Journal of High Energy Physics

, 2019:181 | Cite as

Subsystem trace distance in low-lying states of (1 + 1)-dimensional conformal field theories

  • Jiaju Zhang
  • Paola RuggieroEmail author
  • Pasquale Calabrese
Open Access
Regular Article - Theoretical Physics


We report on a systematic replica approach to calculate the subsystem trace distance for a quantum field theory. This method has been recently introduced in [J. Zhang, P. Ruggiero and P. Calabrese, Phys. Rev. Lett.122 (2019) 141602], of which this work is a completion. The trace distance between two reduced density matrices ρA and σA is obtained from the moments tr(ρA− σA)n and taking the limit n → 1 of the traces of the even powers. We focus here on the case of a subsystem consisting of a single interval of length embedded in the low lying eigenstates of a one-dimensional critical system of length L, a situation that can be studied exploiting the path integral form of the reduced density matrices of two-dimensional conformal field theories. The trace distance turns out to be a scale invariant universal function of ℓ/L. Here we complete our previous work by providing detailed derivations of all results and further new formulas for the distances between several low-lying states in two-dimensional free massless compact boson and fermion theories. Remarkably, for one special case in the bosonic theory and for another in the fermionic one, we obtain the exact trace distance, as well as the Schatten n-distance, for an interval of arbitrary length, while in generic case we have a general form for the first term in the expansion in powers of ℓ/L. The analytical predictions in conformal field theories are tested against exact numerical calculations in XX and Ising spin chains, finding perfect agreement. As a byproduct, new results in two-dimensional CFT are also obtained for other entanglement-related quantities, such as the relative entropy and the fidelity.


Conformal Field Theory Field Theories in Lower Dimensions 


Open Access

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  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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    P. Calabrese, J. Cardy and B. Doyon, Entanglement entropy in extended quantum systems, J. Phys.A 42 (2009) 500301.MathSciNetzbMATHGoogle Scholar
  3. [3]
    N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rept.646 (2016) 1 [arXiv:1512.03388] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    R. Islam, R. Ma, P.M. Preiss, M.E. Tai, A. Lukin, M. Rispoli and M. Greiner, Measuring entanglement entropy in a quantum many-body system, Nature528 (2015) 77 [arXiv:1509.01160].ADSCrossRefGoogle Scholar
  5. [5]
    A.M. Kaufman, M.E. Tai, A. Lukin, M. Rispoli, R. Schittko, P.M. Preiss and M. Greiner, Quantum thermalization through entanglement in an isolated many-body system, Science353 (2016) 794 [arXiv:1603.04409].ADSCrossRefGoogle Scholar
  6. [6]
    A. Elben, B. Vermersch, M. Dalmonte, J.I. Cirac and P. Zoller, Rényi entropies from random quenches in atomic Hubbard and spin models, Phys. Rev. Lett.120 (2018) 050406 [arXiv:1709.05060].ADSCrossRefGoogle Scholar
  7. [7]
    A. Lukin et al., Probing entanglement in a many-body-localized system, Science364 (2019) 256 [arXiv:1805.09819].ADSGoogle Scholar
  8. [8]
    T. Brydges et al., Probing entanglement entropy via randomized measurements, Science364 (2019) 260 [arXiv:1806.05747].ADSGoogle Scholar
  9. [9]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    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].ADSCrossRefGoogle Scholar
  11. [11]
    J.I. Latorre, E. Rico and G. Vidal, Ground state entanglement in quantum spin chains, Quant. Inf. Comput.4 (2004) 48 [quant-ph/0304098] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  12. [12]
    P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.0406 (2004) P06002 [hep-th/0405152] [INSPIRE].zbMATHGoogle Scholar
  13. [13]
    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys.A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  14. [14]
    G. Refael and J.E. Moore, Entanglement Entropy of Random Quantum Critical Points in One Dimension, Phys. Rev. Lett.93 (2004) 260602 [cond-mat/0406737] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    G. Refael and J.E. Moore, Criticality and entanglement in random quantum systems, J. Phys.A 42 (2009) 504010 [arXiv:0908.1986].MathSciNetzbMATHGoogle Scholar
  16. [16]
    N. Laflorencie, Scaling of entanglement entropy in the random singlet phase, Phys. Rev.B 72 (2005) 140408 [cond-mat/0504446].ADSCrossRefGoogle Scholar
  17. [17]
    M. Fagotti, P. Calabrese and J.E. Moore, Entanglement spectrum of random-singlet quantum critical points, Phys. Rev.B 83 (2011) 045110 [arXiv:1009.1614] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett.96 (2006) 110404 [hep-th/0510092] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett.96 (2006) 110405 [cond-mat/0510613] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    M. Haque, O. Zozulya and K. Schoutens, Entanglement entropy in fermionic Laughlin states, Phys. Rev. Lett.98 (2007) 060401 [cond-mat/0609263].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    H. Li and F. 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] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    J.I. Cirac and F. Verstraete, Renormalization and tensor product states in spin chains and lattices, J. Phys.A 42 (2009) 504004 [arXiv:0910.1130] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  23. [23]
    U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals Phys.326 (2011) 96 [arXiv:1008.3477].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    R. Orus, A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Annals Phys.349 (2014) 117 [arXiv:1306.2164] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010 [cond-mat/0503393].MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    M. Cramer, C.M. Dawson, J. Eisert and T.J. Osborne, Exact Relaxation in a Class of Nonequilibrium Quantum Lattice Systems, Phys. Rev. Lett.100 (2008) 030602 [cond-mat/0703314] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    T. Barthel and U. Schollwöck, Dephasing and the steady state in quantum many-particle systems, Phys. Rev. Lett.100 (2008) 100601 [arXiv:0711.4896].ADSCrossRefGoogle Scholar
  28. [28]
    J. Deutsch, H. Li and A. Sharma, Microscopic origin of thermodynamic entropy in isolated systems, Phys. Rev.E 87 (2013) 042135 [arXiv:1202.2403].ADSGoogle Scholar
  29. [29]
    C. Gogolin and J. Eisert, Equilibration, thermalisation and the emergence of statistical mechanics in closed quantum systems, Rept. Prog. Phys.79 (2016) 056001 [arXiv:1503.07538] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    P. Calabrese, F.H. Essler and G. Mussardo, Introduction to ‘quantum integrability in out of equilibrium systems’, J. Stat. Mech. (2016) 064001.CrossRefGoogle Scholar
  31. [31]
    F.H.L. Essler and M. Fagotti, Quench dynamics and relaxation in isolated integrable quantum spin chains, J. Stat. Mech.1606 (2016) 064002 [arXiv:1603.06452] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  32. [32]
    L. Vidmar and M. Rigol, Generalized gibbs ensemble in integrable lattice models, J. Stat. Mech. (2016) 064007 [arXiv:1604.03990].MathSciNetCrossRefGoogle Scholar
  33. [33]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    V. Alba and P. Calabrese, Entanglement dynamics after quantum quenches in generic integrable systems, SciPost Phys.4 (2018) 017 [arXiv:1712.07529] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452 (2008) 854 [arXiv:0708.1324].ADSCrossRefGoogle Scholar
  36. [36]
    N. Lashkari, A. Dymarsky and H. Liu, Eigenstate Thermalization Hypothesis in Conformal Field Theory, J. Stat. Mech.1803 (2018) 033101 [arXiv:1610.00302] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  37. [37]
    A. Dymarsky, N. Lashkari and H. Liu, Subsystem ETH, Phys. Rev.E 97 (2018) 012140 [arXiv:1611.08764] [INSPIRE].ADSGoogle Scholar
  38. [38]
    N. Lashkari, A. Dymarsky and H. Liu, Universality of Quantum Information in Chaotic CFTs, JHEP03 (2018) 070 [arXiv:1710.10458] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys.43 (1975) 199 [Erratum ibid.46 (1976) 206] [INSPIRE].
  40. [40]
    S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev.D 14 (1976) 2460 [INSPIRE].ADSMathSciNetGoogle Scholar
  41. [41]
    S.D. Mathur, The information paradox: A pedagogical introduction, Class. Quant. Grav.26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys.61 (2013) 781 [arXiv:1306.0533] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, On information loss in AdS 3/CFT 2 , JHEP05 (2016) 109 [arXiv:1603.08925] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    H. Chen, C. Hussong, J. Kaplan and D. Li, A Numerical Approach to Virasoro Blocks and the Information Paradox, JHEP09 (2017) 102 [arXiv:1703.09727] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  46. [46]
    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP08 (2006) 045 [hep-th/0605073] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys.A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  49. [49]
    V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP07 (2007) 062 [arXiv:0705.0016] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav.42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. [51]
    A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP08 (2013) 090 [arXiv:1304.4926] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  52. [52]
    T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP11 (2013) 074 [arXiv:1307.2892] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  53. [53]
    T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, JHEP03 (2014) 051 [arXiv:1312.7856] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  54. [54]
    X. Dong, The Gravity Dual of Rényi Entropy, Nature Commun.7 (2016) 12472 [arXiv:1601.06788] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP11 (2016) 028 [arXiv:1607.07506] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  56. [56]
    M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, Lect. Notes Phys.931 (2017) 1 [arXiv:1609.01287].MathSciNetzbMATHCrossRefGoogle Scholar
  57. [57]
    M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information, 10th anniversary ed., Cambridge University Press, Cambridge, U.K., (2010), [].
  58. [58]
    J. Watrous, The theory of quantum information, Cambridge University Press, Cambridge, U.K., (2018), [].
  59. [59]
    M. Fagotti and F.H. Essler, Reduced density matrix after a quantum quench, Phys. Rev.B 87 (2013) 245107 [arXiv:1302.6944].ADSCrossRefGoogle Scholar
  60. [60]
    D.D. Blanco, H. Casini, L.-Y. Hung and R.C. Myers, Relative Entropy and Holography, JHEP08 (2013) 060 [arXiv:1305.3182] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  61. [61]
    V. Balasubramanian, J.J. Heckman and A. Maloney, Relative Entropy and Proximity of Quantum Field Theories, JHEP05 (2015) 104 [arXiv:1410.6809] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  62. [62]
    N. Lashkari, Relative Entropies in Conformal Field Theory, Phys. Rev. Lett.113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    N. Lashkari, Modular Hamiltonian for Excited States in Conformal Field Theory, Phys. Rev. Lett.117 (2016) 041601 [arXiv:1508.03506] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    G. Sárosi and T. Ugajin, Relative entropy of excited states in two dimensional conformal field theories, JHEP07 (2016) 114 [arXiv:1603.03057] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  65. [65]
    G. Sárosi and T. Ugajin, Relative entropy of excited states in conformal field theories of arbitrary dimensions, JHEP02 (2017) 060 [arXiv:1611.02959] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    P. Ruggiero and P. Calabrese, Relative Entanglement Entropies in 1+1-dimensional conformal field theories, JHEP02 (2017) 039 [arXiv:1612.00659] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  67. [67]
    D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  68. [68]
    H. Casini, E. Teste and G. Torroba, Relative entropy and the RG flow, JHEP03 (2017) 089 [arXiv:1611.00016] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  69. [69]
    Y.O. Nakagawa and T. Ugajin, Numerical calculations on the relative entanglement entropy in critical spin chains, J. Stat. Mech.1709 (2017) 093104 [arXiv:1705.07899] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  70. [70]
    S. Murciano, P. Ruggiero and P. Calabrese, Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems, J. Stat. Mech. (2019) 034001 [arXiv:1810.02287] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  71. [71]
    S. Kullback and R.A. Leibler, On information and sufficiency, Annals Math. Statist.22 (1951) 79.MathSciNetzbMATHCrossRefGoogle Scholar
  72. [72]
    J. Zhang, P. Ruggiero and P. Calabrese, Subsystem Trace Distance in Quantum Field Theory, Phys. Rev. Lett.122 (2019) 141602 [arXiv:1901.10993] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett.109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].ADSCrossRefGoogle Scholar
  74. [74]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: A field theoretical approach, J. Stat. Mech.1302 (2013) P02008 [arXiv:1210.5359] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  75. [75]
    P. Calabrese, L. Tagliacozzo and E. Tonni, Entanglement negativity in the critical Ising chain, J. Stat. Mech.1305 (2013) P05002 [arXiv:1302.1113] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  76. [76]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  77. [77]
    M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev.D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].ADSGoogle Scholar
  78. [78]
    P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech.1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].MathSciNetGoogle Scholar
  79. [79]
    M.A. Rajabpour and F. Gliozzi, Entanglement Entropy of Two Disjoint Intervals from Fusion Algebra of Twist Fields, J. Stat. Mech.1202 (2012) P02016 [arXiv:1112.1225] [INSPIRE].MathSciNetGoogle Scholar
  80. [80]
    B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP11 (2013) 164 [arXiv:1309.5453] [INSPIRE].ADSCrossRefGoogle Scholar
  81. [81]
    B. Chen, J.-B. Wu and J.-j. Zhang, Short interval expansion of Rényi entropy on torus, JHEP08 (2016) 130 [arXiv:1606.05444] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  82. [82]
    P. Ruggiero, E. Tonni and P. Calabrese, Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks, J. Stat. Mech.1811 (2018) 113101 [arXiv:1805.05975] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  83. [83]
    F.-L. Lin, H. Wang and J.-j. Zhang, Thermality and excited state Rényi entropy in two-dimensional CFT, JHEP11 (2016) 116 [arXiv:1610.01362] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  84. [84]
    S. He, F.-L. Lin and J.-j. Zhang, Subsystem eigenstate thermalization hypothesis for entanglement entropy in CFT, JHEP08 (2017) 126 [arXiv:1703.08724] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  85. [85]
    S. He, F.-L. Lin and J.-j. Zhang, Dissimilarities of reduced density matrices and eigenstate thermalization hypothesis, JHEP12 (2017) 073 [arXiv:1708.05090] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  86. [86]
    K. Ohmori and Y. Tachikawa, Physics at the entangling surface, J. Stat. Mech.1504 (2015) P04010 [arXiv:1406.4167] [INSPIRE].CrossRefGoogle Scholar
  87. [87]
    J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech.1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  88. [88]
    V. Alba, P. Calabrese and E. Tonni, Entanglement spectrum degeneracy and the Cardy formula in 1+1 dimensional conformal field theories, J. Phys.A 51 (2018) 024001 [arXiv:1707.07532] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  89. [89]
    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
  90. [90]
    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
  91. [91]
    T. Pálmai, Excited state entanglement in one dimensional quantum critical systems: Extensivity and the role of microscopic details, Phys. Rev.B 90 (2014) 161404 [arXiv:1406.3182] [INSPIRE].ADSCrossRefGoogle Scholar
  92. [92]
    T. Pálmai, Entanglement Entropy from the Truncated Conformal Space, Phys. Lett.B 759 (2016) 439 [arXiv:1605.00444] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  93. [93]
    L. Taddia, J.C. Xavier, F.C. Alcaraz and G. Sierra, Entanglement Entropies in Conformal Systems with Boundaries, Phys. Rev.B 88 (2013) 075112 [arXiv:1302.6222].ADSCrossRefGoogle Scholar
  94. [94]
    L. Taddia, F. Ortolani and T. Pálmai, Rényi entanglement entropies of descendant states in critical systems with boundaries: conformal field theory and spin chains, J. Stat. Mech.1609 (2016) 093104 [arXiv:1606.02667] [INSPIRE].CrossRefGoogle Scholar
  95. [95]
    G. Ramirez, J. Rodriguez-Laguna and G. Sierra, Entanglement in low-energy states of the random-hopping model, J. Stat. Mech. (2014) P07003 [arXiv:1402.5015].
  96. [96]
    J. Kurchan, Replica trick to calculate means of absolute values: applications to stochastic equations, J. Phys.A 24 (1991) 4969.ADSMathSciNetGoogle Scholar
  97. [97]
    D.M. Gangardt and A. Kamenev, Replica treatment of the Calogero-Sutherland model, Nucl. Phys.B 610 (2001) 578 [cond-mat/0102405] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  98. [98]
    S.M. Nishigaki, D.M. Gangardt and A. Kamenev, Correlation functions of the BC Calogero-Sutherland model, J. Phys.A 36 (2003) 3137 [cond-mat/0207301] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  99. [99]
    D.M. Gangardt, Universal correlations of trapped one-dimensional impenetrable bosons, J. Phys.A 37 (2004) 9335 [cond-mat/0404104] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  100. [100]
    P. Calabrese and R. Santachiara, Off-diagonal correlations in one-dimensional anyonic models: A replica approach, J. Stat. Mech.0903 (2009) P03002 [arXiv:0811.2991] [INSPIRE].Google Scholar
  101. [101]
    G. Marmorini, M. Pepe and P. Calabrese, One-body reduced density matrix of trapped impenetrable anyons in one dimension, J. Stat. Mech. (2016) 073106 [arXiv:1605.00838].MathSciNetCrossRefGoogle Scholar
  102. [102]
    T. Dupic, B. Estienne and Y. Ikhlef, Entanglement entropies of minimal models from null-vectors, SciPost Phys.4 (2018) 031 [arXiv:1709.09270] [INSPIRE].ADSCrossRefGoogle Scholar
  103. [103]
    F.H.L. Essler, A.M. Läuchli and P. Calabrese, Shell-Filling Effect in the Entanglement Entropies of Spinful Fermions, Phys. Rev. Lett.110 (2013) 115701 [arXiv:1211.2474].ADSCrossRefGoogle Scholar
  104. [104]
    P. Calabrese, F.H.L. Essler and A. Läuchli, Entanglement entropies of the quarter filled Hubbard model, J. Stat. Mech. (2014) P09025 [arXiv:1406.7477].
  105. [105]
    P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer, New York, U.S.A., (1997), [].zbMATHCrossRefGoogle Scholar
  106. [106]
    R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory, Lect. Notes Phys.779 (2009) 1.ADSMathSciNetzbMATHCrossRefGoogle Scholar
  107. [107]
    Z. Li and J.-j. Zhang, On one-loop entanglement entropy of two short intervals from OPE of twist operators, JHEP05 (2016) 130 [arXiv:1604.02779] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  108. [108]
    W.-Z. Guo, F.-L. Lin and J. Zhang, Note on ETH of descendant states in 2D CFT, JHEP01 (2019) 152 [arXiv:1810.01258] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  109. [109]
    M.-C. Chung and I. Peschel, Density-matrix spectra of solvable fermionic systems, Phys. Rev.B 64 (2001) 064412 [cond-mat/0103301] [INSPIRE].ADSCrossRefGoogle Scholar
  110. [110]
    I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys.A 36 (2003) L205 [cond-mat/0212631].ADSMathSciNetzbMATHGoogle Scholar
  111. [111]
    I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys.A 42 (2009) 504003 [arXiv:0906.1663].MathSciNetzbMATHGoogle Scholar
  112. [112]
    I. Peschel, Entanglement in solvable many-particle models, Braz. J. Phys. 42 (2012) 267 [arXiv:1109.0159].ADSCrossRefGoogle Scholar
  113. [113]
    V. Alba, M. Fagotti and P. Calabrese, Entanglement entropy of excited states, J. Stat. Mech.0910 (2009) P10020 [arXiv:0909.1999] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  114. [114]
    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
  115. [115]
    Z. Li and J.-j. Zhang, Holographic Rényi entropy for two-dimensional \( \mathcal{N} \)=(2,2) superconformal field theory, Phys. Rev.D 95 (2017) 126009 [arXiv:1611.00546] [INSPIRE].ADSGoogle Scholar
  116. [116]
    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].ADSCrossRefGoogle Scholar
  117. [117]
    P. Calabrese and F.H.L. Essler, Universal corrections to scaling for block entanglement in spin-1/2 XX chains, J. Stat. Mech. P08029 (2010) [arXiv:1006.3420].
  118. [118]
    M. Fagotti and P. Calabrese, Universal parity effects in the entanglement entropy of XX chains with open boundary conditions, J. Stat. Mech.1101 (2011) P01017 [arXiv:1010.5796] [INSPIRE].MathSciNetGoogle Scholar
  119. [119]
    P. Calabrese, M. Mintchev and E. Vicari, The entanglement entropy of one-dimensional gases, Phys. Rev. Lett.107 (2011) 020601 [arXiv:1105.4756] [INSPIRE].ADSCrossRefGoogle Scholar
  120. [120]
    P. Calabrese, M. Mintchev and E. Vicari, The entanglement entropy of 1D systems in continuous and homogenous space, J. Stat. Mech.1109 (2011) P09028 [arXiv:1107.3985] [INSPIRE].Google Scholar
  121. [121]
    J. Cardy and P. Calabrese, Unusual Corrections to Scaling in Entanglement Entropy, J. Stat. Mech.1004 (2010) P04023 [arXiv:1002.4353] [INSPIRE].Google Scholar
  122. [122]
    M.M. Wilde, A. Winter and D. Yang, Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy, Commun. Math. Phys.331 (2014) 593 [arXiv:1306.1586] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  123. [123]
    M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr and M. Tomamichel, On quantum Rényi entropies: A new generalization and some properties, J. Math. Phys.54 (2013) 122203 [arXiv:1306.3142].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  124. [124]
    C.A. Agon, M. Headrick, D.L. Jafferis and S. Kasko, Disk entanglement entropy for a Maxwell field, Phys. Rev.D 89 (2014) 025018 [arXiv:1310.4886] [INSPIRE].ADSGoogle Scholar
  125. [125]
    C. De Nobili, A. Coser and E. Tonni, Entanglement entropy and negativity of disjoint intervals in CFT: Some numerical extrapolations, J. Stat. Mech.1506 (2015) P06021 [arXiv:1501.04311] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  126. [126]
    T. Mendes-Santos, G. Giudici, M. Dalmonte and M.A. Rajabpour, Entanglement Hamiltonian of quantum critical chains and conformal field theories, arXiv:1906.00471 [INSPIRE].
  127. [127]
    T. Ugajin, Mutual information of excited states and relative entropy of two disjoint subsystems in CFT, JHEP10 (2017) 184 [arXiv:1611.03163] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  128. [128]
    J. Dubail, J.-M. Stéphan, J. Viti and P. Calabrese, Conformal Field Theory for Inhomogeneous One-dimensional Quantum Systems: the Example of Non-Interacting Fermi Gases, SciPost Phys.2 (2017) 002 [arXiv:1606.04401] [INSPIRE].ADSCrossRefGoogle Scholar
  129. [129]
    M. Dalmonte, B. Vermersch and P. Zoller, Quantum Simulation and Spectroscopy of Entanglement Hamiltonians, Nature Phys.14 (2018) 827 [arXiv:1707.04455] [INSPIRE].ADSCrossRefGoogle Scholar
  130. [130]
    G. Giudici, T. Mendes-Santos, P. Calabrese and M. Dalmonte, Entanglement Hamiltonians of lattice models via the Bisognano-Wichmann theorem, Phys. Rev.B 98 (2018) 134403 [arXiv:1807.01322] [INSPIRE].ADSCrossRefGoogle Scholar
  131. [131]
    J.J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys.17 (1976) 303 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  132. [132]
    J.J. Bisognano and E.H. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys.16 (1975) 985 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  133. [133]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  134. [134]
    V. Eisler and I. Peschel, Analytical results for the entanglement Hamiltonian of a free-fermion chain, J. Phys.A 50 (2017) 284003 [arXiv:1703.08126] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  135. [135]
    V. Eisler and I. Peschel, Properties of the entanglement Hamiltonian for finite free-fermion chains, J. Stat. Mech.1810 (2018) 104001 [arXiv:1805.00078] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  136. [136]
    V. Eisler, E. Tonni and I. Peschel, On the continuum limit of the entanglement Hamiltonian, J. Stat. Mech.1907 (2019) 073101 [arXiv:1902.04474] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  137. [137]
    F. Parisen Toldin and F.F. Assaad, Entanglement Hamiltonian of Interacting Fermionic Models, Phys. Rev. Lett.121 (2018) 200602 [arXiv:1804.03163] [INSPIRE].ADSCrossRefGoogle Scholar
  138. [138]
    J. Cardy, Some results on the mutual information of disjoint regions in higher dimensions, J. Phys.A 46 (2013) 285402 [arXiv:1304.7985] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  139. [139]
    L.-Y. Hung, R.C. Myers and M. Smolkin, Twist operators in higher dimensions, JHEP10 (2014) 178 [arXiv:1407.6429] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  140. [140]
    B. Chen and J. Long, Rényi mutual information for a free scalar field in even dimensions, Phys. Rev.D 96 (2017) 045006 [arXiv:1612.00114] [INSPIRE].ADSGoogle Scholar
  141. [141]
    B. Chen, L. Chen, P.-x. Hao and J. Long, On the Mutual Information in Conformal Field Theory, JHEP06 (2017) 096 [arXiv:1704.03692] [INSPIRE].ADSMathSciNetGoogle Scholar
  142. [142]
    B. Chen, Z.-Y. Fan, W.-M. Li and C.-Y. Zhang, Holographic Mutual Information of Two Disjoint Spheres, JHEP04 (2018) 113 [arXiv:1712.05131] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  143. [143]
    E.H. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals Phys.16 (1961) 407 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  144. [144]
    P. Pfeuty, The one-dimensional Ising model with a transverse field, Annals Phys.57 (1970) 79.ADSCrossRefGoogle Scholar
  145. [145]
    P. Calabrese, F.H. Essler and M. Fagotti, Quantum quench in the transverse field ising chain: I. time evolution of order parameter correlators, J. Stat. Mech. (2012) P07016 [arXiv:1204.3911].
  146. [146]
    R. Balian and E. Brézin, Nonunitary bogoliubov transformations and extension of wick’s theorem, Nuovo Cim.B 64 (1969) 37 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

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© The Author(s) 2019

Authors and Affiliations

  1. 1.SISSA and INFNTriesteItaly
  2. 2.International Centre for Theoretical Physics (ICTP)TriesteItaly

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