Journal of High Energy Physics

, 2019:104 | Cite as

Thermodynamic bootstrap program for integrable QFT’s: form factors and correlation functions at finite energy density

  • Axel Cortés Cubero
  • Miłosz PanfilEmail author
Open Access
Regular Article - Theoretical Physics


We study the form factors of local operators of integrable QFT’s between states with finite energy density. These states arise, for example, at finite temperature, or from a generalized Gibbs ensemble. We generalize Smirnov’s form factor axioms, formulating them for a set of particle/hole excitations on top of the thermodynamic background, instead of the vacuum. We show that exact form factors can be found as minimal solutions of these new axioms. The thermodynamic form factors can be used to construct correlation functions on thermodynamic states. The expression found for the two-point function is similar to the conjectured LeClair-Mussardo formula, but using the new form factors dressed by the thermodynamic background, and with all singularities properly regularized. We study the different infrared asymptotics of the thermal two-point function, and show there generally exist two different regimes, manifesting massive exponential decay, or effectively gapless behavior at long distances, respectively. As an example, we compute the few-excitations form factors of vertex operators for the sinh-Gordon model.


Bethe Ansatz Field Theories in Lower Dimensions Integrable Field Theories Nonperturbative Effects 


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]
    A. Altland and B. Simons, Condensed Matter Field Theory, Cambridge University Press, (2007).Google Scholar
  2. [2]
    G. Baym, T. Hatsuda, T. Kojo, P.D. Powell, Y. Song and T. Takatsuka, From hadrons to quarks in neutron stars: a review, Rept. Prog. Phys. 81 (2018) 056902 [arXiv:1707.04966] [INSPIRE].
  3. [3]
    W. Plessas and L. Mathelitsch eds., Lectures on Quark Matter, Springer-Verlag Berlin Heidelberg, (2001).Google Scholar
  4. [4]
    P.B. Arnold, quark-gluon Plasmas and Thermalization, Int. J. Mod. Phys. E 16 (2007) 2555 [arXiv:0708.0812] [INSPIRE].
  5. [5]
    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
  6. [6]
    P. Calabrese, F.H.L. Essler and G. Mussardo, Introduction to ‘Quantum Integrability in Out of Equilibrium Systems’, J. Stat. Mech. 2016 (2016) 064001.Google Scholar
  7. [7]
    F.H.L. Essler, G. Mussardo and M. Panfil, On truncated generalized Gibbs ensembles in the Ising field theory, J. Stat. Mech. 2017 (2017) 013103.Google Scholar
  8. [8]
    L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. 2016 (2016) 064007.Google Scholar
  9. [9]
    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].
  10. [10]
    A. Bastianello and S. Sotiriadis, Quasi locality of the GGE in interacting-to-free quenches in relativistic field theories, J. Stat. Mech. 2017 (2017) 023105 [arXiv:1608.00924] [INSPIRE].
  11. [11]
    E. Vernier and A. Cortés Cubero, Quasilocal charges and progress towards the complete GGE for field theories with nondiagonal scattering, J. Stat. Mech. 2017 (2017) 023101 [arXiv:1609.03220] [INSPIRE].
  12. [12]
    O.A. Castro-Alvaredo, B. Doyon and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6 (2016) 041065 [INSPIRE].
  13. [13]
    B. Bertini, M. Collura, J. De Nardis and M. Fagotti, Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117 (2016) 207201 [arXiv:1605.09790] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    A. Lazarides, A. Das and R. Moessner, Equilibrium states of generic quantum systems subject to periodic driving, Phys. Rev. E 90 (2014) 012110.Google Scholar
  15. [15]
    V. Gritsev and A. Polkovnikov, Integrable Floquet dynamics, SciPost Phys. 2 (2017) 021 [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    A. Cortés Cubero, Integrable Floquet QFT: Elasticity and factorization under periodic driving, SciPost Phys. 5 (2018) 025 [arXiv:1804.07728] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    F.A. Smirnov, Form Factors in Completely Integrable Models of Quantum Field Theory, World Scientific, (1992), [].
  18. [18]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Factorized s Matrices in Two-Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Models, Annals Phys. 120 (1979) 253 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    F.H.L. Essler and R.M. Konik, Application of Massive Integrable Quantum Field Theories to Problems in Condensed Matter Physics, World Scientific Publishing Co, (2005), pp. 684-830.Google Scholar
  20. [20]
    V. Gritsev, A. Polkovnikov and E. Demler, Linear response theory for a pair of coupled one-dimensional condensates of interacting atoms, Phys. Rev. B 75 (2007) 174511.Google Scholar
  21. [21]
    A. Zamolodchikov, Integrable Field Theory from Conformal Field Theory, in Integrable Sys Quantum Field Theory, M. Jimbo, T. Miwa and A. Tsuchiya eds., Academic Press, San Diego, U.S.A., (1989), pp. 641-674,.Google Scholar
  22. [22]
    P. Cooper, S. Dubovsky, V. Gorbenko, A. Mohsen and S. Storace, Looking for Integrability on the Worldsheet of Confining Strings, JHEP 04 (2015) 127 [arXiv:1411.0703] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    M. Karowski and P. Weisz, Exact Form-Factors in (1+1)-Dimensional Field Theoretic Models with Soliton Behavior, Nucl. Phys. B 139 (1978) 455 [INSPIRE].
  25. [25]
    A. Leclair and G. Mussardo, Finite temperature correlation functions in integrable QFT, Nucl. Phys. B 552 (1999) 624 [hep-th/9902075] [INSPIRE].
  26. [26]
    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].
  27. [27]
    J. Balog, Field theoretical derivation of the TBA integral equation, Nucl. Phys. B 419 (1994) 480 [INSPIRE].
  28. [28]
    H. Saleur, A comment on finite temperature correlations in integrable QFT, Nucl. Phys. B 567 (2000) 602 [hep-th/9909019] [INSPIRE].
  29. [29]
    O.A. Castro-Alvaredo and A. Fring, Finite temperature correlation functions from form-factors, Nucl. Phys. B 636 (2002) 611 [hep-th/0203130] [INSPIRE].
  30. [30]
    B. Pozsgay and G. Takács, Form factor expansion for thermal correlators, J. Stat. Mech. 1011 (2010) P11012 [arXiv:1008.3810] [INSPIRE].
  31. [31]
    A. Leclair, F. Lesage, S. Sachdev and H. Saleur, Finite temperature correlations in the one-dimensional quantum Ising model, Nucl. Phys. B 482 (1996) 579 [cond-mat/9606104] [INSPIRE].
  32. [32]
    A.C. Cubero, Nontrivial thermodynamics in ’t Hooft’s large-N limit, Phys. Rev. D 91 (2015) 105025 [arXiv:1503.06139] [INSPIRE].
  33. [33]
    F.H.L. Essler and R.M. Konik, Finite-temperature dynamical correlations in massive integrable quantum field theories, J. Stat. Mech. 0909 (2009) P09018 [arXiv:0907.0779] [INSPIRE].
  34. [34]
    B. Pozsgay and I.M. Szécsényi, LeClair-Mussardo series for two-point functions in Integrable QFT, JHEP 05 (2018) 170 [arXiv:1802.05890] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    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].
  36. [36]
    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].ADSCrossRefGoogle Scholar
  37. [37]
    V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge Monographs on Mathematical Physics, Cambridge University Press, (1993), [].
  38. [38]
    B. Pozsgay, Mean values of local operators in highly excited Bethe states, J. Stat. Mech. 1101 (2011) P01011 [arXiv:1009.4662] [INSPIRE].
  39. [39]
    M. Panfil and J.-S. Caux, Finite-temperature correlations in the Lieb-Liniger one-dimensional Bose gas, Phys. Rev. A 89 (2014) 033605 [arXiv:1308.2887].
  40. [40]
    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
  41. [41]
    J. De 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.Google Scholar
  42. [42]
    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].
  43. [43]
    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].
  44. [44]
    J.D. Nardis and M. Panfil, Density form factors of the 1D Bose gas for finite entropy states, J. Stat. Mech. 2015 (2015) P02019.Google Scholar
  45. [45]
    J.D. Nardis and M. Panfil, Exact correlations in the Lieb-Liniger model and detailed balance out-of-equilibrium, SciPost Phys. 1 (2016) 015.ADSCrossRefGoogle Scholar
  46. [46]
    J. De Nardis and M. Panfil, Particle-hole pairs and density-density correlations in the Lieb-Liniger model, J. Stat. Mech. 1803 (2018) 033102 [arXiv:1712.06581] [INSPIRE].
  47. [47]
    B. Doyon, Finite-temperature form-factors in the free Majorana theory, J. Stat. Mech. 0511 (2005) P11006 [hep-th/0506105] [INSPIRE].
  48. [48]
    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].
  49. [49]
    P. Dorey, Exact S matrices, in Conformal field theories and integrable models. Proceedings, Eotvos Graduate Course, Budapest, Hungary, August 13-18, 1996, pp. 85-125, hep-th/9810026 [INSPIRE].
  50. [50]
    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].
  51. [51]
    C. Acerbi, G. Mussardo and A. Valleriani, On the form-factors of relevant operators and their cluster property, J. Phys. A 30 (1997) 2895 [hep-th/9609080] [INSPIRE].
  52. [52]
    G. Delfino and G. Mussardo, The spin spin correlation function in the two-dimensional Ising model in a magnetic field at T = T(c), Nucl. Phys. B 455 (1995) 724 [hep-th/9507010] [INSPIRE].
  53. [53]
    G. Mussardo, Statistical field theory: an introduction to exactly solved models in statistical physics, Oxford University Press, (2009).Google Scholar
  54. [54]
    A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
  55. [55]
    Z. Bájnok, J. Balog, M. Lájer and C. Wu, Field theoretical derivation of Lüscher’s formula and calculation of finite volume form factors, JHEP 07 (2018) 174 [arXiv:1802.04021] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    A. Shashi, L.I. Glazman, J.-S. Caux and A. Imambekov, Nonuniversal prefactors in the correlation functions of one-dimensional quantum liquids, Phys. Rev. B 84 (2011) 045408.Google Scholar
  57. [57]
    V.E. Korepin and N.A. Slavnov, Time dependance of the denisty-density temperature correlation function of a one-dimensional Bose gas, Nucl. Phys. B 340 (1990) 757.Google Scholar
  58. [58]
    J.-S. Caux, The quench action, J. Stat. Mech. Theor. Exp. 2016 (2016) 064006.Google Scholar
  59. [59]
    N. Kitanine, K.K. Kozlowski, J.M. Maillet, N.A. Slavnov and V. Terras, Form factor approach to dynamical correlation functions in critical models, J. Stat. Mech. 1209 (2012) P09001 [arXiv:1206.2630] [INSPIRE].
  60. [60]
    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].
  61. [61]
    D. Fioretto and G. Mussardo, Quantum Quenches in Integrable Field Theories, New J. Phys. 12 (2010) 055015 [arXiv:0911.3345] [INSPIRE].
  62. [62]
    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].
  63. [63]
    Z. Bájnok, Review of AdS/CFT Integrability, Chapter III.6: Thermodynamic Bethe Ansatz, Lett. Math. Phys. 99 (2012) 299 [arXiv:1012.3995] [INSPIRE].
  64. [64]
    E. Ilievski and T. Prosen, Thermodyamic Bounds on Drude Weights in Terms of Almost-conserved Quantities, Commun. Math. Phys. 318 (2013) 809 [arXiv:1111.3830].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    J. De Nardis, D. Bernard and B. Doyon, Hydrodynamic Diffusion in Integrable Systems, Phys. Rev. Lett. 121 (2018) 160603 [arXiv:1807.02414] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  66. [66]
    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].
  67. [67]
    A. Bastianello, A. De Luca and G. Mussardo, Non relativistic limit of integrable QFT and Lieb-Liniger models, J. Stat. Mech. 1612 (2016) 123104 [arXiv:1608.07548] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  68. [68]
    A. Bastianello and L. Piroli, From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics, J. Stat. Mech. 1811 (2018) 113104 [arXiv:1807.06869] [INSPIRE].CrossRefGoogle Scholar
  69. [69]
    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].
  70. [70]
    S.L. Lukyanov, Form-factors of exponential fields in the sine-Gordon model, Mod. Phys. Lett. A 12 (1997) 2543 [hep-th/9703190] [INSPIRE].
  71. [71]
    S. Negro, On sinh-Gordon Thermodynamic Bethe Ansatz and fermionic basis, Int. J. Mod. Phys. A 29 (2014) 1450111 [arXiv:1404.0619] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institute for Theoretical Physics, Center for Extreme Matter and Emergent PhenomenaUtrecht UniversityUtrechtThe Netherlands
  2. 2.Faculty of PhysicsUniversity of WarsawWarsawPoland

Personalised recommendations