Perturbative post-quench overlaps in quantum field theory

  • Kristóf HódságiEmail author
  • Márton Kormos
  • Gábor Takács
Open Access
Regular Article - Theoretical Physics


In analytic descriptions of quantum quenches, the overlaps between the initial pre-quench state and the eigenstates of the time evolving Hamiltonian are crucial ingredients. We construct perturbative expansions of these overlaps in quantum field theories where either the pre-quench or the post-quench Hamiltonian is integrable. Using the E8 Ising field theory for concrete computations, we give explicit expressions for the overlaps up to second order in the quench size, and verify our results against numerical results obtained using the Truncated Conformal Space Approach. We demonstrate that the expansion using the post-quench basis is very effective, but find some serious limitations for the alternative approach using the pre-quench basis.


Bethe Ansatz Boundary Quantum Field Theory Conformal Field Theory 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

Supplementary material (303 kb)
ESM 1 (ZIP 302 kb)


  1. [1]
    T. Kinoshita, T. Wenger and D.S. Weiss, A quantum Newton’s cradle, Nature 440 (2006) 900.ADSCrossRefGoogle Scholar
  2. [2]
    S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm and J. Schmiedmayer, Non-equilibrium coherence dynamics in one-dimensional Bose gases, Nature 449 (2007) 324 [arXiv:0706.2259].ADSCrossRefGoogle Scholar
  3. [3]
    J. Simon, W.S. Bakr, R. Ma, M.E. Tai, P.M. Preiss and M. Greiner, Quantum simulation of antiferromagnetic spin chains in an optical lattice, Nature 472 (2011) 307 [arXiv:1103.1372].ADSCrossRefGoogle Scholar
  4. [4]
    S. Trotzky et al., Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas, Nature Phys. 8 (2012) 325 [arXiv:1101.2659].ADSCrossRefGoogle Scholar
  5. [5]
    M. Cheneau et al., Light-cone-like spreading of correlations in a quantum many-body system, Nature 481 (2012) 484 [arXiv:1111.0776].ADSCrossRefGoogle Scholar
  6. [6]
    M. Gring et al., Relaxation and prethermalization in an isolated quantum system, Science 337 (2012) 1318 [arXiv:1112.0013].ADSCrossRefGoogle Scholar
  7. [7]
    T. Langen, R. Geiger, M. Kuhnert, B. Rauer and J. Schmiedmayer, Local emergence of thermal correlations in an isolated quantum many-body system, Nature Phys. 9 (2013) 640 [arXiv:1305.3708].ADSCrossRefGoogle Scholar
  8. [8]
    T. Fukuhara et al., Microscopic observation of magnon bound states and their dynamics, Nature 502 (2013) 76 [arXiv:1305.6598].ADSCrossRefGoogle Scholar
  9. [9]
    F. Meinert et al., Quantum quench in an atomic one-dimensional Ising chain, Phys. Rev. Lett. 111 (2013) 053003 [arXiv:1304.2628].
  10. [10]
    T. Langen et al., Experimental observation of a generalized gibbs ensemble, Science 348 (2015) 207 [arXiv:1411.7185].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A.M. Kaufman et al., Quantum thermalization through entanglement in an isolated many-body system, Science 353 (2016) 794 [arXiv:1603.04409].ADSCrossRefGoogle Scholar
  12. [12]
    A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83 (2011) 863 [arXiv:1007.5331] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    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].
  14. [14]
    P. Calabrese, F.H.L. Essler and G. Mussardo, Introduction to ‘quantum integrability in out of equilibrium systems’, J. Stat. Mech. 6 (2016) 064001.CrossRefGoogle Scholar
  15. [15]
    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].
  16. [16]
    P. Calabrese and J. Cardy, Quantum quenches in extended systems, J. Stat. Mech. 0706 (2007) P06008 [arXiv:0704.1880] [INSPIRE].
  17. [17]
    J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.ADSCrossRefGoogle Scholar
  18. [18]
    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50 (1994) 888 [cond-mat/9403051].
  19. [19]
    M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324].ADSCrossRefGoogle Scholar
  20. [20]
    M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons, Phys. Rev. Lett. 98 (2007) 050405 [cond-mat/0604476].
  21. [21]
    B. Wouters, J.D. Nardis, M. Brockmann, D. Fioretto, M. Rigol and J.-S. Caux, Quenching the anisotropic heisenberg chain: exact solution and generalized Gibbs ensemble predictions, Phys. Rev. Lett. 113 (2014) 117202 [arXiv:1405.0172].ADSCrossRefGoogle Scholar
  22. [22]
    B. Pozsgay et al., Correlations after quantum quenches in the XXZ spin chain: failure of the generalized Gibbs ensemble, Phys. Rev. Lett. 113 (2014) 117203 [arXiv:1405.2843].ADSCrossRefGoogle Scholar
  23. [23]
    G. Goldstein and N. Andrei, Failure of the local generalized Gibbs ensemble for integrable models with bound states, Phys. Rev. A 90 (2014) 043625 [arXiv:1405.4224].
  24. [24]
    B. Pozsgay, Failure of the generalized eigenstate thermalization hypothesis in integrable models with multiple particle species, J. Stat. Mech. 9 (2014) 09026 [arXiv:1406.4613].
  25. [25]
    T. Prosen, Quasilocal conservation laws in XXZ spin-1/2 chains: open, periodic and twisted boundary conditions, Nucl. Phys. B 886 (2014) 1177 [arXiv:1406.2258] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    E. Ilievski, J.D. Nardis, B. Wouters, J.-S. Caux, F. Essler and T. Prosen, Complete generalized Gibbs ensembles in an interacting theory, Phys. Rev. Lett. 115 (2015) 157201 [arXiv:1507.02993].ADSCrossRefGoogle Scholar
  27. [27]
    E. Ilievski, E. Quinn, J. De Nardis and M. Brockmann, String-charge duality in integrable lattice models, J. Stat. Mech. 1606 (2016) 063101 [arXiv:1512.04454] [INSPIRE].
  28. [28]
    E. Ilievski, M. Medenjak, T. Prosen and L. Zadnik, Quasilocal charges in integrable lattice systems, J. Stat. Mech. 1606 (2016) 064008 [arXiv:1603.00440] [INSPIRE].
  29. [29]
    F.H.L. Essler, G. Mussardo and M. Panfil, Generalized Gibbs ensembles for quantum field theories, Phys. Rev. A 91 (2015) 051602 [arXiv:1411.5352] [INSPIRE].
  30. [30]
    M.A. Cazalilla, Effect of suddenly turning on interactions in the Luttinger model, Phys. Rev. Lett. 97 (2006) 156403 [cond-mat/0606236].
  31. [31]
    A. Silva, Statistics of the work done on a quantum critical system by quenching a control parameter, Phys. Rev. Lett. 101 (2008) 120603 [arXiv:0806.4301].ADSCrossRefGoogle Scholar
  32. [32]
    S. Sotiriadis, P. Calabrese and J. Cardy, Quantum quench from a thermal initial state, EPL (Europhys. Lett.) 87 (2009) 20002.ADSCrossRefGoogle Scholar
  33. [33]
    D. Fioretto and G. Mussardo, Quantum quenches in integrable field theories, New J. Phys. 12 (2010) 055015 [arXiv:0911.3345] [INSPIRE].
  34. [34]
    B. Dóra, M. Haque and G. Zaránd, Crossover from adiabatic to sudden interaction quench in a Luttinger liquid, Phys. Rev. Lett. 106 (2011) 156406 [arXiv:1011.6655].ADSCrossRefGoogle Scholar
  35. [35]
    P. Calabrese, F.H.L. Essler and M. Fagotti, Quantum quench in the transverse field Ising chain, Phys. Rev. Lett. 106 (2011) 227203 [arXiv:1104.0154] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    P. Calabrese, F.H.L. Essler and M. Fagotti, Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators, J. Statist. Mech. 7 (2012) 07016 [arXiv:1204.3911].
  37. [37]
    P. Calabrese, F.H.L. Essler and M. Fagotti, Quantum quenches in the transverse field Ising chain: II. Stationary state properties, J. Stat. Mech. 7 (2012) 07022 [arXiv:1205.2211].
  38. [38]
    F.H.L. Essler, S. Evangelisti and M. Fagotti, Dynamical correlations after a quantum quench, Phys. Rev. Lett. 109 (2012) 247206 [arXiv:1208.1961].ADSCrossRefGoogle Scholar
  39. [39]
    M. Collura, S. Sotiriadis and P. Calabrese, Equilibration of a Tonks-Girardeau gas following a trap release, Phys. Rev. Lett. 110 (2013) 245301 [arXiv:1303.3795].ADSCrossRefGoogle Scholar
  40. [40]
    M. Heyl, A. Polkovnikov and S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model, Phys. Rev. Lett. 110 (2013) 135704 [arXiv:1206.2505].ADSCrossRefGoogle Scholar
  41. [41]
    L. Bucciantini, M. Kormos and P. Calabrese, Quantum quenches from excited states in the Ising chain, J. Phys. A 47 (2014) 175002 [arXiv:1401.7250] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  42. [42]
    M. Kormos, M. Collura and P. Calabrese, Analytic results for a quantum quench from free to hard-core one dimensional bosons, Phys. Rev. A 89 (2014) 013609 [arXiv:1307.2142] [INSPIRE].
  43. [43]
    S. Sotiriadis and P. Calabrese, Validity of the GGE for quantum quenches from interacting to noninteracting models, J. Stat. Mech. 1407 (2014) P07024 [arXiv:1403.7431] [INSPIRE].
  44. [44]
    S. Sotiriadis, Memory-preserving equilibration after a quantum quench in a one-dimensional critical model, Phys. Rev. A 94 (2016) 031605 [arXiv:1507.07915] [INSPIRE].
  45. [45]
    M. Collura, M. Kormos and P. Calabrese, Quantum quench in a harmonically trapped one-dimensional Bose gas, Phys. Rev. A 97 (2018) 033609 [arXiv:1710.11615].
  46. [46]
    J.-S. Caux and F.H.L. Essler, Time evolution of local observables after quenching to an integrable model, Phys. Rev. Lett. 110 (2013) 257203 [arXiv:1301.3806] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    J.D. Nardis, L. Piroli and J.-S. Caux, Relaxation dynamics of local observables in integrable systems, J. Phys. A 48 (2015) 43FT01 [arXiv:1505.03080].
  48. [48]
    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].
  49. [49]
    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].
  50. [50]
    K.K. Kozlowski and B. Pozsgay, Surface free energy of the open XXZ spin-1/2 chain, J. Stat. Mech. 1205 (2012) P05021 [arXiv:1201.5884] [INSPIRE].
  51. [51]
    J.D. Nardis, B. Wouters, M. Brockmann and J.-S. Caux, Solution for an interaction quench in the Lieb-Liniger Bose gas, Phys. Rev. A 89 (2014) 033601 [arXiv:1308.4310].
  52. [52]
    M. Brockmann, B. Wouters, D. Fioretto, J.D. Nardis, R. Vlijm and J.-S. Caux, Quench action approach for releasing the Néel state into the spin-1/2 XXZ chain, J. Stat. Mech. 12 (2014) 12009 [arXiv:1408.5075].CrossRefGoogle Scholar
  53. [53]
    M. Brockmann, J.D. Nardis, B. Wouters and J.-S. Caux, A Gaudin-like determinant for overlaps of Néel and XXZ Bethe states, J. Phys. A 47 (2014) 145003 [arXiv:1401.2877].ADSzbMATHGoogle Scholar
  54. [54]
    B. Pozsgay, Overlaps between eigenstates of the XXZ spin-1/2 chain and a class of simple product states, J. Stat. Mech. 6 (2014) 06011 [arXiv:1309.4593].
  55. [55]
    M. Brockmann, Overlaps of q-raised Néel states with XXZ Bethe states and their relation to the Lieb-Liniger Bose gas, J. Stat. Mech. 5 (2014) 05006 [arXiv:1402.1471].
  56. [56]
    M. Brockmann, J.D. Nardis, B. Wouters and J.-S. Caux, Néel-XXZ state overlaps: odd particle numbers and Lieb-Liniger scaling limit, J. Phys. A 47 (2014) 345003 [arXiv:1403.7469].zbMATHGoogle Scholar
  57. [57]
    L. Piroli, P. Calabrese and F.H. Essler, Multiparticle bound-state formation following a quantum quench to the one-dimensional Bose gas with attractive interactions, Phys. Rev. Lett. 116 (2016) 070408 [arXiv:1509.08234].
  58. [58]
    M. Mestyán, B. Bertini, L. Piroli and P. Calabrese, Exact solution for the quench dynamics of a nested integrable system, J. Stat. Mech. 1708 (2017) 083103 [arXiv:1705.00851] [INSPIRE].
  59. [59]
    B. Pozsgay, Overlaps with arbitrary two-site states in the XXZ spin chain, J. Stat. Mech. 1805 (2018) 053103 [arXiv:1801.03838] [INSPIRE].
  60. [60]
    M. de Leeuw, C. Kristjansen and K. Zarembo, One-point functions in defect CFT and integrability, JHEP 08 (2015) 098 [arXiv:1506.06958] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    O. Foda and K. Zarembo, Overlaps of partial Néel states and Bethe states, J. Stat. Mech. 1602 (2016) 023107 [arXiv:1512.02533] [INSPIRE].
  62. [62]
    I. Buhl-Mortensen, M. de Leeuw, C. Kristjansen and K. Zarembo, One-point functions in AdS/dCFT from matrix product states, JHEP 02 (2016) 052 [arXiv:1512.02532] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    M. de Leeuw, C. Kristjansen and S. Mori, AdS/dCFT one-point functions of the SU(3) sector, Phys. Lett. B 763 (2016) 197 [arXiv:1607.03123] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  64. [64]
    L. Piroli, B. Pozsgay and E. Vernier, What is an integrable quench?, Nucl. Phys. B 925 (2017) 362 [arXiv:1709.04796] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    B. Pozsgay, L. Piroli and E. Vernier, Integrable matrix product states from boundary integrability, SciPost Phys. 6 (2019) 062 [arXiv:1812.11094] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    S. Sotiriadis, G. Takács and G. Mussardo, Boundary state in an integrable quantum field theory out of equilibrium, Phys. Lett. B 734 (2014) 52 [arXiv:1311.4418] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  67. [67]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    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].ADSCrossRefzbMATHGoogle Scholar
  70. [70]
    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].ADSCrossRefGoogle Scholar
  71. [71]
    S. Sotiriadis, D. Fioretto and G. Mussardo, Zamolodchikov-Faddeev algebra and quantum quenches in integrable field theories, J. Stat. Mech. 1202 (2012) P02017 [arXiv:1112.2963] [INSPIRE].MathSciNetGoogle Scholar
  72. [72]
    G. Mussardo, Infinite-time average of local fields in an integrable quantum field theory after a quantum quench, Phys. Rev. Lett. 111 (2013) 100401 [arXiv:1308.4551] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    D. Schuricht, Quantum quenches in integrable systems: constraints from factorisation, J. Stat. Mech. 1511 (2015) P11004 [arXiv:1509.00435] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  74. [74]
    A. Cortés Cubero, G. Mussardo and M. Panfil, Quench dynamics in two-dimensional integrable SUSY models, J. Stat. Mech. 1603 (2016) 033115 [arXiv:1511.02712] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  75. [75]
    B. Bertini, L. Piroli and P. Calabrese, Quantum quenches in the sinh-Gordon model: steady state and one point correlation functions, J. Stat. Mech. 1606 (2016) 063102 [arXiv:1602.08269] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  76. [76]
    A. Cortés Cubero, Planar quantum quenches: computation of exact time-dependent correlation functions at large N , J. Stat. Mech. 1608 (2016) 083107 [arXiv:1604.03879] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  77. [77]
    F.H.L. Essler, G. Mussardo and M. Panfil, On truncated generalized Gibbs ensembles in the Ising field theory, J. Stat. Mech. 1701 (2017) 013103 [arXiv:1610.02495] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  78. [78]
    G. Delfino, Quantum quenches with integrable pre-quench dynamics, J. Phys. A 47 (2014) 402001 [arXiv:1405.6553] [INSPIRE].zbMATHGoogle Scholar
  79. [79]
    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].
  80. [80]
    A.B. Zamolodchikov, Integrals of motion and S matrix of the (scaled) T = T c Ising model with magnetic field, Int. J. Mod. Phys. A 4 (1989) 4235 [INSPIRE].
  81. [81]
    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].
  82. [82]
    S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841 [Erratum ibid. A 9 (1994) 4353] [hep-th/9306002] [INSPIRE].
  83. [83]
    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].MathSciNetCrossRefGoogle Scholar
  84. [84]
    B. Pozsgay and G. Takács, Form-factors in finite volume I: form-factor bootstrap and truncated conformal space, Nucl. Phys. B 788 (2008) 167 [arXiv:0706.1445] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  85. [85]
    M. Kormos and B. Pozsgay, One-point functions in massive integrable QFT with boundaries, JHEP 04 (2010) 112 [arXiv:1002.2783] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    B. Pozsgay and G. Takács, Form factors in finite volume. II. Disconnected terms and finite temperature correlators, Nucl. Phys. B 788 (2008) 209 [arXiv:0706.3605] [INSPIRE].
  87. [87]
    G. Delfino, G. Mussardo and P. Simonetti, Nonintegrable quantum field theories as perturbations of certain integrable models, Nucl. Phys. B 473 (1996) 469 [hep-th/9603011] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  88. [88]
    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].
  89. [89]
    V.A. Fateev and A.B. Zamolodchikov, Conformal field theory and purely elastic S matrices, Int. J. Mod. Phys. A 5 (1990) 1025 [INSPIRE].
  90. [90]
    F.A. Smirnov, Form factors in completely integrable models of quantum field theory, Adv. Ser. Math. Phys. 14 (1992) 1 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  91. [91]
    G. Delfino, P. Grinza and G. Mussardo, Decay of particles above threshold in the Ising field theory with magnetic field, Nucl. Phys. B 737 (2006) 291 [hep-th/0507133] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  92. [92]
  93. [93]
    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].ADSCrossRefGoogle Scholar
  94. [94]
    V.P. Yurov and A.B. Zamolodchikov, Truncated fermionic space approach to the critical 2D Ising model with magnetic field, Int. J. Mod. Phys. A 6 (1991) 4557 [INSPIRE].ADSCrossRefGoogle Scholar
  95. [95]
    M. Kormos, Boundary renormalisation group flows of the supersymmetric Lee-Yang model and its extensions, Nucl. Phys. B 772 (2007) 227 [hep-th/0701061] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  96. [96]
    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].ADSMathSciNetCrossRefGoogle Scholar
  97. [97]
    M. Lencsés and G. Takács, Confinement in the q-state Potts model: an RG-TCSA study, JHEP 09 (2015) 146 [arXiv:1506.06477] [INSPIRE].ADSCrossRefGoogle Scholar
  98. [98]
    M. Lencsés, J. Viti and G. Takács, Chiral entanglement in massive quantum field theories in 1 + 1 dimensions, JHEP 01 (2019) 177 [arXiv:1811.06500] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  99. [99]
    E.H. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals Phys. 16 (1961) 407 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  100. [100]
    P. Pfeuty, The one-dimensional Ising model with a transverse field, Annals Phys. 57 (1970) 79.Google Scholar
  101. [101]
    B.M. McCoy and T.T. Wu, Two-dimensional Ising field theory in a magnetic field: breakup of the cut in the two point function, Phys. Rev. D 18 (1978) 1259 [INSPIRE].ADSGoogle Scholar
  102. [102]
    P. Fonseca and A. Zamolodchikov, Ising spectroscopy I: mesons at T < T c, hep-th/0612304 [INSPIRE].
  103. [103]
    A. Zamolodchikov, Ising spectroscopy II: particles and poles at T > T c, arXiv:1310.4821 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsBudapest University of Technology and EconomicsBudapestHungary
  2. 2.BME “Momentum” Statistical Field Theory Research GroupBudapestHungary

Personalised recommendations