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We study the effect of interfacial phenomena in two-dimensional perfect and random (or disordered) $q$-state Potts models with continuous phase transitions, using, mainly, Monte Carlo techniques. In particular, for the total interfacial adsorption, the critical behavior, including corrections to scaling, are analyzed. The role of randomness is scrutinized. Results are discussed applying scaling arguments and invoking findings for bulk critical properties. In all studied cases, i.e., $q = 3$, $4$, and $q = 8$, the spread of the interfacial adsorption profiles is observed to increase linearly with the lattice size at the bulk transition point.
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We study the Blume-Capel model on the square lattice. To allow for wetting and interfacial adsorption, the spins on opposite boundaries are fixed in two different states, +1 and -1, with reduced couplings at one of the boundaries. Using mainly Monte Carlo techniques, of Metropolis and Wang-Landau type, phase diagrams showing bulk and wetting transitions are determined. The role of the non-boundary state, 0, adsorbed preferably at the interface between -1 and +1 rich regions, is elucidated.
A phenomenological approach to the ferromagnetic two dimensional Potts model on square lattice is proposed. Our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q-state Potts model. The duality symmetry of the 2D Potts model together with the known results on its critical exponent {alpha} allow to fix consistently the details of the proposed expression for the free energy. The agreement of the analytic ansatz with numerical data in the q=3 case is very good at high and low temperatures as well as at the critical point. It is shown that the q>4 cases naturally fit into the same scheme and that one should also expect a good agreement with numerical data. The limiting q=4 case is shortly discussed.
Phase transition of the two- and three-state quantum Potts models on the Sierpinski pyramid are studied by means of a tensor network framework, the higher-order tensor renormalization group method. Critical values of the transverse magnetic field and the magnetic exponent $beta$ are evaluated. Despite the fact that the Hausdorff dimension of the Sierpinski pyramid is exactly two $( = log_2^{~} 4)$, the obtained critical properties show that the effective dimension is lower than two.
We report a surprising result, established by numerical simulations and analytical arguments for a one-dimensional lattice model of random sequential adsorption, that even an arbitrarily small imprecision in the lattice-site localization changes the convergence to jamming from fast, exponential, to slow, power-law, with, for some parameter values, a discontinuous jump in the jamming coverage value. This finding has implications for irreversible deposition on patterned substrates with pre-made landing sites for particle attachment. We also consider a general problem of the particle (depositing object) size not an exact multiple of the lattice spacing, and the lattice sites themselves imprecise, broadened into allowed-deposition intervals. Regions of exponential vs. power-law convergence to jamming are identified, and certain conclusions regarding the jamming coverage are argued for analytically and confirmed numerically.
We study the stochastic dynamics of infinitely many globally interacting $q$-state units on a ring that is externally driven. While repulsive interactions always lead to uniform occupations, attractive interactions give rise to much richer phenomena: We analytically characterize a Hopf bifurcation which separates a high-temperature regime of uniform occupations from a low-temperature one where all units coalesce into a single state. For odd $q$ below the critical temperature starts a synchronization regime which ends via a second phase transition at lower temperatures, while for even $q$ this intermediate phase disappears. We find that interactions have no effects except below critical temperature for attractive interactions. A thermodynamic analysis reveals that the dissipated work is reduced in this regime, whose temperature range is shown to decrease as $q$ increases. The $q$-dependence of the power-efficiency trade-off is also analyzed.
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