Communications in Mathematical Physics

, Volume 330, Issue 3, pp 1339–1394 | Cite as

Entropy-Driven Phase Transition in Low-Temperature Antiferromagnetic Potts Models

  • Roman KoteckýEmail author
  • Alan D. Sokal
  • Jan M. Swart


We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane quadrangulations, including the diced lattice. More precisely, we show the existence of (at least) three infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one sublattice have a higher probability to be in one state than in either of the other two states. For the special case of the diced lattice, we give a good rigorous lower bound on this probability, based on computer-assisted calculations that are not available for the other lattices.


Planar Graph Gibbs Measure Simple Circuit Graph Automorphism Minimal Cutset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alm S.E., Parviainen R.: Bounds for the connective constant of the hexagonal lattice. J. Phys. A: Math. Gen. 37, 549–560 (2004)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Ashley, J., Grünbaum, B., Shephard, G.C., Stromquist, W.: Self-duality groups and ranks of self-dualities. In: Applied Geometry and Discrete Mathematics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 11–50. American Mathematical Society, Providence (1991)Google Scholar
  3. 3.
    Babai, L.: The growth rate of vertex-transitive planar graphs. In: Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 564–573. Association for Computing Machinery, New York (1997)Google Scholar
  4. 4.
    Banavar J.R., Grest G.S., Jasnow D.: Antiferromagnetic Potts and Ashkin–Teller models in three dimensions. Phys. Rev. B 25, 4639–4650 (1982)ADSCrossRefGoogle Scholar
  5. 5.
    Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic press, London (1982)zbMATHGoogle Scholar
  6. 6.
    Bonnington C.P., Imrich W., Watkins M.E.: Separating double rays in locally finite planar graphs. Discrete Math. 145, 61–72 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Blöte H.W.J., Hilhorst H.J.: Roughening transitions and the zero-temperature triangular Ising antiferromagnet. J. Phys. A: Math. Gen. 15, L631–L637 (1982)ADSCrossRefGoogle Scholar
  8. 8.
    Brightwell, G.R., Winkler, P.: Random colorings of a Cayley tree. In: Bollobás, B. (eds) Contemporary Combinatorics, pp. 247–276. Bolyai Mathematical Society, Springer, Budapest, Berlin (2002)Google Scholar
  9. 9.
    Bruhn H., Diestel R.: Duality in infinite graphs. J. Comb. Theory. B 96, 225–239 (2006)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bruhn H., Stein M.: Duality of ends. Comb. Probab. Comput. 19, 47–60 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Burton, Jr. J.K., Henley, C.L.: A constrained Potts antiferromagnet model with an interface representation. J. Phys. A: Math. Gen. 30, 8385–8413 (1997). (cond-mat/9708171)Google Scholar
  12. 12.
    Coxeter H.S.M.: Regular compound tessellations of the hyperbolic plane. Proc. R. Soc. London A 278, 147–167 (1964)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Deng, Y., Chen, K., Huang, Y.: Private communication (2012)Google Scholar
  14. 14.
    Deng, Y., Huang, Y., Jacobsen, J.L., Salas, J., Sokal, A.D.: Finite-temperature phase transition in a class of four-state Potts antiferromagnets. Phys. Rev. Lett. 107, 150601 (2011)., arXiv:1108.1743 Google Scholar
  15. 15.
    Dickman R.F., McCoy R.A.: The Freudenthal compactification. Dissertationes Math. (Rozprawy Mat.) 262, 35 (1988)MathSciNetGoogle Scholar
  16. 16.
    Diestel, R.: Graph Theory. 4th edn. Springer, New York (2010).
  17. 17.
    Diestel, R.: Locally finite graphs with ends: a topological approach, arXiv:0912.4213. An earlier version of this survey was published in Discrete Math. 311v3, 1423–1447 (2011) and 310, 2750–2765 (2010)
  18. 18.
    Diestel R., Kühn D.: Graph-theoretical versus topological ends of graphs. J. Comb. Theory B 87, 197–206 (2003)CrossRefGoogle Scholar
  19. 19.
    Diestel R., Kühn D.: On infinite cycles I. Combinatorica 24, 69–89 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Diestel R., Kühn D.: On infinite cycles II. Combinatorica 24, 91–116 (2004)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Duminil-Copin, H., Smirnov, S.: The connective constant of the honeycomb lattice equals \({\sqrt{2 + \sqrt{2}}}\). Ann. Math. 175, 1653–1665 (2012). arXiv:1007.0575v2
  22. 22.
    Durrett R.: Lecture Notes on Particle Systems and Percolation. Wadsworth, Pacific Grove (1988)zbMATHGoogle Scholar
  23. 23.
    Edwards R.G., Sokal A.D.: Generalization of the Fortuin–Kasteleyn–Swendsen–Wang representation and Monte Carlo algorithm. Phys. Rev. D 38, 2009–2012 (1988)ADSCrossRefMathSciNetGoogle Scholar
  24. 24.
    Ferreira S.J., Sokal A.D.: Antiferromagnetic Potts models on the square lattice: a high-precision Monte Carlo study. J. Stat. Phys. 96, 461–530 (1999)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Freudenthal H.: Über die Enden topologischer Räume und Gruppen. Math. Z. 33, 692–713 (1931)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Freudenthal H.: Über die Enden diskreter Räume und Gruppen. Comment. Math. Helv. 17, 1–38 (1945)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Galvin, D., Kahn, J., Randall, D., Sorkin, G.B.: Phase coexistence and torpid mixing in the 3-coloring model on \({\mathbb{Z}^d}\), arXiv:1210.4232
  28. 28.
    Georgakopoulos, A.: Graph topologies induced by edge lengths. Discrete Math. 311, 1523–1542 (2011)., arXiv:0903.1744 Google Scholar
  29. 29.
    Georgakopoulos, A.: The planar cubic Cayley graphs, preprint (2011). arXiv:1102.2087v2
  30. 30.
    Georgii H.-O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin (1988)CrossRefzbMATHGoogle Scholar
  31. 31.
    Gottlob A.P., Hasenbusch M.: Three-state anti-ferromagnetic Potts model in three dimensions: Universality and critical amplitudes. Physica A 210, 217–236 (1994)ADSCrossRefGoogle Scholar
  32. 32.
    Gottlob A.P., Hasenbusch M.: The XY model and the three-state antiferromagnetic Potts model in three dimensions: critical properties from fluctuating boundary conditions. J. Stat. Phys. 77, 919–930 (1994)ADSCrossRefGoogle Scholar
  33. 33.
    Grimmett G.R.: The Random-Cluster Model. Springer, Berlin (2006)CrossRefzbMATHGoogle Scholar
  34. 34.
    Halin R.: Über unendliche Wege in Graphen. Math. Ann. 157, 125–137 (1964)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002).
  36. 36.
    Huse D.A., Rutenberg A.D.: Classical antiferromagnets on the Kagomé lattice. Phys. Rev. B 45, 7536–7539 (1992)ADSCrossRefGoogle Scholar
  37. 37.
    Itakura M.: Monte Carlo simulation of the antiferromagnetic four-state Potts model on simple cubic and body-centered-cubic lattices. Phys. Rev. B 60, 6558–6565 (1999)ADSCrossRefGoogle Scholar
  38. 38.
    Jacobsen, J.L.: Conformal field theory applied to loop models. In: Guttmann A.J. (ed.) Polygons, Polyominoes and Polycubes. Lecture Notes in Physics #775, Chapter 14, pp. 347–424. Springer, Dordrecht (2009)Google Scholar
  39. 39.
    Jacobsen, J.L., Sokal, A.D.: Mapping of a graph-homomorphism (RSOS) model onto a multivariate Tutte polynomial (Potts model) (in preparation)Google Scholar
  40. 40.
    Jensen, I.: A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice. J. Phys. A: Math. Gen. 36, 5731–5745 (2003) (cond-mat/0301468)Google Scholar
  41. 41.
    Jensen, I.: Honeycomb lattice polygons and walks as a test of series analysis techniques. J. Phys.: Conf. Ser. 42, 163–178 (2006). Google Scholar
  42. 42.
    Jonasson J.: Uniqueness of uniform random colorings of regular trees. Stat. Probab. Lett. 57, 243–248 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 43.
    Kolafa J.: Monte Carlo study of the three-state square Potts antiferromagnet. J. Phys. A: Math. Gen. 17, L777–L781 (1984)ADSCrossRefGoogle Scholar
  44. 44.
    Kondev J., Henley C.L.: Four-coloring model on the square lattice: a critical ground state. Phys. Rev. B 52, 6628–6639 (1995)ADSCrossRefGoogle Scholar
  45. 45.
    Kondev J., Henley C.L.: Kac-Moody symmetries of critical ground states. Nucl. Phys. B 464, 540–575 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Kotecký R.: Long-range order for antiferromagnetic Potts models. Phys. Rev. B 31, 3088–3092 (1985)ADSCrossRefGoogle Scholar
  47. 47.
    Kotecký, R., Salas, J., Sokal, A.D.: Phase transition in the three-state Potts antiferromagnet on the diced lattice. Phys. Rev. Lett. 101, 030601 (2008). arXiv:0802.2270 Google Scholar
  48. 48.
    Krön, B.: Introduction to ends of graphs, preprint (2005).
  49. 49.
    Krön, B.: Infinite faces and ends of almost transitive plane graphs, Hamburger Beiträge zur Mathematik, Heft 257, preprint (2006).
  50. 50.
    Krön, B., Teufl, E.: Ends—group theoretical and topological aspects, preprint (2009).
  51. 51.
    Liggett T.M.: Interacting Particle Systems. Springer, New York (1985)CrossRefzbMATHGoogle Scholar
  52. 52.
    Liggett T.M.: Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, Berlin (1999)zbMATHGoogle Scholar
  53. 53.
    Madras N., Slade G.: The Self-Avoiding Walk. Birkhäuser, Boston (1993)zbMATHGoogle Scholar
  54. 54.
    Madras N., Wu C.C.: Self-avoiding walks on hyperbolic graphs. Comb. Probab. Comput. 14, 523–548 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 55.
    Moore, C., Newman, M.E.J.: Height representation, critical exponents, and ergodicity in the four-state triangular Potts antiferromagnet. J. Stat. Phys. 99, 629–660 (2000). (cond-mat/9902295)Google Scholar
  56. 56.
    Nienhuis B.: Exact critical point and critical exponents of O(n) models in two dimensions. Phys. Rev. Lett. 49, 1062–1065 (1982)ADSCrossRefMathSciNetGoogle Scholar
  57. 57.
    Nienhuis B., Hilhorst H.J., Blöte H.W.J.: Triangular SOS models and cubic-crystal shapes. J. Phys. A: Math. Gen. 17, 3559–3581 (1984)ADSCrossRefGoogle Scholar
  58. 58.
    den Nijs M.P.M., Nightingale M.P., Schick M.: Critical fan in the antiferromagnetic three-state Potts model. Phys. Rev. B 26, 2490–2500 (1982)ADSCrossRefGoogle Scholar
  59. 59.
    Northshield S.: Geodesics and bounded harmonic functions on infinite planar graphs. Proc. Am. Math. Soc. 113, 229–233 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  60. 60.
    O’Keeffe M.: Self-dual plane nets in crystal chemistry. Aust. J. Chem. 45, 1489–1498 (1992)CrossRefGoogle Scholar
  61. 61.
    O’Keeffe, M., Hyde, B.G.: Crystal Structures I. Patterns and Symmetry, Section 5.3.7. Mineralogical Society of America, Washington DC (1996).
  62. 62.
    Peled, R.: High-dimensional Lipschitz functions are typically flat, preprint (2010), arXiv:1005.4636
  63. 63.
    Pönitz, A., Tittman, P.: Improved upper bounds for self-avoiding walks in \({\mathbb{Z}^d}\). Electron. J. Comb. 7, Research Paper 21 (2000)Google Scholar
  64. 64.
    Richter R.B., Thomassen C.: 3-connected planar spaces uniquely embed in the sphere. Trans. Am. Math. Soc. 354, 4585–4595 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  65. 65.
    Salas, J., Sokal, A.D.: The three-state square-lattice Potts antiferromagnet at zero temperature, J. Stat. Phys. 92, 729–753 (1998) (cond-mat/9801079)Google Scholar
  66. 66.
    Scullard C.R.: Exact site percolation thresholds using a site-to-bond transformation and the star-triangle transformation. Phys. Rev. E 73, 016107 (2006)ADSCrossRefMathSciNetGoogle Scholar
  67. 67.
    Servatius, B., Servatius, H.: Symmetry, automorphisms, and self-duality of infinite planar graphs and tilings. In: Bálint, V. (University of Žilina, Žilina, 1998) Proceedings of the International Scientific Conference on Mathematics (Žilina, 30 June–3 July 1998), pp. 83–116.
  68. 68.
    Thomassen C.: Planarity and duality of finite and infinite graphs. J. Comb. Theory B 29, 244–271 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  69. 69.
    Thomassen C.: Duality of infinite graphs. J. Comb. Theory B 33, 137–160 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  70. 70.
    Wang J.-S., Swendsen R.H., Kotecký R.: Antiferromagnetic Potts models. Phys. Rev. Lett. 63, 109–112 (1989)ADSCrossRefGoogle Scholar
  71. 71.
    Wang J.-S., Swendsen R.H., Kotecký R.: Three-state antiferromagnetic Potts models: a Monte Carlo study. Phys. Rev. B 42, 2465–2474 (1990)ADSCrossRefGoogle Scholar
  72. 72.
    Wierman, J.C.: Construction of infinite self-dual graphs. In: Proceedings of the 5th Hawaii International Conference on Statistics, Mathematics and Related Fields (2006)Google Scholar
  73. 73.
    Zeng, C., Henley, C.L.: Zero-temperature phase transitions of an antiferromagnetic Ising model of general spin on a triangular lattice. Phys. Rev. B 55, 14935–14947 (1997). (cond-mat/9609007)Google Scholar
  74. 74.
    Ziff, R.M., Scullard, C.R., Wierman, J.C., Sedlock, M.R.A.: The critical manifolds of inhomogeneous bond percolation on bow-tie and checkerboard lattices. J. Phys. A: Math. Theor. 45, 494005 (2012)., arXiv:1210.6609 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Roman Kotecký
    • 1
    • 2
    Email author
  • Alan D. Sokal
    • 3
    • 4
  • Jan M. Swart
    • 5
  1. 1.Department of MathematicsUniversity of WarwickCoventryUK
  2. 2.Center for Theoretical StudyCharles UniversityPragueCzech Republic
  3. 3.Department of PhysicsNew York UniversityNew YorkUSA
  4. 4.Department of MathematicsUniversity College LondonLondonUK
  5. 5.Institute of Information Theory and Automation of the ASCR (ÚTIA)Prague 8Czech Republic

Personalised recommendations