Systems with a bulk first-order transition can display diverging correlation lengths close to a surface. This surface induced disordering yields a special type of surface criticality. Using extensive numerical simulations we study surface quantities in the two-dimensional Potts model with a large number of states
q which undergoes a discontinuous bulk transition. The surface critical exponents are thereby found to depend on the value of
q, which is in contrast to prior claims that these exponents should be universal and independent of
q. It follows that surface-induced disordering at first-order transitions is characterized by exponents that depend on the details of the model.
Phase transitions in the two-dimensional ferromagnetic Potts model with the number of spin states q = 3 on a triangular lattice were studied using Monte-Carlo method. The systems of linear size L = 20–120 were considered. Using the method of the fourth-order Binder cumulants, it was demonstrated that the second order phase transition is observed in the two-dimensional ferromagnetic Potts model. The static critical exponents for the heat capacity α, susceptibility γ, magnetization β and correlation radius ν were calculated based on the finite-dimensional scaling theory.
We study the three-state antiferromagnetic Potts model on the simple-cubic lattice, paying attention to the surface critical behaviors. When the nearest-neighboring interactions of the surface is tuned, we obtain a phase diagram similar to the XY model, owing to the emergent O(2) symmetry of the bulk critical point. For the ordinary transition, we get ${y}_{h1}=0.780(3), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=1.44(1)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=0.736(6)$; for the special transition, we get ${y}_{s}=0.59(1), {y}_{h1}=1.693(2), {\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.391(4)$, and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.179(5)$; in the extraordinary-log phase, the surface correlation function ${C}_{\ensuremath{\parallel}}(r)$ decays logarithmically with decaying exponent
q=0.60(2), however, the correlation ${C}_{\ensuremath{\perp}}(r)$ still decays algebraically with critical exponent ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.442(5)$. If the ferromagnetic next-nearest-neighboring surface interactions are added, we find two transition points, the first one is a special point between the ordinary phase and the extraordinary-log phase, the second one is a transition between the extraordinary-log phase and the
Z6 symmetry-breaking phase, with critical exponent
ys=0.41(2). The scaling behaviors of the second transition is very interesting, the surface spin-correlation function ${C}_{\ensuremath{\parallel}}(r)$, and the surface squared staggered magnetization at this point decays logarithmically with exponent
q=0.37(1); however, the surface structure factor with the smallest wave vector and the correlation function ${C}_{\ensuremath{\perp}}(r)$ satisfy power-law decaying, with critical exponents ${\ensuremath{\eta}}_{\ensuremath{\parallel}}=\ensuremath{-}0.69(1)$ and ${\ensuremath{\eta}}_{\ensuremath{\perp}}=\ensuremath{-}0.37(1)$, respectively.
We study two-dimensional q -state random-bond Potts models for both q=8 and q=5 with a linearly varying temperature. By applying a successive Monte Carlo renormalization group procedure, both the static and dynamic critical exponents are obtained for randomness amplitudes (the strong to weak coupling ratio) of r_{0}=3 , 10, 15, and 20. The correlation length exponent nu increases with disorder from less than to larger than unity and this variation is justified by the good collapse of the specific heat near the critical region. The specific heat exponent is obtained by the usual hyperscaling relation alpha=2-dnu and thus indicates no possibility of the activated dynamic scaling. Both r_{0} and q have effects on the critical dynamics of the disordered systems, which can be seen from variations of the rate exponent, the hysteresis exponent, and the dynamic critical exponent. Implications of these results are discussed.
A summary of my research activity is presented in this thesis. The first chapter starts with a short presentation of the known critical properties of the pure Potts model, mainly in two dimensions. The influence of disorder coupled to the energy density is then discussed in the general context of temperature-driven phase transitions. A physical explanation for the smoothing of the transition, due to Imry and Wortis, is presented and then applied to explain the phase diagram of the homogeneous random two and three-dimensional Potts models. It is then extended to the Potts model on irregular graphs and with layered randomness. In the third section, the critical properties at the random fixed point are examined in the light of the series expansion obtained by Renormalisation Group. The multifractal spectrum of scaling dimensions is discussed in the two-dimensional case and a new symmetry is proposed. The predictions of conformal invariance for profiles and correlation functions are compared with numerical data for the two-dimensional random Potts model in strip or square geometries. Finally, the geometrical properties of interfaces induced by symmetry-breaking boundary conditions are analysed in the context of the recent Stochastic Schramm-Loewner Evolution theory. The second chapter is devoted to the aging properties of homogeneous or frustrated Potts models submitted to a quench below or at their critical temperature. The scaling theory of two-time functions, as well as the more recent Local Scale Invariance theory , are presented in the first section. The violation of the Fluctuation-Dissipation theorem is described and the interpretation of the so-called Fluctuation-Dissipation Ratio as an out-of-equilibrium effective temperature is discussed. These ideas are applied to different spin models. First, one-dimensional systems with the Ising symmetry are considered. The equivalence of the KDH model with a gas of immobile particles subjected to pair-annihilation is exploited to obtain new analytical results. The question of the universality of dynamical exponents and ratios is then addressed in the two-dimensional case. Several lattices and several models in different universality classes are compared. A new model with the same symmetry as the three-state Potts model but with an irreversible dynamics is introduced. Its phase diagram in the steady state is determined and its aging properties on the critical line are studied. Finally, the aging of the Fully-Frustrated Ising and XY models are examined. Emphasis is put on the presence of logarithmic corrections due to the existence of topological defects. Two applications of the Jarzynski relation to the Potts model are presented at the end of the chapter. A short third chapter introduces the numerical simulations employed to analyse different set of experimental data. Monte Carlo simulations were used to reproduced some magnetic phase transitions in layered nanostructures. Spectroscopic data of surface reconstructions were compared to the electronic band structure obtained by computing the first eigenvectors of the Bloch-Schrödinger equation with a projective technique combined with a multiscale approach.
The critical behavior of the disordered two-dimensional antiferromagnetic Potts model with the number of spin states q= 3 on a triangular lattice with disorder in the form of nonmagnetic impurities is studied by the Monte Carlo method. The critical exponents for the susceptibility γ, magnetization β, specific heat α, and correlation radius ν are calculated in the framework of the finite-size scaling theory at spin concentrations p = 0.90, 0.80, 0.70, and 0.65. It is found that the critical exponents increase with the degree of disorder, whereas the ratios and do not change, thus holding the scaling equality
ν2β+νγ=d . Such behavior of the critical exponents is related to the weak universality of the critical behavior characteristic of disordered systems. All results are obtained using independent Monte Carlo algorithms, such as the Metropolis and Wolff algorithms.