Thermalization in 2D critical quench and UV/IR mixing

  • Gautam Mandal
  • Shruti Paranjape
  • Nilakash Sorokhaibam
Open Access
Regular Article - Theoretical Physics


We consider quantum quenches in models of free scalars and fermions with a generic time-dependent mass m(t) that goes from m0 to zero. We prove that, as anticipated in MSS [1], the post-quench dynamics can be described in terms of a state of the generalized Calabrese-Cardy form |ψ〉 = exp[−κ2H − ∑ n >2 κ n W n ]|Bd〉. The W n (n = 2, 3, . . ., W2 = H) here represent the conserved W charges and |Bd〉 represents a conformal boundary state. Our result holds irrespective of whether the pre-quench state is a ground state or a squeezed state, and is proved without recourse to perturbation expansion in the κ n ’s as in MSS. We compute exact time-dependent correlators for some specific quench protocols m(t). The correlators explicitly show thermalization to a generalized Gibbs ensemble (GGE), with inverse temperature β = 4κ2, and chemical potentials μ n = 4κ n . In case the pre-quench state is a ground state, it is possible to retrieve the exact quench protocol m(t) from the final GGE, by an application of inverse scattering techniques. Another notable result, which we interpret as a UV/IR mixing, is that the long distance and long time (IR) behaviour of some correlators depends crucially on all κ n ’s, although they are highly irrelevant couplings in the usual RG parlance. This indicates subtleties in RG arguments when applied to non-equilibrium dynamics.


Conformal Field Theory Field Theories in Lower Dimensions 


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]
    G. Mandal, R. Sinha and N. Sorokhaibam, Thermalization with chemical potentials and higher spin black holes, JHEP 08 (2015) 013 [arXiv:1501.04580] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  2. [2]
    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].CrossRefADSGoogle Scholar
  3. [3]
    R. Nandkishore and D.A. Huse, Many body localization and thermalization in quantum statistical mechanics, Ann. Rev. Condensed Matter Phys. 6 (2015) 15 [arXiv:1404.0686] [INSPIRE].CrossRefADSGoogle Scholar
  4. [4]
    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].CrossRefADSGoogle Scholar
  5. [5]
    S. Bhattacharyya and S. Minwalla, Weak Field Black Hole Formation in Asymptotically AdS Spacetimes, JHEP 09 (2009) 034 [arXiv:0904.0464] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  6. [6]
    P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 102 (2009) 211601 [arXiv:0812.2053] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  7. [7]
    V. Balasubramanian et al., Thermalization of Strongly Coupled Field Theories, Phys. Rev. Lett. 106 (2011) 191601 [arXiv:1012.4753] [INSPIRE].CrossRefADSGoogle Scholar
  8. [8]
    S.R. Das, T. Nishioka and T. Takayanagi, Probe Branes, Time-dependent Couplings and Thermalization in AdS/CFT, JHEP 07 (2010) 071 [arXiv:1005.3348] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  9. [9]
    V. Balasubramanian et al., Holographic Thermalization, Phys. Rev. D 84 (2011) 026010 [arXiv:1103.2683] [INSPIRE].
  10. [10]
    J. Cardy, Quantum Quenches to a Critical Point in One Dimension: some further results, J. Stat. Mech. 1602 (2016) 023103 [arXiv:1507.07266] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  11. [11]
    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].
  12. [12]
    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].CrossRefADSGoogle Scholar
  13. [13]
    P. Calabrese, F.H.L. Essler and M. Fagotti, Quantum quenches in the transverse field Ising chain: II. Stationary state properties, J. Stat. Mech. Theor. Exp. 7 (2012) 22 [arXiv:1205.2211].
  14. [14]
    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].CrossRefADSGoogle Scholar
  15. [15]
    T. Barthel and U. Schollwöck, Dephasing and the Steady State in Quantum Many-Particle Systems, Phys. Rev. Lett. 100 (2008) 100601 [arXiv:0711.4896].CrossRefADSGoogle Scholar
  16. [16]
    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].
  17. [17]
    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].
  18. [18]
    M. Rigol, V. Dunjko and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452 (2008) 854 [arXiv:0708.1324].CrossRefADSGoogle Scholar
  19. [19]
    A. Iucci and M.A. Cazalilla, Quantum quench dynamics of the Luttinger model, Phys. Rev. A 80 (2009) 063619 [arXiv:1003.5170].
  20. [20]
    D. Fioretto and G. Mussardo, Quantum Quenches in Integrable Field Theories, New J. Phys. 12 (2010) 055015 [arXiv:0911.3345] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  21. [21]
    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].CrossRefADSGoogle Scholar
  22. [22]
    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. Stat. Mech. Theor. Exp. 7 (2012) 07016 [arXiv:1204.3911].
  23. [23]
    G. Mandal and T. Morita, Quantum quench in matrix models: Dynamical phase transitions, Selective equilibration and the Generalized Gibbs Ensemble, JHEP 10 (2013) 197 [arXiv:1302.0859] [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    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].MathSciNetCrossRefGoogle Scholar
  25. [25]
    M. Scully and M. Zubairy, Quantum Optics, Cambridge University Press, (1997).Google Scholar
  26. [26]
    E. Tiesinga and P.R. Johnson, Collapse and revival dynamics of number-squeezed superfluids of ultracold atoms in optical lattices, Phys. Rev. A 83 (2011) 063609.CrossRefADSGoogle Scholar
  27. [27]
    S.R. Das, D.A. Galante and R.C. Myers, Universal scaling in fast quantum quenches in conformal field theories, Phys. Rev. Lett. 112 (2014) 171601 [arXiv:1401.0560] [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    S.R. Das, D.A. Galante and R.C. Myers, Universality in fast quantum quenches, JHEP 02 (2015) 167 [arXiv:1411.7710] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  29. [29]
    N. Birrell and P. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press, (1984).Google Scholar
  30. [30]
    D. Das and S.R. Das, unpublished.Google Scholar
  31. [31]
    I. Bakas and E. Kiritsis, Bosonic Realization of a Universal W -Algebra and Z Parafermions, Nucl. Phys. B 343 (1990) 185 [Erratum ibid. B 350 (1991) 512] [INSPIRE].
  32. [32]
    S.R. Das, D.A. Galante and R.C. Myers, Smooth and fast versus instantaneous quenches in quantum field theory, JHEP 08 (2015) 073 [arXiv:1505.05224] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  33. [33]
    S. Sotiriadis and J. Cardy, Quantum quench in interacting field theory: A self-consistent approximation, Phys. Rev. B 81 (2010) 134305 [arXiv:1002.0167] [INSPIRE].CrossRefADSGoogle Scholar
  34. [34]
    M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, (1965).Google Scholar
  35. [35]
    A. Perelomov, Generalized Coherent States and Their Applications, Modern Methods of Plant Analysis, Springer-Verlag, (1986).Google Scholar
  36. [36]
    A. Duncan, Explicit Dimensional Renormalization of Quantum Field Theory in Curved Space-Time, Phys. Rev. D 17 (1978) 964 [INSPIRE].MathSciNetADSGoogle Scholar
  37. [37]
    G. Festuccia and H. Liu, The arrow of time, black holes and quantum mixing of large-N Yang-Mills theories, JHEP 12 (2007) 027 [hep-th/0611098] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  38. [38]
    E. Koelink, Scattering theory, Lecture notes, section 4, .
  39. [39]
    A. Cohen and T. Kappeler, Scattering and inverse scattering for steplike potentials in the schroedinger equation, Indiana Univ. Math. J. 34 (1985) 127.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    P. Deift and E. Trubowitz, Inverse scattering on the line, Commun. Pure Appl. Math. 32 (1979) 121.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    A. Cabo-Bizet, E. Gava, V.I. Giraldo-Rivera and K.S. Narain, Black Holes in the 3D Higher Spin Theory and Their Quasi Normal Modes, JHEP 11 (2014) 013 [arXiv:1407.5203] [INSPIRE].
  42. [42]
    M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
  43. [43]
    L. Landau and E. Lifshitz, Quantum Mechanics: Non-relativistic Theory, Butterworth-Heinemann, (1977).Google Scholar
  44. [44]
    S.D. Mathur, Is the Polyakov path integral prescription too restrictive?, hep-th/9306090 [INSPIRE].
  45. [45]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York, U.S.A., (1997), ISBN: 038794785X.Google Scholar
  46. [46]
    B. Craps, D-branes and boundary states in closed string theories, hep-th/0004198 [INSPIRE].
  47. [47]
    E. Bergshoeff, C.N. Pope, L.J. Romans, E. Sezgin and X. Shen, The Super W (infinity) Algebra, Phys. Lett. B 245 (1990) 447 [INSPIRE].CrossRefMATHADSGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsTata Institute of Fundamental ResearchMumbaiIndia

Personalised recommendations