Lifshitz scaling, microstate counting from number theory and black hole entropy

  • Dmitry Melnikov
  • Fábio NovaesEmail author
  • Alfredo Pérez
  • Ricardo Troncoso
Open Access
Regular Article - Theoretical Physics


Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation Ekz and dynamical exponent z > 1. The asymptotic growth of the number of states of these theories can be described by an extension of Cardy formula that depends on z. We show that this result can be recovered by counting the partitions of an integer into z-th powers, as proposed by Hardy and Ramanujan a century ago. This gives a novel duality relationship between the characteristic energy of the dispersion relation with the cylinder radius and the ground state energy. For free bosons with Lifshitz scaling, this relationship is shown to be identically fulfilled by virtue of the reflection property of the Riemann ζ-function. The quantum Benjamin-Ono2 (BO2) integrable system, relevant in the AGT correspondence, is also analyzed. As a holographic realization, we provide a special set of boundary conditions for which the reduced phase space of Einstein gravity with a couple of U (1) fields on AdS3 is described by the BO2 equations. This suggests that the phase space can be quantized in terms of quantum BO2 states. Indeed, in the semiclassical limit, the ground state energy of BO2 coincides with the energy of global AdS3, and the Bekenstein-Hawking entropy for BTZ black holes is recovered from the anisotropic extension of Cardy formula.


Integrable Field Theories Space-Time Symmetries Classical Theories of Gravity Gauge-gravity correspondence 


Open Access

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  1. [1]
    M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
  2. [2]
    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S.A. Hartnoll, Horizons, holography and condensed matter, in Black holes in higher dimensions, G.T. Horowitz ed., (2012), pg. 387 [arXiv:1106.4324] [INSPIRE].
  4. [4]
    M. Taylor, Lifshitz holography, Class. Quant. Grav. 33 (2016) 033001 [arXiv:1512.03554] [INSPIRE].
  5. [5]
    E. Bettelheim, A.G. Abanov and P. Wiegmann, Quantum shock waves: the case for non-linear effects in dynamics of electronic liquids, Phys. Rev. Lett. 97 (2006) 246401 [cond-mat/0606778] [INSPIRE].
  6. [6]
    P. Wiegmann, Non-linear hydrodynamics and fractionally quantized solitons at fractional quantum Hall edge, Phys. Rev. Lett. 108 (2012) 206810 [arXiv:1112.0810] [INSPIRE].CrossRefGoogle Scholar
  7. [7]
    S. Sotiriadis, Equilibration in one-dimensional quantum hydrodynamic systems, J. Phys. A 50 (2017) 424004 [arXiv:1612.00373] [INSPIRE].
  8. [8]
    H.A. Gonzalez, D. Tempo and R. Troncoso, Field theories with anisotropic scaling in 2D, solitons and the microscopic entropy of asymptotically Lifshitz black holes, JHEP 11 (2011) 066 [arXiv:1107.3647] [INSPIRE].
  9. [9]
    A. Pérez, D. Tempo and R. Troncoso, Boundary conditions for general relativity on AdS 3 and the KdV hierarchy, JHEP 06 (2016) 103 [arXiv:1605.04490] [INSPIRE].
  10. [10]
    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, NY, U.S.A. (1997) [INSPIRE].
  11. [11]
    A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
  12. [12]
    F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
  13. [13]
    A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \) -deformed 2D quantum field theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
  14. [14]
    L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470] [INSPIRE].
  15. [15]
    P. Kraus, J. Liu and D. Marolf, Cutoff AdS 3 versus the \( T\overline{T} \) deformation, JHEP 07 (2018) 027 [arXiv:1801.02714] [INSPIRE].
  16. [16]
    S. Datta and Y. Jiang, \( T\overline{T} \) deformed partition functions, JHEP 08 (2018) 106 [arXiv:1806.07426] [INSPIRE].
  17. [17]
    O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov, Modular invariance and uniqueness of \( T\overline{T} \) deformed CFT, JHEP 01 (2019) 086 [arXiv:1808.02492] [INSPIRE].
  18. [18]
    J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
  19. [19]
    D. Grumiller, A. Perez, D. Tempo and R. Troncoso, Log corrections to entropy of three dimensional black holes with soft hair, JHEP 08 (2017) 107 [arXiv:1705.10605] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    E. Shaghoulian, A Cardy formula for holographic hyperscaling-violating theories, JHEP 11 (2015) 081 [arXiv:1504.02094] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  21. [21]
    T.M. Apostol, Modular functions and Dirichlet series in number theory, Springer, New York, NY, U.S.A. (1990).Google Scholar
  22. [22]
    E. Ayon-Beato, A. Garbarz, G. Giribet and M. Hassaine, Lifshitz black hole in three dimensions, Phys. Rev. D 80 (2009) 104029 [arXiv:0909.1347] [INSPIRE].
  23. [23]
    E. Ayón-Beato, M. Bravo-Gaete, F. Correa, M. Hassaïne, M.M. Juárez-Aubry and J. Oliva, First law and anisotropic Cardy formula for three-dimensional Lifshitz black holes, Phys. Rev. D 91 (2015) 064006 [Addendum ibid. D 96 (2017) 049903] [arXiv:1501.01244] [INSPIRE].
  24. [24]
    M. Bravo-Gaete, S. Gomez and M. Hassaine, Cardy formula for charged black holes with anisotropic scaling, Phys. Rev. D 92 (2015) 124002 [arXiv:1510.04084] [INSPIRE].
  25. [25]
    H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft hairy horizons in three spacetime dimensions, Phys. Rev. D 95 (2017) 106005 [arXiv:1611.09783] [INSPIRE].
  26. [26]
    H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
  27. [27]
    M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Chemical potentials in three-dimensional higher spin anti-de Sitter gravity, JHEP 12 (2013) 048 [arXiv:1309.4362] [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    C. Bunster, M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Generalized black holes in three-dimensional spacetime, JHEP 05 (2014) 031 [arXiv:1404.3305] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    G.H. Hardy and S. Ramanujan, Asymptotic formulæ in combinatory analysis, Proc. Lond. Math. Soc. s2-17 (1918) 75.Google Scholar
  30. [30]
    E.M. Wright, Asymptotic partition formulae. III. Partitions into k-th powers, Acta Math. 63 (1934) 143.Google Scholar
  31. [31]
    R.C. Vaughan, Squares: additive questions and partitions, Int. J. Number Theor. 11 (2015) 1367.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    A. Gafni, Power partitions, J. Number Theor. 163 (2016) 19 [arXiv:1506.06124].
  33. [33]
    S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].CrossRefGoogle Scholar
  34. [34]
    S.W. Hawking, M.J. Perry and A. Strominger, Superrotation charge and supertranslation hair on black holes, JHEP 05 (2017) 161 [arXiv:1611.09175] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    F. Luca and D. Ralaivaosaona, An explicit bound for the number of partitions into roots, J. Number Theor. 169 (2016) 250.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Y.-L. Li and Y.-G. Chen, On the r-th root partition function, II, J. Number Theor. 188 (2018) 392.Google Scholar
  37. [37]
    E. Lifshitz, On the theory of second-order phase transitions I, Zh. Eksp. Teor. Fiz. 11 (1941) 255.Google Scholar
  38. [38]
    E. Lifshitz, On the theory of second-order phase transitions II, Zh. Eksp. Teor. Fiz. 11 (1941) 269.Google Scholar
  39. [39]
    D. Melnikov, F. Novaes, A. Pérez and R. Troncoso, work in progress.Google Scholar
  40. [40]
    Y. Matsuno, Bilinear transformation method, volume 174, Academic Press, New York, NY, U.S.A. (1984).Google Scholar
  41. [41]
    D.R. Lebedev and A.O. Radul, Generalized internal long waves equations: construction, Hamiltonian structure and conservation laws, Commun. Math. Phys. 91 (1983) 543 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    A. Degasperis, D. Lebedev, M. Olshanetsky, S. Pakuliak, A. Perelomov and P. Santini, Nonlocal integrable partners to generalized MKdV and two-dimensional Toda lattice equations in the formalism of a dressing method with quantized spectral parameter, Commun. Math. Phys. 141 (1991) 133 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    A. Degasperis et al., Generalized intermediate long-wave hierarchy in zero-curvature representation with noncommutative spectral parameter, J. Math. Phys. 33 (1992) 3783.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    A.G. Abanov and P.B. Wiegmann, Quantum hydrodynamics, quantum Benjamin-Ono equation and Calogero model, Phys. Rev. Lett. 95 (2005) 076402 [cond-mat/0504041] [INSPIRE].
  45. [45]
    A.G. Abanov, E. Bettelheim and P. Wiegmann, Integrable hydrodynamics of Calogero-Sutherland model: bidirectional Benjamin-Ono equation, J. Phys. A 42 (2009) 135201 [arXiv:0810.5327] [INSPIRE].
  46. [46]
    A. Imambekov, T.L. Schmidt and L.I. Glazman, One-dimensional quantum liquids: beyond the Luttinger liquid paradigm, Rev. Mod. Phys. 84 (2012) 1253 [arXiv:1110.1374].CrossRefGoogle Scholar
  47. [47]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    A.V. Litvinov, On spectrum of ILW hierarchy in conformal field theory, JHEP 11 (2013) 155 [arXiv:1307.8094] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    I.M. Gel’fand and I.Y. Dorfman, Hamiltonian operators and algebraic structures related to them, Funct. Anal. Appl. 13 (1980) 248.CrossRefzbMATHGoogle Scholar
  50. [50]
    A. Das, Integrable models, World Sci. Lect. Notes Phys. 30 (1989) 1 [INSPIRE].
  51. [51]
    P.J. Olver, Applications of Lie groups to differential equations, volume 107, Springer, U.S.A. (1986).Google Scholar
  52. [52]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
  56. [56]
    B. Feigin, M. Jimbo and E. Mukhin, Integrals of motion from quantum toroidal algebras, J. Phys. A 50 (2017) 464001 [arXiv:1705.07984] [INSPIRE].
  57. [57]
    A. Achucarro and P.K. Townsend, Extended supergravities in d = (2 + 1) as Chern-Simons theories, Phys. Lett. B 229 (1989) 383 [INSPIRE].
  58. [58]
    O. Fuentealba et al., Integrable systems with BMS 3 Poisson structure and the dynamics of locally flat spacetimes, JHEP 01 (2018) 148 [arXiv:1711.02646] [INSPIRE].
  59. [59]
    E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
  60. [60]
    O. Coussaert, M. Henneaux and P. van Driel, The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [INSPIRE].
  61. [61]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Annals Phys. 88 (1974) 286 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
    M. Henneaux, L. Maoz and A. Schwimmer, Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity, Annals Phys. 282 (2000) 31 [hep-th/9910013] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  64. [64]
    M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].

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© The Author(s) 2019

Authors and Affiliations

  • Dmitry Melnikov
    • 1
    • 2
  • Fábio Novaes
    • 1
    Email author
  • Alfredo Pérez
    • 3
  • Ricardo Troncoso
    • 3
  1. 1.International Institute of PhysicsFederal University of Rio Grande do NorteNatalBrazil
  2. 2.Institute for Theoretical and Experimental PhysicsMoscowRussia
  3. 3.Centro de Estudios Científicos (CECs)ValdiviaChile

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