Abstract
We present a new description of ordering and phase transitions in terms of genuine local connectivity, i.e., physical connections and disconnections which lead to global order and disorder, respectively. It is generally applicable to complex spin models. We apply it to a simple case of thed-dimensionalQ-state general clock (GCL) model with two interaction energy parameters (0⩽ε1⩽ε2). This model was previously studied forQ=6 ind=3 by the Monte Carlo twist method. The following are the main results. There are novel types of ordered phases (called IOPs) which are ferromagnetic but dominated by two- or three-spin states and exhibit much softer behavior, with stiffness exponent ψ≈1.2, than the low-temperature ferromagnetic phase, with ψ=2, and one of their phase transitions occurs without symmetry breaking. The physical connections and disconnections are expressed in terms of new variables, link (l-), hinge (h-), and vacant (v-) bonds. We introduce a new version of the GCL model with ε2=∞ (called RGCL model) which cannot be disordered, since it has nov-bonds. It is proved to be equivalent to the restricted SOS model forQ>4 in the hypercubic lattice. Then we prove that at least one percolated phase ofh-bonds exists at high temperature (at any temperature for ε1=0) in thed-dimensional RGCL model for ∞>d>1. For the GCL model with ε1=0 where ε2<∞, we then prove the existence of it at low enough temperatures. Based on these results and from the numerical study mentioned above, we obtain that the IOPs are percolated states ofh-bonds, and the phase transition without symmetry breaking is purely topological. Also, for the SOS models ind>2 given by ℋ=Σ|H i −H j |k, we show there is a boundaryk c (≈5) that separates them into two regimes, a preroughening transition fork>k c and no transitions otherwise. An algorithm for the GCL model and order parameters of these percolated phases are given in terms of clusters ofl- andh-bonds. The IOPs are also discussed in detail.
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References
D. Stauffer,Phys. Rep. 54:1–74 (1979).
P. W. Kasteleyn and C. M. Fortuin,J. Phys. Soc. Jpn. 26(suppl.):11 (1969);Physica 57:536 (1972).
A. Coniglio and W. Klein,J. Phys. A 13:2775 (1980); C.-K. Hu,Phys. Rev. B 29:5103 (1984); C.-K. Hu and C.-N. Chen,Phys. Rev. B 38:2765 (1988).
R. H. Swendsen and J.-S. Wang,Phys. Rev. Lett. 58:86 (1987).
R. G. Edwards and A. D. Sokal,Phys. Rev. D 38:2009 (1988); A. D. Sokal, Monte Carlo methods in statistical mechanics: Foundations and algorithms, Lecture Notes, Troisième Cycle de la Physique en Suisse Romande (1989).
M. D. De Meo, D. W. Heermann, and K. Binder,J. Stat. Phys. 60:585 (1990).
A. N. Berker and L. P. Kadanoff,J. Phys. A 13:L259 (1980); J. Banavar, G. S. Grest, and D. Jasnow,Phys. Rev. B 25:4639 (1982); I. Ono,Prog. Theor. Phys. Suppl. 87:102 (1986).
Y. Ueno, G. Sun, and I. Ono,J. Phys. Soc. Jpn. 58:1162 (1989); Errata,J. Phys. Soc. Jpn. 61:4672 (1992).
J.-S. Wang, R. H. Swendsen, and R. Kotecký,Phys. Rev. B 42:2465 (1990).
M. Mekata,J. Phys. Soc. Jpn. 42:76 (1977); F. Matsubara and S. Inawashiro,J. Phys. Soc. Jpn. 53:4373 (1984); D. Blanckschtein, M. Ma, and A. Berker,Phys. Rev. B 29:5250 (1984).
K. Mitsubo, G. Sun, and Y. Ueno, inCooperative Dynamics in Complex Systems, H. Takayama, ed. (Springer, Berlin, 1989), p. 49; K. Mitsubo and Y. Ueno, Unpublished.
Y. Ueno and K. Mitsubo,Phys. Rev. B 43:8654 (1991); P. D. Scholten and L. J. Irakliotis,Phys. Rev. B 48:1291 (1993).
Y. Ueno and K. Kasono,Phys. Rev. B 48:16471 (1993).
O. Nagai, Y. Yamada, and H. T. Diep,Phys. Rev. B 32:480 (1985); G. Sun and Y. Ueno,Z. Phys. 82:425 (1991).
J. L. Cardy,J. Phys. A 13:1507 (1980).
H. Shioda and Y. Ueno,J. Phys. Soc. Jpn. 62:970 (1993).
J. D. Weeks, inOrdering in Strongly Fluctuating Condensed Matter Systems, T. Riste, ed. (Plenum Press, New York, 1980), p. 293.
M. den Nijs,J. Phys. A 18:L549 (1985).
S. T. Chui and J. D. Weeks,Phys. Rev. B 14:4978 (1976); H. J. F. Knops,Phys. Rev. Lett. 39:766 (1977); J. V. José, L. P. Kadanoff, S. Kirkpatrick, and D. R. Nelson,Phys. Rev. B 16:1217 (1977).
J. M. Kosterlitz and D. J. Thouless,J. Phys. C 6:1181 (1973); J. M. Kosterlitz,J. Phys. C 7:1047 (1974).
M. Göpfert and G. Mack,Commun. Math. Phys. 82:545 (1982).
K. Rommelse and M. den Nijs,Phys. Rev. Lett. 59:2578 (1987).
F. D. M. Haldane,Phys. Lett. 93A:464 (1983);Phys. Rev. Lett. 50:1153 (1983); M. den Nijs and K. Rommelse,Phys. Rev. B 40:4709 (1989).
H. Tasaki,Phys. Rev. Lett. 66:798 (1991).
H. Shioda and Y. Ueno,J. Phys. Soc. Jpn. 62:4224 (1993).
D. Hamuro, Y. Ueno, and G. Sun, unpublished.
S. Ostlund,Phys. Rev. B 24:398 (1981).
M. E. Fisher and D. S. Fisher,Phys. Rev. B 25:239 (1982); O. A. Huse and M. E. Fisher,Phys. Rev. B 29:239 (1984).
Y. Ueno,J. Phys. Soc. Jpn. 55:2586 (1986); G. Sun, Y. Ueno, and Y. Ozeki,J. Phys. Soc. Jpn. 57:156 (1988).
J. C. Le Guillon and J. Zinn-Justin,Phys. Rev. B 21:3976 (1980).
T. Ohyama and H. Shiba,J. Phys. Soc. Jpn. 61:4174 (1992); Y. Okabe and M. Kikuchi, Unpublished.
Y. Ajiro, T. Inami, and H. Kadowaki,J. Phys. Soc. Jpn. 59:4142 (1990).
H. Kadowaki, T. Inami, Y. Ajiro, and Y. Endoh,J. Phys. Soc. Jpn. 60:1708 (1991).
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Ueno, Y. Description of ordering and phase transitions in terms of local connectivity: Proof of a novel type of percolated state in the general clock model. J Stat Phys 80, 841–873 (1995). https://doi.org/10.1007/BF02178558
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DOI: https://doi.org/10.1007/BF02178558