Bootstrap approach to geometrical four-point functions in the two-dimensional critical Q-state Potts model: a study of the s-channel spectra

  • Jesper Lykke JacobsenEmail author
  • Hubert Saleur
Open Access
Regular Article - Theoretical Physics


We revisit in this paper the problem of connectivity correlations in the Fortuin-Kasteleyn cluster representation of the two-dimensional Q-state Potts model conformal field theory. In a recent work [1], results for the four-point functions were obtained, based on the bootstrap approach, combined with simple conjectures for the spectra in the different fusion channels. In this paper, we test these conjectures using lattice algebraic considerations combined with extensive numerical studies of correlations on infinite cylinders. We find that the spectra in the scaling limit are much richer than those proposed in [1]: they involve in particular fields with conformal weight hr,s where r is dense on the real axis.


Conformal Field Theory Lattice Integrable Models 


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  1. [1]
    M. Picco, S. Ribault and R. Santachiara, A conformal bootstrap approach to critical percolation in two dimensions, SciPost Phys. 1 (2016) 009 [arXiv:1607.07224] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    C.M. Fortuin and P.W. Kasteleyn, On the Random cluster model. 1. Introduction and relation to other models, Physica 57 (1972) 536 [INSPIRE].
  3. [3]
    R.B. Potts, Some generalized order-disorder transformations, Math. Proc. Cambr. Phil. Soc. 48 (1952) 106.Google Scholar
  4. [4]
    G. Delfino and J. Viti, On three-point connectivity in two-dimensional percolation, J. Phys. A 44 (2011) 032001 [arXiv:1009.1314] [INSPIRE].ADSzbMATHGoogle Scholar
  5. [5]
    M. Picco, R. Santachiara, J. Viti and G. Delfino, Connectivities of Potts Fortuin-Kasteleyn clusters and time-like Liouville correlator, Nucl. Phys. B 875 (2013) 719 [arXiv:1304.6511] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  6. [6]
    Y. Ikhlef, J.L. Jacobsen and H. Saleur, Three-point functions in c ≤ 1 Liouville theory and conformal loop ensembles, Phys. Rev. Lett. 116 (2016) 130601 [arXiv:1509.03538] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    P. Di Francesco, D. Sénéchal and P. Mathieu, Conformal field theory, Springer, Germany (1997).CrossRefzbMATHGoogle Scholar
  8. [8]
    G. Gori and J. Viti, Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance, JHEP 12 (2018) 131 [arXiv:1806.02330] [INSPIRE].CrossRefGoogle Scholar
  9. [9]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    L. Luo, B.-Q. Xia and Y.-F. Cao, Peakon solutions to supersymmetric Camassa-Holm equation and Degasperis-Procesi equation, Commun. Theor. Phys. 59 (2013) 73 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  11. [11]
    P. Di Francesco, H. Saleur and J.B. Zuber, Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models, J. Stat. Phys. 49 (1987) 57.ADSCrossRefzbMATHGoogle Scholar
  12. [12]
    R.J. Baxter, Potts model at the critical temperature, J. Phys. C 6 (1973) L445.ADSGoogle Scholar
  13. [13]
    R.J. Baxter, S.B. Kelland and F.Y. Wu, Equivalence of the Potts model or Whitney polynomial with an ice-type model, J. Phys. A 9 (1976) 397.ADSzbMATHGoogle Scholar
  14. [14]
    H.N.V. Temperley and E.T. Lieb, Relation between the ‘percolation’ and ‘colouring’ problem and other graph-theoretical problems associated with planar lattices: some exact results for the ‘percolation’ problem, Proc. Roy. Soc. London A 322 (1971) 251.ADSCrossRefzbMATHGoogle Scholar
  15. [15]
    R. Vasseur and J.L. Jacobsen, Critical properties of joint spin and Fortuin-Kasteleyn observables in the two-dimensional Potts model, J. Phys. A 45 (2012) 165001 [arXiv:1111.4033].ADSzbMATHGoogle Scholar
  16. [16]
    J. Dubail, J.L. Jacobsen and H. Saleur, Critical exponents of domain walls in the two-dimensional Potts model, J. Phys. A 43 (2010) 482002 [arXiv:1008.1216].zbMATHGoogle Scholar
  17. [17]
    J. Dubail, J.L. Jacobsen and H. Saleur, Bulk and boundary critical behaviour of thin and thick domain walls in the two-dimensional Potts model, J. Stat. Mech. 12 (2010) 12026 [arXiv:1010.1700].CrossRefGoogle Scholar
  18. [18]
    G. Delfino and J. Viti, Potts q-color field theory and scaling random cluster model, Nucl. Phys. B 852 (2011) 149 [arXiv:1104.4323] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Statist. Phys. 34 (1984) 731 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    B. Duplantier and H. Saleur, Exact critical properties of two-dimensional dense self-avoiding walks, Nucl. Phys. B 290 (1987) 291.ADSCrossRefGoogle Scholar
  21. [21]
    R. Vasseur, J.L. Jacobsen and H. Saleur, Logarithmic observables in critical percolation, J. Stat. Mech. 1207 (2012) L07001 [arXiv:1206.2312] [INSPIRE].Google Scholar
  22. [22]
    R. Vasseur and J.L. Jacobsen, Operator content of the critical Potts model indimensions and logarithmic correlations, Nucl. Phys. B 880 (2014) 435 [arXiv:1311.6143] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    R. Couvreur, J. Lykke Jacobsen and R. Vasseur, Non-scalar operators for the Potts model in arbitrary dimension, J. Phys. A 50 (2017) 474001 [arXiv:1704.02186] [INSPIRE].ADSzbMATHGoogle Scholar
  24. [24]
    T. Halverson and A. Ram, Partition algebras, Eur. J. Combin. 26 (2005) 869.CrossRefzbMATHGoogle Scholar
  25. [25]
    R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, London U.K. (1982).zbMATHGoogle Scholar
  26. [26]
    P.P. Martin, Potts models and related problems in statistical mechanics, World Scientific, Singapore (1991).CrossRefzbMATHGoogle Scholar
  27. [27]
    P. Martin and H. Saleur, The Blob algebra and the periodic Temperley-Lieb algebra, Lett. Math. Phys. 30 (1994) 189 [hep-th/9302094] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  28. [28]
    J.J. Graham and G.I. Lehrer, The representation theory of affine Temperley-Lieb algebras, L’Ens. Math. 44 (1998) 173.zbMATHGoogle Scholar
  29. [29]
    A.M. Gainutdinov, N. Read, H. Saleur and R. Vasseur, The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk LCFT at c = 0, JHEP 05 (2015) 114 [arXiv:1409.0167].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    A.M. Gainutdinov, J.L. Jacobsen and H. Saleur, A fusion for the periodic Temperley-Lieb algebra and its continuum limit, JHEP 11 (2018) 117 [arXiv:1712.07076] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    V.F.R. Jones, Quotient of the affine Hecke algebra in the Brauer algebra, L’Ens. Math. 40 (1994) 313.Google Scholar
  32. [32]
    N. Read and H. Saleur, Enlarged symmetry algebras of spin chains, loop models and S-matrices, Nucl. Phys. B 777 (2007) 263 [cond-mat/0701259] [INSPIRE].
  33. [33]
    J.F. Richard and J.L. Jacobsen, Eigenvalue amplitudes of the Potts model on a torus, Nucl. Phys. B 769 (2007) 256 [math-ph/0608055].
  34. [34]
    J.L. Jacobsen and J. Salas, Phase diagram of the chromatic polynomial on a torus, Nucl. Phys. B 783 (2007) 238 [cond-mat/0703228] [INSPIRE].
  35. [35]
    J.L. Jacobsen and P. Zinn-Justin, A transfer matrix for the backbone exponent of two-dimensional percolation, J. Phys. A 35 (2002) 2131 [cond-mat/0111374].
  36. [36]
    J.L. Jacobsen and P. Zinn-Justin, Monochromatic path crossing exponents and graph connectivity in two-dimensional percolation, Phys. Rev. E 66 (2002) 055102(R) [cond-mat/0207063].
  37. [37]
    Y. Deng, H.W.J. Blöte and B. Nienhuis, Backbone exponents of the two-dimensional q-state Potts model: a Monte Carlo investigation, Phys. Rev. E 69 (2004) 026114.ADSGoogle Scholar
  38. [38]
    Z. Zhou, J. Yang, Y. Deng and R.M. Ziff, Shortest-path fractal dimension for percolation in two and three dimensions, Phys. Rev. E 86 (2012) 061101 [arXiv:1112.3428].ADSGoogle Scholar
  39. [39]
    A.M. Gainutdinov, N. Read and H. Saleur, Continuum limit and symmetries of the periodic gl(1|1) spin chain, Nucl. Phys. B 871 (2013) 245 [arXiv:1112.3403] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  40. [40]
    A.M. Gainutdinov, N. Read and H. Saleur, Bimodule structure in the periodic gl(1|1) spin chain, Nucl. Phys. B 871 (2013) 289 [arXiv:1112.3407] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  41. [41]
    A.M. Gainutdinov, N. Read and H. Saleur, Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the \( \mathfrak{g}\mathfrak{l} \)(1|1) periodic spin chain, Howe duality and the interchiral algebra, Commun. Math. Phys. 341 (2016) 35 [arXiv:1207.6334] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  42. [42]
    A.M. Gainutdinov, J.L. Jacobsen, H. Saleur and R. Vasseur, A physical approach to the classification of indecomposable Virasoro representations from the blob algebra, Nucl. Phys. B 873 (2013) 614 [arXiv:1212.0093] [INSPIRE].
  43. [43]
    A.M. Gainutdinov et al., Logarithmic conformal field theory: a lattice approach, J. Phys. A 46 (2013) 494012 [arXiv:1303.2082] [INSPIRE].
  44. [44]
    J. Belletête et al., On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories, J. Phys. A 50 (2017) 484002 [arXiv:1705.07769] [INSPIRE].zbMATHGoogle Scholar
  45. [45]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  46. [46]
    V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B 330 (1990) 523 [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    F.C. Alcaraz, U. Grimm and V. Rittenberg, The XXZ Heisenberg chain, conformal invariance and the operator content of c < 1 systems, Nucl. Phys. B 316 (1989) 735 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  48. [48]
    M. den Nijs, Extended scaling relations for the magnetic critical exponents of the Potts model, Phys. Rev. B 27 (1983) 1674 [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    J.L. Jacobsen and H. Saleur, Combinatorial aspects of boundary loop models, J. Stat. Mech. 1 (2008) 01021 [arXiv:0709.0812].
  50. [50]
    J. de Gier, A. Ponsaing and J.L. Jacobsen, Finite-size corrections for universal boundary entropy in bond percolation, SciPost Phys. 1 (2016) 012 [arXiv:1610.04006] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    B. Estienne and Y. Ikhlef, Correlation functions in loop models, arXiv:1505.00585 [INSPIRE].
  52. [52]
    S. Migliaccio and S. Ribault, The analytic bootstrap equations of non-diagonal two-dimensional CFT, JHEP 05 (2018) 169 [arXiv:1711.08916] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  53. [53]
    I. Runkel and G.M.T. Watts, A Nonrational CFT with c = 1 as a limit of minimal models, JHEP 09 (2001) 006 [hep-th/0107118] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    S. Ribault and R. Santachiara, Liouville theory with central charge less than one, JHEP 08 (2015) 109 [arXiv:1503.02067].
  55. [55]
    W.M. Koo and H. Saleur, Representations of the Virasoro algebra from lattice models, Nucl. Phys. B 426 (1994) 459 [hep-th/9312156] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    J.L. Jacobsen, S. Ribault, H. Saleur and L.G. Samuelsson, in preparation.Google Scholar
  57. [57]
    H.W.J. Blöte and M.P. Nightingale, Critical behaviour of the two-dimensional Potts model with a continuous number of states: a finite size scaling analysis, Physica A 112 (1982) 405.ADSCrossRefGoogle Scholar
  58. [58]
    J. Salas and A.D. Sokal, Transfer matrices and partition function zeros for antiferromagnetic Potts models. 1. General theory and square lattice chromatic polynomial, J. Statist. Phys. 104 (2001) 609 [cond-mat/0004330] [INSPIRE].
  59. [59]
    S.C. Chang, J. Salas and R. Shrock, Exact Potts model partition functions for strips of the square lattice, J. Stat. Phys. 107 (2002) 1207 [cond-mat/0108144].
  60. [60]
    F.M. Gomes and D.C. Sorensen, Arpack++. An object-oriented version of ARPACK eigenvalue package,
  61. [61]
    B. Haible and R.B. Kreckel, CLN — Class Library for Numbers,
  62. [62]
    J.L. Jacobsen, Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley-Lieb algebras, J. Phys. A 48 (2015) 454003 [arXiv:1507.03027].zbMATHGoogle Scholar
  63. [63]
    J.L. Jacobsen, C.R. Scullard and A.J. Guttmann, On the growth constant for square-lattice self-avoiding walks, J. Phys. A 49 (2016) 494004 [arXiv:1607.02984].zbMATHGoogle Scholar
  64. [64]
    J.L. Jacobsen, J. Salas and C.R. Scullard, Phase diagram of the triangular-lattice Potts antiferromagnet, J. Phys. A 50 (2017) 345002 [arXiv:1702.02006].zbMATHGoogle Scholar
  65. [65]
    B. Mc Coy and T.T. Wu, The two-dimensional Ising model, Harvard University Press, U.S.A. (1973).CrossRefGoogle Scholar
  66. [66]
    A. Rocha-Caridi, Vacuum vector representations of the Virasoro algebra, in Vertex operators in mathematics and physics, J. Lepowsky et al. eds., Springer, Germany (1984).Google Scholar
  67. [67]
    P. Reinicke and T. Vescan, Finite-size corrections to matrix elements in a conformal theory. Applications to the magnetisation of the three-state Potts model, J. Phys. A 20 (1987) L653.ADSGoogle Scholar
  68. [68]
    S.-K. Yang, Modular invariant partition function of the Ashkin-Teller model on the critical line and N = 2 superconformal invariance, Nucl. Phys. B 285 (1987) 183 [INSPIRE].ADSCrossRefGoogle Scholar
  69. [69]
    Al.B. Zamolodchikov, Two-dimensional conformal symmetry and critical four-spin correlation functions in the Ashkin-Teller model, Sov. Phys. JETP 63 (1986) 1061.Google Scholar
  70. [70]
    J.L. Jacobsen, J. Salas and A.D. Sokal, Spanning forests and the q state Potts model in the limit q → 0, J. Statist. Phys. 119 (2005) 1153 [cond-mat/0401026] [INSPIRE].
  71. [71]
    S. Caracciolo et al., Fermionic field theory for trees and forests, Phys. Rev. Lett. 93 (2004) 080601 [cond-mat/0403271] [INSPIRE].
  72. [72]
    J.L. Jacobsen and H. Saleur, The Arboreal gas and the supersphere σ-model, Nucl. Phys. B 716 (2005) 439 [cond-mat/0502052] [INSPIRE].
  73. [73]
    H. Saleur, Polymers and percolation in two-dimensions and twisted N = 2 supersymmetry, Nucl. Phys. B 382 (1992) 486 [hep-th/9111007] [INSPIRE].ADSCrossRefGoogle Scholar
  74. [74]
    E.V. Ivashkevich, Correlation functions of dense polymers and c = −2 conformal field theory, J. Phys. A 32 (1999) 1691 [cond-mat/9801183].
  75. [75]
    G. Kirchhoff, Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird, Ann. Phys. Chem. 72 (1847) 497.ADSCrossRefGoogle Scholar
  76. [76]
    V.B. Priezzhev, The dimer problem and the Kirchhoff theorem, Sov. Phys. Usp. 28 (1985) 1125.ADSCrossRefGoogle Scholar
  77. [77]
    H.G. Kausch, Curiosities at c = −2, hep-th/9510149 [INSPIRE].
  78. [78]
    H.G. Kausch, Symplectic fermions, Nucl. Phys. B 583 (2000) 513 [hep-th/0003029] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  79. [79]
    J.M. Luck, Finite size scaling and the two-dimensional XY model, J. Phys. A 15 (1982) L169 [INSPIRE].ADSGoogle Scholar

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Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique, Département de Physique de l’ENS, École Normale Supérieure, Sorbonne Université, CNRSPSL Research UniversityParisFrance
  2. 2.Sorbonne Université, École Normale Supérieure, CNRS, Laboratoire de Physique Théorique (LPT ENS)ParisFrance
  3. 3.Institut de Physique Théorique, Université Paris Saclay, CEA, CNRSGif-sur-YvetteFrance
  4. 4.Department of PhysicsUniversity of Southern CaliforniaLos AngelesU.S.A.

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