Finite-volume spectra of the Lee-Yang model

  • Zoltan Bajnok
  • Omar el Deeb
  • Paul A. Pearce
Open Access
Regular Article - Theoretical Physics


We consider the non-unitary Lee-Yang minimal model \( \mathrm{\mathcal{M}}\left(2,\;5\right) \) in three different finite geometries: (i) on the interval with integrable boundary conditions labelled by the Kac labels (r, s) = (1, 1), (1, 2), (ii) on the circle with periodic boundary conditions and (iii) on the periodic circle including an integrable purely transmitting defect. We apply φ1,3 integrable perturbations on the boundary and on the defect and describe the flow of the spectrum. Adding a Φ1,3 integrable perturbation to move off-criticality in the bulk, we determine the finite size spectrum of the massive scattering theory in the three geometries via Thermodynamic Bethe Ansatz (TBA) equations. We derive these integral equations for all excitations by solving, in the continuum scaling limit, the TBA functional equations satisfied by the transfer matrices of the associated A4 RSOS lattice model of Forrester and Baxter in Regime III. The excitations are classified in terms of (m, n) systems. The excited state TBA equations agree with the previously conjectured equations in the boundary and periodic cases. In the defect case, new TBA equations confirm previously conjectured transmission factors.


Lattice Integrable Models Field Theories in Lower Dimensions Lattice Quantum 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.


  1. [1]
    C.-N. Yang, Some exact results for the many body problems in one dimension with repulsive delta function interaction, Phys. Rev. Lett. 19 (1967) 1312 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    C.-N. Yang and C.P. Yang, Thermodynamics of one-dimensional system of bosons with repulsive delta function interaction, J. Math. Phys. 10 (1969) 1115 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in relativistic models. Scaling three state potts and Lee-Yang models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    A.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358 (1991) 497 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    A.B. Zamolodchikov, TBA equations for integrable perturbed SU(2)k × SU(2) /SU(2)k+ coset models, Nucl. Phys. B 366 (1991) 122 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    P. Dorey and R. Tateo, Excited states by analytic continuation of TBA equations, Nucl. Phys. B 482 (1996) 639 [hep-th/9607167] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    C.-N. Yang and T.D. Lee, Statistical theory of equations of state and phase transitions. 1. Theory of condensation, Phys. Rev. 87 (1952) 404 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    T.D. Lee and C.-N. Yang, Statistical theory of equations of state and phase transitions. 2. Lattice gas and Ising model, Phys. Rev. 87 (1952) 410 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    M.E. Fisher, Yang-Lee edge singularity and ϕ 3 field theory, Phys. Rev. Lett. 40 (1978) 1610 [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    J.L. Cardy, Conformal invariance and the Yang-Lee edge singularity in two-dimensions, Phys. Rev. Lett. 54 (1985) 1354 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    J.L. Cardy and G. Mussardo, S matrix of the Yang-Lee edge singularity in two-dimensions, Phys. Lett. B 225 (1989) 275 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    P.A. Pearce and A. Klümper, Finite size corrections and scaling dimensions of solvable lattice models: an analytic method, Phys. Rev. Lett. 66 (1991) 974 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    A. Klümper and P.A. Pearce, Analytic calculation of scaling dimensions: tricritical hard squares and critical hard hexagons, J. Stat. Phys. 64 (1991) 13.ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    A. Klümper and P.A. Pearce, Conformal weights of RSOS lattice models and their fusion hierarchies, Physica A 183 (1992) 304 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press (1982).Google Scholar
  17. [17]
    D.L. O’Brien, P.A. Pearce and S.O. Warnaar, Calculation of conformal partition funtions: tricritical hard squares with fixed boundaries, Nucl. Phys. B 501 (1997) 773.ADSCrossRefMATHGoogle Scholar
  18. [18]
    P.A. Pearce and B. Nienhuis, Scaling limit of RSOS lattice models and TBA equations, Nucl. Phys. B 519 (1998) 579 [hep-th/9711185] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    P.A. Pearce, L. Chim and C.-r. Ahn, Excited TBA equations. 1. Massive tricritical Ising model, Nucl. Phys. B 601 (2001) 539 [hep-th/0012223] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    P.A. Pearce, L. Chim and C. Ahn, Excited TBA equations. 2. Massless flow from tricritical to critical Ising model, Nucl. Phys. B 660 (2003) 579 [hep-th/0302093] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    C. Destri and H.J. de Vega, New thermodynamic Bethe ansatz equations without strings, Phys. Rev. Lett. 69 (1992) 2313 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    C. Destri and H.J. de Vega, Nonlinear integral equation and excited states scaling functions in the sine-Gordon model, Nucl. Phys. B 504 (1997) 621 [hep-th/9701107] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  23. [23]
    D. Fioravanti, A. Mariottini, E. Quattrini and F. Ravanini, Excited state Destri-De Vega equation for sine-Gordon and restricted sine-Gordon models, Phys. Lett. B 390 (1997) 243 [hep-th/9608091] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable quantum field theories in finite volume: Excited state energies, Nucl. Phys. B 489 (1997) 487 [hep-th/9607099] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  25. [25]
    A. LeClair, G. Mussardo, H. Saleur and S. Skorik, Boundary energy and boundary states in integrable quantum field theories, Nucl. Phys. B 453 (1995) 581 [hep-th/9503227] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    P. Dorey, A. Pocklington, R. Tateo and G. Watts, TBA and TCSA with boundaries and excited states, Nucl. Phys. B 525 (1998) 641 [hep-th/9712197] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    P. Dorey, I. Runkel, R. Tateo and G. Watts, g-function flow in perturbed boundary conformal field theories, Nucl. Phys. B 578 (2000) 85 [hep-th/9909216] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    Z. Bajnok and Z. Simon, Solving topological defects via fusion, Nucl. Phys. B 802 (2008) 307 [arXiv:0712.4292] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    D.A. Huse, Exact exponents for infinitely many new multicritical points, Phys. Rev. B 30 (1984) 3908 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  31. [31]
    H. Riggs, Solvable lattice models with minimal and nonunitary critical behavior in two-dimensions, Nucl. Phys. B 326 (1989) 673 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    G.E. Andrews, R.J. Baxter and P.J. Forrester, Eight vertex SOS model and generalized Rogers-Ramanujan type identities, J. Statist. Phys. 35 (1984) 193 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    P.J. Forrester and R.J. Baxter, Further exact solutions of the eight-vertex SOS model and generalizations of the Rogers-Ramanujan identities, J. Statist. Phys. 38 (1985) 435 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    G. Feverati, P.A. Pearce and F. Ravanini, Exact φ 1,3 boundary flows in the tricritical Ising model, Nucl. Phys. B 675 (2003) 469 [hep-th/0308075] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, series and products, Academic Press (1980).Google Scholar
  36. [36]
    D. Bianchini, E. Ercolessi, P.A. Pearce and F. Ravanini, RSOS quantum chains associated with off-critical minimal models and Zn parafermions, in preparation.Google Scholar
  37. [37]
    R.J. Baxter and P.A. Pearce, Hard hexagons: interfacial tension and correlation length, J. Phys. A 15 (1982) 897.ADSMathSciNetGoogle Scholar
  38. [38]
    R.J. Baxter and P.A. Pearce, Hard squares with diagonal attractions, J. Phys. A 16 (1983) 2239.ADSMathSciNetGoogle Scholar
  39. [39]
    R.J. Baxter, Hard hexagons: exact solution, J. Phys. A 13 (1980) L61.ADSMathSciNetGoogle Scholar
  40. [40]
    H.W.J. Blöte, J.L. Cardy and M.P. Nightingale, Conformal invariance, the central charge and universal finite size amplitudes at criticality, Phys. Rev. Lett. 56 (1986) 742 [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    Y. Stroganov, A new calculation method for partition functions in some lattice models, Phys. Lett. A 74 (1979) 116 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    R.J. Baxter, The inversion relation method for some two-dimensional exactly solved models in lattice statistics, J. Statist. Phys. 28 (1982) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    D.L. O’Brien and P.A. Pearce, Surface free energies, interfacial tensions and correlation lengths of the ABF models, J. Phys. A 30 (1997) 2353 [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  44. [44]
    C. Chui, C. Mercat, W.P. Orrick and P.A. Pearce, Integrable lattice realisations of conformal twisted boundary conditions, Phys. Lett. B 517 (2001) 429ADSCrossRefMATHGoogle Scholar
  45. [45]
    R.E. Behrend, P.A. Pearce and D.L. O’Brien, Interaction-round-a-face models with fixed boundary conditions: the ABF fusion hierarchy, J. Statist. Phys. 84 (1996) 1 [hep-th/9507118] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    R.E. Behrend and P.A. Pearce, Integrable and conformal boundary conditions for sl(2) A-D-E lattice models and unitary minimal conformal field theories, J. Statist. Phys. 102 (2001) 577 [hep-th/0006094] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    B.L. Feigin, T. Nakanishi and H. Ooguri, The annihilating ideals of minimal models, Int. J. Mod. Phys. A 7S1A (1992) 217 [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    O. Foda and T.A. Welsh, On the combinatorics of Forrester-Baxter models, in Physical combinatorics, Progress in Mathematics volume 191, Birkhauser, Boston U.S.A. (2000), math/0002100.
  49. [49]
    G. Feverati, P.A. Pearce and N.S. Witte, Physical combinatorics and quasiparticles, J. Stat. Mech. (2009) P10013.Google Scholar
  50. [50]
    R.J. Baxter, Corner transfer matrices of the eight-vertex model. 1. Low-temperature expansions and conjectured properties, J. Statist. Phys. 15 (1976) 485 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    R.J. Baxter, Corner transfer matrices of the eight-vertex model. 2. The Ising model case, J. Statist. Phys. 17 (1977) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    J.F. Fortin, P. Jacob and P. Mathieu, SM (2, 4κ) fermionic characters and restricted jagged partitions, J. Phys. A 38 (2005) 1699 [hep-th/0406194] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  53. [53]
    W. Nahm, A. Recknagel and M. Terhoeven, Dilogarithm identities in conformal field theory, Mod. Phys. Lett. A 8 (1993) 1835 [hep-th/9211034] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    E. Melzer, Fermionic character sums and the corner transfer matrix, Int. J. Mod. Phys. A 9 (1994) 1115 [hep-th/9305114] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    A. Berkovich, Fermionic counting of RSOS states and Virasoro character formulas for the unitary minimal series M (νν + 1), Nucl. Phys. B 431 (1994) 315 [hep-th/9403073] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    G. Feverati and P.A. Pearce, Critical RSOS and minimal models: Fermionic paths, Virasoro algebra and fields, Nucl. Phys. B 663 (2003) 409 [hep-th/0211185] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  57. [57]
    G. Feverati, P.A. Pearce and N.S. Witte, Physical combinatorics and quasiparticles, J. Stat. Mech. (2009) P10013.Google Scholar
  58. [58]
    I. Affleck and A.W.W. Ludwig, Universal nonintegerground state degeneracyin critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  59. [59]
    Z. Bajnok, L. Holló and G. Watts, Defect scaling Lee-Yang model from the perturbed DCFT point of view, Nucl. Phys. B 886 (2014) 93 [arXiv:1307.4536] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  60. [60]
    C.H. Otto Chui, C. Mercat and P.A. Pearce, Integrable boundaries and universal TBA functional equations, Prog. Math. Phys. 23 (2002) 391 [hep-th/0108037] [INSPIRE].MathSciNetMATHGoogle Scholar
  61. [61]
    Z. Bajnok and O. el Deeb, Form factors in the presence of integrable defects, Nucl. Phys. B 832 (2010) 500 [arXiv:0909.3200] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  62. [62]
    Z. Bajnok, F. Buccheri, L. Hollo, J. Konczer and G. Takács, Finite volume form factors in the presence of integrable defects, Nucl. Phys. B 882 (2014) 501 [arXiv:1312.5576] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  63. [63]
    A. De Luca and F. Franchini, Approaching the restricted solid-on-solid critical points through entanglement: one model for many universalities, Phys. Rev. B 87 (2013) 045118 [arXiv:1205.6426] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    D. Bianchini, Entanglement entropy in restricted integrable spin chains, M.Sc. Thesis, University of Bologna, Bologna, Italy (2013).Google Scholar
  65. [65]
    D. Bianchini and F. Ravanini, in preparation.Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Zoltan Bajnok
    • 1
  • Omar el Deeb
    • 1
    • 2
  • Paul A. Pearce
    • 3
  1. 1.MTA Lendület Holographic QFT GroupWigner Research Centre for PhysicsBudapest 114Hungary
  2. 2.Physics Department, Faculty of ScienceBeirut Arab University (BAU)BeirutLebanon
  3. 3.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia

Personalised recommendations