Journal of Statistical Physics

, Volume 80, Issue 3–4, pp 841–873 | Cite as

Description of ordering and phase transitions in terms of local connectivity: Proof of a novel type of percolated state in the general clock model

  • Yohtaro Ueno


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.

Key Words

Local connectivity bond variables clusters general clock model SOS model incompletely ordered phase percolated state topological phase transition algorithm 


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Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • Yohtaro Ueno
    • 1
  1. 1.Department of PhysicsTokyo Institute of TechnologyMeguro, TokyoJapan

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