DESCRIPTION OF ORDERING AND PHASE-TRANSITIONS IN TERMS OF LOCAL CONNECTIVITY - PROOF OF A NOVEL TYPE OF PERCOLATED STATE IN THE GENERAL CLOCK MODEL

We present a new description of ordering and phase transitions in term s of genuine local connectivity, i.e., physical connections and discon nections which lead to global order and disorder, respectively. It is generally applicable to complex spin models. We apply it to a simple c ase of the d-dimensional Q-state general clock (GCL) model with two in teraction energy parameters (0 less than or equal to epsilon(1) less t han or equal to epsilon(2)). This model was previously studied for Q = 6 in d = 3 by the Monte Carlo twist method. The following are the mai n results. There are novel types of ordered phases (called IOPs) which are ferromagnetic but dominated by two or three-spin states and exhib it much softer behavior, with stiffness exponent psi approximate to 1. 2, than the low-temperature ferromagnetic phase, with psi = 2, and one of their phase transitions occurs without symmetry breaking. The phys ical connections and disconnections are expressed in terms of new vari ables, link (l-), hinge (h-), and vacant (v-) bonds. We introduce a ne w version of the GCL model with epsilon(2) = infinity (called RGCL mod el) which cannot be disordered, since it has no v-bonds. It is proved to be equivalent to the restricted SOS model for Q > 4 in the hypercub ic lattice. Then we prove that at least one percolated phase of h-bond s exists at high temperature (at any temperature for epsilon(1) = 0) i n the d-dimensional RGCL model for infinity > d > 1. For the GCL model with epsilon(1) = 0 where epsilon(2) < infinity, We then prove the ex istence of it at low enough temperatures. Based on these results and f rom the numerical study mentioned above, we obtain that the IOPs are p ercolated states of h-bonds, and the phase transition without symmetry breaking is purely topological. Also, for the SOS models in d > 2 giv en by H = Sigma \H-i - H-j\(k), we show there is a boundary k(c) (appr oximate to 5) that separates them into two regimes, a preroughening tr ansition for k > k(c) and no transitions otherwise. An algorithm for t he GCL model and order parameters of these percolated phases are given in terms of clusters of l- and h-bonds. The IOPs are also discussed i n detail.