [arXiv:1604.03571].Google Scholar
  16. [16]
    M. Collura, M. Kormos and G. Takács, Dynamical manifestation of the Gibbs paradox after a quantum quench, Phys. Rev. A 98 (2018) 053610 [arXiv:1801.05817] [INSPIRE].Google Scholar
  17. [17]
    O. Pomponio, L. Pristyák and G. Takács, Quasi-particle spectrum and entanglement generation after a quench in the quantum Potts spin chain, arXiv:1810.05539 [INSPIRE].
  18. [18]
    L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, Quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  19. [19]
    M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  20. [20]
    S. Das and S. Shankaranarayanan, How robust is the entanglement entropy: Area relation?, Phys. Rev. D 73 (2006) 121701 [gr-qc/0511066] [INSPIRE].
  21. [21]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  22. [22]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  23. [23]
    K.G. Wilson and J.B. Kogut, The Renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
  24. [24]
    A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [INSPIRE].Google Scholar
  25. [25]
    J. Polchinski, Scale and Conformal Invariance in Quantum Field Theory, Nucl. Phys. B 303 (1988) 226 [INSPIRE].
  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].MathSciNetzbMATHGoogle Scholar
  27. [27]
    J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  28. [28]
    A. Cappelli, C. Itzykson and J.B. Zuber, Modular Invariant Partition Functions in Two-Dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  29. [29]
    A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].MathSciNetGoogle Scholar
  30. [30]
    A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
  31. [31]
    V.P. Yurov and A.B. Zamolodchikov, Truncated conformal space approach to scaling Lee-Yang model, Int. J. Mod. Phys. A 5 (1990) 3221 [INSPIRE].Google Scholar
  32. [32]
    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].MathSciNetzbMATHGoogle Scholar
  33. [33]
    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].MathSciNetzbMATHGoogle Scholar
  34. [34]
    O.A. Castro-Alvaredo, Massive Corrections to Entanglement in Minimal E8 Toda Field Theory, SciPost Phys. 2 (2017) 008 [arXiv:1610.07040] [INSPIRE].Google Scholar
  35. [35]
    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].zbMATHGoogle Scholar
  36. [36]
    T. Pálmai, Entanglement Entropy from the Truncated Conformal Space, Phys. Lett. B 759 (2016) 439 [arXiv:1605.00444] [INSPIRE].
  37. [37]
    D. Gaiotto, Domain Walls for Two-Dimensional Renormalization Group Flows, JHEP 12 (2012) 103 [arXiv:1201.0767] [INSPIRE].
  38. [38]
    A. Konechny, RG boundaries and interfaces in Ising field theory, J. Phys. A 50 (2017) 145403 [arXiv:1610.07489] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  39. [39]
    X.-L. Qi, H. Katsura and A.W.W. Ludwig, General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States, Phys. Rev. Lett. 108 (2012) 196402 [arXiv:1103.5437].
  40. [40]
    J. Dubail, N. Read and E.H. Rezayi, Edge state inner products and real-space entanglement spectrum of trial quantum Hall states, Phys. Rev. B 86 (2012) 245310 [arXiv:1207.7119] [INSPIRE].Google Scholar
  41. [41]
    L.A. Pando Zayas and N. Quiroz, Left-Right Entanglement Entropy of Boundary States, JHEP 01 (2015) 110 [arXiv:1407.7057] [INSPIRE].zbMATHGoogle Scholar
  42. [42]
    D. Das and S. Datta, Universal features of left-right entanglement entropy, Phys. Rev. Lett. 115 (2015) 131602 [arXiv:1504.02475] [INSPIRE].
  43. [43]
    L.A. Pando Zayas and N. Quiroz, Left-Right Entanglement Entropy of Dp-branes, JHEP 11 (2016) 023 [arXiv:1605.08666] [INSPIRE].
  44. [44]
    M.P. Zaletel and R.S.K. Mong, Exact matrix product states for quantum Hall wave functions, Phys. Rev. B 86 (2012) 245305 [arXiv:1208.4862].Google Scholar
  45. [45]
    B. Estienne, Z. Papić, N. Regnault and B.A. Bernevig, Matrix product states for trial quantum Hall states, Phys. Rev. B 87 (2013) 161112 [arXiv:1211.3353].Google Scholar
  46. [46]
    J. Cardy, Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories, SciPost Phys. 3 (2017) 011 [arXiv:1706.01568] [INSPIRE].Google Scholar
  47. [47]
    A.J.A. James, R.M. Konik, P. Lecheminant, N.J. Robinson and A.M. Tsvelik, Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods, Rept. Prog. Phys. 81 (2018) 046002 [arXiv:1703.08421] [INSPIRE].MathSciNetGoogle Scholar
  48. [48]
    H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
  49. [49]
    H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  50. [50]
    N. Lashkari, Relative Entropies in Conformal Field Theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].Google Scholar
  51. [51]
    E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].Google Scholar
  52. [52]
    B.-Q. Jin and V.E. Korepin, Quantum Spin Chain, Toeplitz Determinants and the Fisher-Hartwig Conjecture, J. Stat. Phys. 116 (2004) 79 [quant-ph/0304108].
  53. [53]
    G.Y. Cho, A.W.W. Ludwig and S. Ryu, Universal entanglement spectra of gapped one-dimensional field theories, Phys. Rev. B 95 (2017) 115122 [arXiv:1603.04016] [INSPIRE].Google Scholar
  54. [54]
    J.J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].MathSciNetGoogle Scholar
  55. [55]
    J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
  56. [56]
    P. Calabrese and J.L. Cardy, Time-dependence of correlation functions following a quantum quench, Phys. Rev. Lett. 96 (2006) 136801 [cond-mat/0601225] [INSPIRE].
  57. [57]
    J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].MathSciNetGoogle Scholar
  58. [58]
    E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].zbMATHGoogle Scholar
  59. [59]
    D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372 (1992) 654 [INSPIRE].MathSciNetGoogle Scholar
  60. [60]
    I. Runkel, Boundary structure constants for the A series Virasoro minimal models, Nucl. Phys. B 549 (1999) 563 [hep-th/9811178] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  61. [61]
    I. Affleck and A.W.W. Ludwig, Universal noninteger ’ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].MathSciNetzbMATHGoogle Scholar
  62. [62]
    A.B. Zamolodchikov, Conformal Symmetry and Multicritical Points in Two-Dimensional Quantum Field Theory. (In Russian), Sov. J. Nucl. Phys. 44 (1986) 529 [INSPIRE].
  63. [63]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York U.S.A. (1997).Google Scholar
  64. [64]
    F.Y. Wu, The Potts model, Rev. Mod. Phys. 54 (1982) 235 [Erratum ibid. 55 (1983) 315] [INSPIRE].
  65. [65]
    D. Friedan, Z.-a. Qiu and S.H. Shenker, Superconformal Invariance in Two-Dimensions and the Tricritical Ising Model, Phys. Lett. 151B (1985) 37 [INSPIRE].MathSciNetGoogle Scholar
  66. [66]
    P. Christe and G. Mussardo, Integrable Systems Away from Criticality: The Toda Field Theory and S Matrix of the Tricritical Ising Model, Nucl. Phys. B 330 (1990) 465 [INSPIRE].MathSciNetGoogle Scholar
  67. [67]
    I. Affleck, Edge critical behavior of the two-dimensional tricritical Ising model, J. Phys. A 33 (2000) 6473 [cond-mat/0005286] [INSPIRE].
  68. [68]
    J.L. Cardy, Scaling and renormalization in statistical physics, Cambridge Lecture Notes in Physics, Cambridge University Press, Cambridge U.K. (1996).Google Scholar
  69. [69]
    A. Feiguin et al., Interacting anyons in topological quantum liquids: The golden chain, Phys. Rev. Lett. 98 (2007) 160409 [cond-mat/0612341] [INSPIRE].
  70. [70]
    V.A. Fateev, The Exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories, Phys. Lett. B 324 (1994) 45 [INSPIRE].MathSciNetGoogle Scholar
  71. [71]
    A.B. Zamolodchikov, Resonance factorized scattering and roaming trajectories, J. Phys. A 39 (2006) 12847.Google Scholar
  72. [72]
    W.M. Koo and H. Saleur, Representations of the Virasoro algebra from lattice models, Nucl. Phys. B 426 (1994) 459 [hep-th/9312156] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  73. [73]
    A. Milsted and G. Vidal, Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula, Phys. Rev. B 96 (2017) 245105 [arXiv:1706.01436] [INSPIRE].Google Scholar
  74. [74]
    T. Rakovszky, M. Mestyán, M. Collura, M. Kormos and G. Takács, Hamiltonian truncation approach to quenches in the Ising field theory, Nucl. Phys. B 911 (2016) 805 [arXiv:1607.01068] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  75. [75]
    D.X. Horváth and G. Takács, Overlaps after quantum quenches in the sine-Gordon model, Phys. Lett. B 771 (2017) 539 [arXiv:1704.00594] [INSPIRE].zbMATHGoogle Scholar
  76. [76]
    K. Hódsági, M. Kormos and G. Takács, Quench dynamics of the Ising field theory in a magnetic field, SciPost Phys. 5 (2018) 027 [arXiv:1803.01158] [INSPIRE].Google Scholar
  77. [77]
    I. Kukuljan, S. Sotiriadis and G. Takács, Correlation Functions of the Quantum sine-Gordon Model in and out of Equilibrium, Phys. Rev. Lett. 121 (2018) 110402 [arXiv:1802.08696] [INSPIRE].Google Scholar
  78. [78]
    D.X. Horváth, I. Lovas, M. Kormos, G. Takács and G. Zaránd, Non-equilibrium time evolution and rephasing in the quantum sine-Gordon model, arXiv:1809.06789 [INSPIRE].
  79. [79]
    D. Schuricht and F.H.L. Essler, Dynamics in the Ising field theory after a quantum quench, J. Stat. Mech. 1204 (2012) P04017 [arXiv:1203.5080] [INSPIRE].Google Scholar
  80. [80]
    B. Bertini, D. Schuricht and F.H.L. Essler, Quantum quench in the sine-Gordon model, J. Stat. Mech. 1410 (2014) P10035 [arXiv:1405.4813] [INSPIRE].MathSciNetGoogle Scholar
  81. [81]
    A. Cortés Cubero and D. Schuricht, Quantum quench in the attractive regime of the sine-Gordon model, J. Stat. Mech. 1710 (2017) 103106 [arXiv:1707.09218] [INSPIRE].MathSciNetGoogle Scholar
  82. [82]
    G. Delfino, Quantum quenches with integrable pre-quench dynamics, J. Phys. A 47 (2014) 402001 [arXiv:1405.6553] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  83. [83]
    G. Delfino and J. Viti, On the theory of quantum quenches in near-critical systems, J. Phys. A 50 (2017) 084004 [arXiv:1608.07612] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  84. [84]
    M. Kormos and G. Zaránd, Quantum quenches in the sine-Gordon model: a semiclassical approach, Phys. Rev. E 93 (2016) 062101 [arXiv:1507.02708] [INSPIRE].MathSciNetGoogle Scholar
  85. [85]
    D.X. Horváth, S. Sotiriadis and G. Takács, Initial states in integrable quantum field theory quenches from an integral equation hierarchy, Nucl. Phys. B 902 (2016) 508 [arXiv:1510.01735] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  86. [86]
    D.X. Horváth, M. Kormos and G. Takács, Overlap singularity and time evolution in integrable quantum field theory, JHEP 08 (2018) 170 [arXiv:1805.08132] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  87. [87]
    V.S. Dotsenko and V.A. Fateev, Operator Algebra of Two-Dimensional Conformal Theories with Central Charge C ≤ 1, Phys. Lett. B 154 (1985) 291 [INSPIRE].MathSciNetGoogle Scholar
  88. [88]
    G. Feverati, K. Graham, P.A. Pearce, G.Z. Tóth and G. Watts, A Renormalisation group for the truncated conformal space approach, J. Stat. Mech. 0803 (2008) P03011 [hep-th/0612203] [INSPIRE].Google Scholar
  89. [89]
    R.M. Konik and Y. Adamov, A Numerical Renormalization Group for Continuum One-Dimensional Systems, Phys. Rev. Lett. 98 (2007) 147205 [cond-mat/0701605] [INSPIRE].
  90. [90]
    P. Giokas and G. Watts, The renormalisation group for the truncated conformal space approach on the cylinder, arXiv:1106.2448 [INSPIRE].
  91. [91]
    M. Lencsés and G. Takács, Excited state TBA and renormalized TCSA in the scaling Potts model, JHEP 09 (2014) 052 [arXiv:1405.3157] [INSPIRE].Google Scholar
  92. [92]
    M. Hogervorst, S. Rychkov and B.C. van Rees, Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?, Phys. Rev. D 91 (2015) 025005 [arXiv:1409.1581] [INSPIRE].MathSciNetGoogle Scholar
  93. [93]
    S. Rychkov and L.G. Vitale, Hamiltonian truncation study of the φ 4 theory in two dimensions, Phys. Rev. D 91 (2015) 085011 [arXiv:1412.3460] [INSPIRE].MathSciNetGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.International Institute of Physics, UFRNLagoa NovaBrazil
  2. 2.Escola de Ciência e Tecnologia, UFRNLagoa NovaBrazil
  3. 3.BME “Momentum” Statistical Physics Research Group, Department of Theoretical PhysicsBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations