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

Partial Breaking of Three-Fold Symmetry via Percolation of a Domain Wall

“…Therefore, the Ising symmetry gets restored, the symmetry of the system remains only partially broken and the spins exhibit a partial order instead of complete order. We note that this partial order is similar to the up-down-up-down height profile of layers grown in the disordered-flat phase of crystal growth [27,28]. This type of partial order has also been reported for Z 4 models in three dimensions and is termed as a σ phase in the literature for the Ashkin-Teller model [29,30].…”

“…Since the two-dimensional system cannot accomodate the simultaneous percolation of state 0 and state 1 clusters, and yet the temperature is ripe for domain wall proliferation, the (0|1) domain walls do not percolate but remain only at a percolation threshold. While the threshold behavior in two dimensions has not been reported before, the percolation behavior in three dimensions has been discussed in the context of a six-state model [28].…”

Ordered phase and phase transitions in the three-dimensional generalized six-state clock model

“…This ordered phase is called completely ordered phase (COP). Since the COP has six degenerate states, the order parameter space of this model is illustrated by a hexagon as in Ueno rigorously proved that for ε 1 = 0, at least one kind of IOPs can exist at T < ε 2 = ε 3 [13]. For ε 1 > 0 , Ueno and Kasono proposed that this model has two incompletely ordered phases (IOP1, IOP2) besides the COP by using the Monte Carlo twist method (MCTM) [4].…”

“…This degenerate state is IOP1 and corresponds to the ground state of the 3AFP model. Taking the approach of physical percolation to phase transitions, Ueno rigorously proved that for ε 1 = 0, at least one kind of IOPs can exist at T < ε 2 = ε 3 [13]. For ε 1 > 0 , Ueno and Kasono proposed that this model has two incompletely ordered phases (IOP1, IOP2) besides the COP by using the Monte Carlo twist method (MCTM) [4].…”

Discrete approximations to vector spin models

“…It would be interesting to see if the restoration of continuous symmetries which happens for the two-dimensional clock model would have a higher-dimensional analogue, in that in some intermediate-temperature regime there might exist a continuum of Gibbs measures, even for the nearest-neighbor clock model. We conjecture that the intermediate phase studied in [15,20,23,24,26,27], might be of this type. In the terminology of Ueno et al [27] we would have a continuum of "Incompletely Ordered Phases", where the order can be in the two spin directions n, n + 1, where n mod q ∈ {1, · · · , q}, with continuously varying weights of these directions.…”