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Potts model with invisible states: a review

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Abstract

The Potts model with invisible states was introduced to explain discrepancies between theoretical predictions and experimental observations of phase transitions in some systems where \(Z_q\) symmetry is spontaneously broken. It differs from the ordinary q-state Potts model in that each spin, besides the usual q visible states, can be also in any of r so-called invisible states. Spins in an invisible state do not interact with their neighbours, but they do contribute to the entropy of the system. As a consequence, an increase in r may cause a phase transition to change from second to first order. Potts models with invisible states describe a number of systems of interest in physics and beyond and have been treated by various tools of statistical and mathematical physics. In this paper, we aim to give a review of this fundamental topic.

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Notes

  1. Negative numbers of invisible states merely represent an analytical continuations of r to negative values in the exact solution.

References

  1. R. Tamura, S. Tanaka, N. Kawashima, Phase transition in Potts model with invisible states. Progress Theoret. Phys. 124(2), 381–388 (2010). https://doi.org/10.1143/PTP.124.381

    Article  ADS  MATH  Google Scholar 

  2. S. Tanaka, R. Tamura, N. Kawashima, Phase transition of generalized ferromagnetic Potts model - effect of invisible states. J. Phys. Conf. Ser. 297, 012022 (2011). https://doi.org/10.1088/1742-6596/297/1/012022

    Article  Google Scholar 

  3. R. Tamura, N. Kawashima, First-order transition to incommensurate phase with broken lattice rotation symmetry in frustrated Heisenberg model. J. Phys. Soc. Jpn. 77(10), 103002 (2008). https://doi.org/10.1143/JPSJ.77.103002

    Article  ADS  Google Scholar 

  4. E.M. Stoudenmire, S. Trebst, L. Balents, Quadrupolar correlations and spin freezing in \(s=1\) triangular lattice antiferromagnets. Phys. Rev. B 79, 214436 (2009). https://doi.org/10.1103/PhysRevB.79.214436

    Article  ADS  Google Scholar 

  5. S. Okumura, H. Kawamura, T. Okubo, Y. Motome, Novel spin-liquid states in the frustrated Heisenberg antiferromagnet on the honeycomb lattice. J. Phys. Soc. Jpn. 79(11), 114705 (2010). https://doi.org/10.1143/JPSJ.79.114705

    Article  ADS  Google Scholar 

  6. F.Y. Wu, The Potts model. Rev. Mod. Phys. 54, 235–268 (1982). https://doi.org/10.1103/RevModPhys.54.235

    Article  ADS  MathSciNet  Google Scholar 

  7. A.C.D. van Enter, G. Iacobelli, S. Taati, First-order transition in Potts models with ‘invisible’ states: Rigorous proofs. Progress Theoret. Phys. 126(5), 983–991 (2011). https://doi.org/10.1143/PTP.126.983

    Article  ADS  MATH  Google Scholar 

  8. A.C.D. van Enter, G. Iacobelli, S. Taati, Potts Model with Invisible Colors: Random-Cluster Representation and Pirogov-Sinai Analysis. Rev. Math. Phys. 24(2), 1250004–1125000442 (2012). https://doi.org/10.1142/S0129055X12500043. arXiv:1109.0189 [math-ph]

    Article  MathSciNet  MATH  Google Scholar 

  9. D.A. Johnston, R.P.K.C.M. Ranasinghe, Potts models with (17) invisible states on thin graphs. J. Phys. A Math. Theor. 46(22), 225001 (2013). https://doi.org/10.1088/1751-8113/46/22/225001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. T. Mori, Microcanonical analysis of exactness of the mean-field theory in long-range interacting systems. J. Stat. Phys. 147(5), 1020–1040 (2012). https://doi.org/10.1007/s10955-012-0511-0

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. M. Krasnytska, P. Sarkanych, B. Berche, Y. Holovatch, R. Kenna, Marginal dimensions of the Potts model with invisible states. J. Phys. A Math. Theor. 49(25), 255001 (2016). https://doi.org/10.1088/1751-8113/49/25/255001

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. P. Sarkanych, Y. Holovatch, R. Kenna, Exact solution of a classical short-range spin model with a phase transition in one dimension: the Potts model with invisible states. Phys. Lett. A 381(41), 3589–3593 (2017). https://doi.org/10.1016/j.physleta.2017.08.063

    Article  ADS  MATH  Google Scholar 

  13. P. Sarkanych, Y. Holovatch, R. Kenna, Classical phase transitions in a one-dimensional short-range spin model. J. Phys. A Math. Theor. 51(50), 505001 (2018). https://doi.org/10.1088/1751-8121/aaea02

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Sarkanych, M. Krasnytska, Ising model with invisible states on scale-free networks. Phys. Lett. A 383(27), 125844 (2019). https://doi.org/10.1016/j.physleta.2019.125844

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Tanaka, R. Tamura, Dynamical properties of Potts model with invisible states. J. Phys. Conf. Ser. 320, 012025 (2011). https://doi.org/10.1088/1742-6596/320/1/012025

    Article  Google Scholar 

  16. G. Iacobelli, Metastates, non-gibbsianness and phase transitions: a stroll through statistical mechanics. PhD thesis, University of Groningen (2012). Relation: http://www.rug.nl/ Rights: University of Groningen. https://research.rug.nl/en/publications/metastates-non-gibbsianness-and-phase-transitions-a-stroll-throug

  17. B. Király, Phase transitions in evolutionary potential games. PhD thesis, Institute of Technical Physics and Materials Science Centre for Energy Research (2019). https://repozitorium.omikk.bme.hu/bitstream/handle/10890/13392/ertekezes.pdf

  18. D. Lee, W. Choi, J. Kertész, B. Kahng, Universal mechanism for hybrid percolation transitions. Sci. Rep. 7(1), 5723 (2017). https://doi.org/10.1038/s41598-017-06182-3

    Article  ADS  Google Scholar 

  19. R.B. Potts, Some generalized order-disorder transformations. Math. Proc. Cambridge Philos. Soc. 48(1), 106–109 (1952). https://doi.org/10.1017/S0305004100027419

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. H.W. Capel, Phase transitions in spin-one Ising systems. Phys. Lett. 23(5), 327–328 (1966). https://doi.org/10.1016/0031-9163(66)90023-0

    Article  ADS  MathSciNet  Google Scholar 

  21. M. Blume, Theory of the first-order magnetic phase change in UO\(_{2}\). Phys. Rev. 141, 517–524 (1966). https://doi.org/10.1103/PhysRev.141.517

    Article  ADS  Google Scholar 

  22. M. Blume, V.J. Emery, R.B. Griffiths, Ising model for the \({\lambda }\) transition and phase separation in He\(^{3}\)-He\(^{4}\) mixtures. Phys. Rev. A 4, 1071–1077 (1971). https://doi.org/10.1103/PhysRevA.4.1071

    Article  ADS  Google Scholar 

  23. J. Wajnflasz, R. Pick, Transitions low spin-high spin dans les complexes de Fe2+. J. Phys. Colloques 32(C1), 1–91192 (1971). https://doi.org/10.1051/jphyscol:1971127

    Article  Google Scholar 

  24. L. Laanait, A. Messager, S. Miracle-Solé, J. Ruiz, S. Shlosman, Interfaces in the Potts model. I. Pirogov-Sinai theory of the Fortuin-Kasteleyn representation. Commun. Math. Phys. 140(1), 81–91 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. C. Borgs, J.T. Chayes, J.H. Kim, A. Frieze, P. Tetali, E. Vigoda, V.H. Vu, Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics. Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (New York), 218–229 (1999)

  26. N. Ananikian, N. S. Izmailyan, D. A. Johnston, R. Kenna, R. P. K. C. M. Ranasinghe, Potts models with invisible states on general Bethe lattices. J. Phys. A Math. Theor. 46(38), 385002 (2013). https://doi.org/10.1088/1751-8113/46/38/385002

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. L. D. Landau, E. M. Lifshitz, Statistical physics: volume 5, vol. 5 (Elsevier, Oxford, 2013)

    Google Scholar 

  28. P. Sarkanych, M. Krasnytska, Potts model with invisible states on a scale-free network. Condens. Matter Phys. 26(1), 13507 (2023). https://doi.org/10.5488/CMP.26.13507. arXiv:2211.14048

    Article  ADS  MATH  Google Scholar 

  29. D. Achlioptas, R.M. D’Souza, J. Spencer, Explosive percolation in random networks. Science 323(5920), 1453–1455 (2009). https://doi.org/10.1126/science.1167782

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. O. Riordan, L. Warnke, Explosive percolation is continuous. Science 333(6040), 322–324 (2011). https://doi.org/10.1126/science.1206241

    Article  ADS  Google Scholar 

  31. N. Bastas, P. Giazitzidis, M. Maragakis, K. Kosmidis, Explosive percolation: unusual transitions of a simple model. Physica A 407, 54–65 (2014). https://doi.org/10.1016/j.physa.2014.03.085

    Article  ADS  Google Scholar 

  32. J. Adler, Bootstrap percolation. Physica A 171(3), 453–470 (1991). https://doi.org/10.1016/0378-4371(91)90295-N

    Article  ADS  MathSciNet  Google Scholar 

  33. S.V. Buldyrev, R. Parshani, P. Gerald, H.E. Stanley, S. Havlin, Catastrophic cascade of failures in interdependent networks. Nature 464, 1025–1028 (2010). https://doi.org/10.1038/nature08932

    Article  ADS  Google Scholar 

  34. D. Lee, S. Choi, M. Stippinger, J. Kertész, B. Kahng, Hybrid phase transition into an absorbing state: percolation and avalanches. Phys. Rev. E 93, 042109 (2016). https://doi.org/10.1103/PhysRevE.93.042109

    Article  ADS  Google Scholar 

  35. H.J. Herrmann, Discontinuous percolation. J. Phys. Conf. Ser. 681(1), 012003 (2016). https://doi.org/10.1088/1742-6596/681/1/012003

    Article  Google Scholar 

  36. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008). https://doi.org/10.1103/RevModPhys.80.1275

    Article  ADS  Google Scholar 

  37. S.H. Lee, M. Ha, H. Jeong, J.D. Noh, H. Park, Critical behavior of the Ising model in annealed scale-free networks. Phys. Rev. E 80, 051127 (2009)

    Article  ADS  Google Scholar 

  38. M. Krasnytska, B. Berche, Y. Holovatch, Phase transitions in the Potts model on complex networks. Condens. Matter Phys. 16, 23602 (2013). https://doi.org/10.5488/CMP.16.23602

    Article  ADS  Google Scholar 

  39. F. Iglói, L. Turban, First- and second-order phase transitions in scale-free networks. Phys. Rev. E 66, 036140 (2002). https://doi.org/10.1103/PhysRevE.66.036140

    Article  ADS  Google Scholar 

  40. R. Albert, A.-L. Barabási, Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002). https://doi.org/10.1103/RevModPhys.74.47

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. A.V. Goltsev, S.N. Dorogovtsev, J.F.F. Mendes, Critical phenomena in networks. Phys. Rev. E 67, 026123 (2003). https://doi.org/10.1103/PhysRevE.67.026123

    Article  ADS  Google Scholar 

  42. S.N. Dorogovtsev, A.V. Goltsev, J.F.F. Mendes, Potts model on complex networks. Eur. Phys. J. B 38, 177–182 (2004). https://doi.org/10.1140/epjb/e2004-00019-y

    Article  ADS  Google Scholar 

  43. P.W. Kasteleyn, C.M. Fortuin, Phase transitions in lattice systems with random local properties. Physical Society of Japan Journal Supplement. Proceedings of the International Conference on Statistical Mechanics held 9-14 September, 1968 in Koyto 26, 11 (1969)

  44. D. Stauffer, A. Aharony, Introduction to percolation theory (Taylor & Francis, London, 1994)

    MATH  Google Scholar 

  45. N.S. Ananikyan, S.A. Hajryan, E.S. Mamasakhlisov, V.F. Morozov, Helix-coil transition in polypeptides: a microscopical approach. Biopolymers 30(3–4), 357–367 (1990). https://doi.org/10.1002/bip.360300313

    Article  Google Scholar 

  46. S.A. Hairyan, E.S. Mamasakhlisov, V.F. Morozov, The helix-coil transition in polypeptides: a microscopic approach. II. Biopolymers 35(1), 75–84 (1995). https://doi.org/10.1002/bip.360350108

    Article  Google Scholar 

  47. V.F. Morozov, A.V. Badasyan, A.V. Grigoryan, M.A. Sahakyan, Y.S. Mamasakhlisov, Stacking and hydrogen bonding: DNA cooperativity at melting. Biopolymers 75(5), 434–439 (2004). https://doi.org/10.1002/bip.20143

    Article  Google Scholar 

  48. V. Morozov, A. Badasyan, A. Grigorian, M. Sahakyan, E. Mamasakhlisov, Stacking decreases the cooperativity of melting of homopolymeric DNA. Mod. Phys. Lett. B 19(01n02), 79–83 (2005). https://doi.org/10.1142/S0217984905008062

    Article  ADS  Google Scholar 

  49. A.V. Badasyan, A.V. Grigoryan, E.S. Mamasakhlisov, A.S. Benight, V.F. Morozov, The helix-coil transition in heterogeneous double stranded DNA: microcanonical method. J. Chem. Phys. 123(19), 194701 (2005). https://doi.org/10.1063/1.2107507

    Article  ADS  Google Scholar 

  50. A.V. Grigoryan, E.S. Mamasakhlisov, T.Y. Buryakina, A.V. Tsarukyan, A.S. Benight, V.F. Morozov, Stacking heterogeneity: a model for the sequence dependent melting cooperativity of duplex DNA. J. Chem. Phys. 126(16), 165101 (2007). https://doi.org/10.1063/1.2727456

    Article  ADS  Google Scholar 

  51. A.V. Badasyan, A. Giacometti, Y.S. Mamasakhlisov, V.F. Morozov, A.S. Benight, Microscopic formulation of the Zimm-Bragg model for the helix-coil transition. Phys. Rev. E 81, 021921 (2010). https://doi.org/10.1103/PhysRevE.81.021921

    Article  ADS  Google Scholar 

  52. A.V. Badasyan, S.A. Tonoyan, Y.S. Mamasakhlisov, A. Giacometti, A.S. Benight, V.F. Morozov, Competition for hydrogen-bond formation in the helix-coil transition and protein folding. Phys. Rev. E 83, 051903 (2011). https://doi.org/10.1103/PhysRevE.83.051903

    Article  ADS  Google Scholar 

  53. A. Badasyan, S. Tonoyan, A. Giacometti, R. Podgornik, V.A. Parsegian, Y. Mamasakhlisov, V. Morozov, Osmotic pressure induced coupling between cooperativity and stability of a helix-coil transition. Phys. Rev. Lett. 109, 068101 (2012). https://doi.org/10.1103/PhysRevLett.109.068101

    Article  ADS  Google Scholar 

  54. N. Schreiber, R. Cohen, G. Amir, S. Haber, Changeover phenomenon in randomly colored Potts models. J. Stat. Mech. Theory Exp. 2022(4), 043205 (2022). https://doi.org/10.1088/1742-5468/ac603a

    Article  MathSciNet  MATH  Google Scholar 

  55. S. Tanaka, R. Tamura, I. Sato, K. Kurihara, Hybrid quantum annealing for cluster problems, pp. 169–192. https://doi.org/10.1142/9789814425988_0006. https://www.worldscientific.com/doi/abs/10.1142/9789814425988_0006

  56. K. Kurihara, S. Tanaka, S. Miyashita, Quantum annealing for clustering. In: Proceedings of the 25th Annual Conference on Uncertainty in Artificial Intelligence (2009)

  57. I. Sato, K. Kurihara, S. Tanaka, H. Nakagawa, S., Miyashita, Quantum annealing for variational Bayes inference. In: Proceedings of the 25th Annual Conference on Uncertainty in Artificial Intelligence (2009)

  58. R. Tamura, S. Tanaka, A method to change phase transition nature – toward annealing methods, pp. 135–163 (2014). https://doi.org/10.1142/9789814602372_0009. https://www.worldscientific.com/doi/abs/10.1142/9789814602372_0009

  59. S. Tanaka, R. Tamura, Quantum annealing: from viewpoints of statistical physics, condensed matter physics, and computational physics, pp. 1–59. https://doi.org/10.1142/9789814425193_0001. https://www.worldscientific.com/doi/abs/10.1142/9789814425193_0001

  60. M. Henkel, H. Hinrichsen, S. Lűbeck, Absorbing Phase Transitions, Non-Equilibrium Phase Transitions, vol. 1. (Springer, Dordrecht, 2008), p.385

    MATH  Google Scholar 

  61. M. Henkel, M. Pleimling, Ageing and dynamical scaling far from equilibrium, Non-equilibrium phase transitions, vol. 2. (Springer, Dordrecht, 2010), p.544. https://doi.org/10.1007/978-90-481-2869-3

    Book  MATH  Google Scholar 

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Acknowledgements

It is our pleasure and honour to contribute this paper to the Festschrift dedicated to Malte Henkel on the occasion of his jubilee. Doing so, we acknowledge Malte’s contributions to many problems in the field of phase transitions and criticality mentioned (and not mentioned) in this review [60, 61]. MK thanks the National Academy of Sciences of Ukraine, grant for research laboratories/groups of young scientists No 07/01-2022(4); the PAUSE program and hospitality at LPCT, Lorraine University.

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Authors and Affiliations

  1. Laboratoire de Physique et Chimie Théoriques, Université de Lorraine - CNRS, UMR, 7019, Nancy, France

    Mariana Krasnytska & Bertrand Berche

  2. Institute for Condensed Matter Physics, National Acad. Sci. of Ukraine, Lviv, 79011, Ukraine

    Mariana Krasnytska, Petro Sarkanych & Yurij Holovatch

  3. L4 Collaboration and Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry, D-04009, Leipzig, Germany

    Mariana Krasnytska, Petro Sarkanych, Bertrand Berche, Yurij Holovatch & Ralph Kenna

  4. Centre for Fluid and Complex Systems, Coventry University, Coventry, CV1 5FB, UK

    Yurij Holovatch & Ralph Kenna

  5. Complexity Science Hub Vienna, Vienna, 1080, Austria

    Yurij Holovatch

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  1. Mariana Krasnytska
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Correspondence to Mariana Krasnytska.

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Krasnytska, M., Sarkanych, P., Berche, B. et al. Potts model with invisible states: a review. Eur. Phys. J. Spec. Top. 232, 1681–1691 (2023). https://doi.org/10.1140/epjs/s11734-023-00843-3

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