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Logarithmic minimal models

Profile image of Paul PearcePaul Pearce

2006, Journal of Statistical Mechanics: Theory and Experiment

https://doi.org/10.1088/1742-5468/2006/11/P11017
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Abstract

Working in the dense loop representation, we use the planar Temperley-Lieb algebra to build integrable lattice models called logarithmic minimal models LM(p, p ′ ). Specifically, we construct Yang-Baxter integrable Temperley-Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley-Lieb algebra are inherently non-local and not (timereversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1 -6(p-p ′ ) 2 pp ′ where p, p ′ = 1, 2, . . . are coprime. The first few members of the principal series LM(m, m + 1) are critical dense polymers (m = 1, c = -2), critical percolation (m = 2, c = 0) and logarithmic Ising model (m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights ∆ r,s = ((m+1)r-ms) 2 -1 4m(m+1) , r, s = 1, 2, . . .. The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.

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References (78)

  1. V. Gurarie, hep-th/9303160, Nucl. Phys. B410, 535 (1993).
  2. F. Rohsiepe, On reducible but indecomposable representations of the Virasoro algebra, hep-th/9611160 (1996).
  3. M. Flohr, hep-th/9605151, Int. J. Mod. Phys. A12, 1943-1958 (1997); Null vectors in logarithmic conformal field theory, hep-th/0009137 (2000);
  4. M. Flohr, hep-th/0111228, Int. J. Mod. Phys. A18, 4497-4592 (2003).
  5. M.R. Gaberdiel and H.G. Kausch, hep-th/9604026, Nucl. Phys. B477, 293-318 (1996); hep-th/9807091, Nucl. Phys. B538, 631 (1999).
  6. M.R. Gaberdiel, hep-th/0111260, Int. J. Mod. Phys. A18, 4593-4638 (2003).
  7. S. Moghimi-Araghi and S. Rouhani, hep-th/0002142, Lett. Math. Phys. 53, 49-57 (2000);
  8. I.I. Kogan and J.F. Wheater, hep-th/0003184, Phys. Lett. B486, 353 (2000);
  9. S. Kawai and J.F. Wheater, hep-th/0103197, Phys. Lett. B508, 203 (2001);
  10. A. Bredthauer and M. Flohr, hep-th/0204154, Nucl. Phys. B639, 450-470 (2002);
  11. S. Kawai, hep-th/0204169, Int. J. Mod. Phys. A18, 4655-4684 (2003).
  12. V. Pasquier and H. Saleur, Nucl. Phys. B330, 523 (1990).
  13. M. Marcu, J. Math. Phys. 21, 1277-1283; 1284-1292 (1980).
  14. L. Rozansky and H. Saleur, Nucl. Phys. B376, 461 (1992); hep-th/9203069, Nucl. Phys. B389, 365 (1993).
  15. Z. Maassarani and D. Serban, hep-th/9605062, Nucl. Phys. B489, 603-625 (1997).
  16. H. Saleur, hep-th/9111007, Nucl. Phys. B382, 486 (1992); hep-th/9111008, Nucl. Phys. B382, 532 (1992).
  17. H.G. Kausch, Curiosities at c = -2, hep-th/9510149 (1995).
  18. J. Kondev and J.B. Marston, Supersymmetry and Localization in the Quantum Hall Effect, cond-mat/9612223 (1996).
  19. E.V. Ivashkevich, cond-mat/9801183, J. Phys A32, 1691 (1999).
  20. N. Read and H. Saleur, hep-th/0106124, Nucl. Phys. B613, 409 (2001).
  21. H. Saleur, J. Phys. A20, 455-470 (1987).
  22. B. Duplantier, J. Phys. A19, L1009-1014 (1986);
  23. H. Saleur and B. Duplantier, Phys. Rev. Lett. 58, 2325 (1987).
  24. J. Cardy, Logarithmic correlations in quenched random magnets and polymers, cond-mat/9911024 (1999).
  25. V. Gurarie and A.W.W. Ludwig, cond-mat/9911392, J.Phys. A35, L377-384 (2002).
  26. H.G. Kausch, Phys. Lett. B259, 448-455 (1991).
  27. M. Flohr, hep-th/9509166, Int. J. Mod. Phys. A11, 4147-4172 (1996).
  28. J. Fjelstad, J. Fuchs, S. Hwang, A.M. Semikhatov and I.Yu. Tipunin, Nucl. Phys. B633, 379-413 (2002).
  29. P. Calabrese, M. Caselle, A. Celi, A. Pelissetto and E. Vicari, hep-th/0005254, J. Phys. A33, 8155-8170 (2000);
  30. L.-P. Arguin and Y. Saint-Aubin, hep-th/0109138, Phys. Lett. B541, 384-389 (2002).
  31. J. Rasmussen, hep-th/0405257, Nucl. Phys. B701, 516-528 (2004); Jordan cells in loga- rithmic limits of conformal field theory, hep-th/0406110 (2004).
  32. H. Eberle and M. Flohr, Virasoro representations and fusion for general augmented minimal models, hep-th/0604097 (2006).
  33. H.N.V. Temperley and E.H. Lieb, Proc. Roy. Soc. (London) A322, 251 (1971).
  34. V.F.R. Jones, Planar algebras I, math.QA/9909027 (1999).
  35. G.E. Andrews, R.J. Baxter and P.J. Forrester, J. Stat. Phys. 35, 193-266 (1984);
  36. P.J. For- rester and R.J. Baxter, J. Stat. Phys. 38, 435-472 (1985).
  37. I.G. Enting, J. Phys. A13, 3713-3722 (1980);
  38. B. Derrida, J. Phys. A14, L5 (1981);
  39. H. Blöte, M.P. Nightingale and B. Derrida, J. Phys. A14, L45 (1981);
  40. H. Blöte and M.P. Nightingale, Physica A112, 405 (1982);
  41. M.P. Nightingale and H. Blöte, J. Phys. A16, L657 (1983);
  42. H. Blöte and B. Nienhuis, J. Phys. A22, 1415-1438 (1989).
  43. P.A. Pearce, V. Rittenberg and J. de Gier, cond-mat/0108051 (2001);
  44. P.A. Pearce, V. Rit- tenberg, J. de Gier and B. Nienhuis, J. Phys. A35, L661-668 (2002).
  45. B. Nienhuis, Phys. Rev. Lett. 49, 1062-1065 (1982);
  46. J. Stat. Phys. 34, 731-761 (1984).
  47. S-C. Chang and R. Schrock, cond-mat/0404524, Physica A347, 314-352 (2005);
  48. J.L. Ja- cobsen and J. Salas, Transfer matrices and partition-function zeros for antiferromagnetic Potts models, cond-mat/0407444 (2004);
  49. J.-F. Richard and J.L. Jacobsen, Character de- composition of Potts model partition functions I. Cyclic geometry and II Toroidal geometry, math-ph/0605015, 0605016 (2006).
  50. J. Cardy, Nucl. Phys. B240, 514 (1984).
  51. H. Saleur and M. Bauer, Nucl. Phys. B320, 591 (1989).
  52. J. Cardy, Nucl. Phys. B324, 581 (1989).
  53. R.E. Behrend and P.A. Pearce, hep-th/0006094, J. Stat. Phys. 102, 577-640 (2001).
  54. W. Nahm, hep-th/9402039, Int. J. Mod. Phys. B8, 3693 (1994).
  55. E. Melzer, Int. J. Mod. Phys. A9, 1115 (1994).
  56. A. Berkovich, Nucl. Phys. B431, 315 (1994).
  57. R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982).
  58. P.P. Martin, Potts models and related problems in statistical mechanics, Series on Advances in Statistical Mechanics, Volume 5, World Scientific, Singapore (1991).
  59. L. Kauffman, Topology 26, 395 (1987).
  60. F.M. Goodman and H. Wenzl, Pacific Journal of Mathematics 161, 307-334 (1993).
  61. E.K. Sklyanin, J. Phys. A21, 2375-2389 (1988).
  62. R.E. Behrend, P.A. Pearce and D.L. O'Brien, hep-th/9507118, J. Stat. Phys. 84, 1-48 (1996).
  63. P.P. Kulish, J. Phys. A36, L489-L493 (2003).
  64. C.N. Yang and C.P. Yang, Phys. Rev. 150, 321; 327 (1966).
  65. F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R.J. Baxter and G.R.W. Quispel, J. Phys. A20, 6397-6409 (1987).
  66. R.J. Baxter, J. Stat. Phys. 28, 1 (1982).
  67. D.L. O'Brien, P.A. Pearce and R.E. Behrend, cond-mat/9511081, J. Phys. A30, 2353-2366 (1997).
  68. R. Nepomechie and P.A. Pearce, Boundary S matrices of unitary minimal models, in prepa- ration (2006).
  69. J.M. van den Broeck and L.W. Schwartz, Siam J. Math. Anal. 10, 639 (1979).
  70. H.W.J. Blöte, J.L. Cardy, M.P. Nightingale, Phys. Rev. Lett. 56, 742-745 (1986).
  71. I. Affleck, Phys. Rev. Lett. 56, 746-748 (1986).
  72. D.L. O'Brien, P.A. Pearce and S.O. Warnaar, Physica A228, 63-77 (1996).
  73. P.A. Pearce and J. Rasmussen, Solvable critical dense polymers, hep-th/0610273; Physical combinatorics of critical dense polymers, in preparation (2006).
  74. A.V. Razumov and Yu G. Stroganov, cond-mat/0012141, J. Phys. A34, 3185 (2001).
  75. M. Flohr, hep-th/9509166, Int. J. Mod. Phys. A11, 4147 (1996);
  76. M.R. Gaberdiel and H.G. Kausch, hep-th/9606050, Phys. Lett. B386, 131-137 (1996);
  77. B.L. Feigin, A.M. Gain- utdinov, A.M. Semikhatov and I.Yu. Tipunin, Logarithmic extensions of minimal mod- els: characters and modular transformations, hep-th/0606196 (2006); Kazhdan-Lusztig dual quantum group for logarithmic extensions of Virasoro minimal models, math.QA/0606506 (2006) and further references therein.
  78. B.L. Feigin and D.B. Fuchs, Funct. Anal. Appl. 17, 241 (1983).

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