Four-state Potts Model Research Articles

Unusual disordering processes of oxygen overlayers on Rh(111): A combined diffraction study using thermal He atoms and low-energy electrons

The temperature-dependent behavior of the Rh(111)-(2\ifmmode\times\else\texttimes\fi{}2)-1O phase was investigated by He-atom scattering (HAS) and low-energy electron diffraction. The adsorption system undergoes an order-disorder phase transition at ${\mathrm{T}}_{\mathrm{c}}$=280\ifmmode\pm\else\textpm\fi{}5 K, with critical exponents found to be consistent with the four-state Potts model. Beyond the phase transition the HAS specular peak intensity exhibits a strong and reversible increase. This finding points toward a reduction of the surface charge-density corrugation induced by the phase transition itself. Around 160 K, hydrogen adsorbed on the Rh(111)-(2x2)-1O surface reacts with oxygen to form water, and drives the overlayer in an out-of-equilibrium condition which is characterized by a dramatic domain-wall proliferation.

Star-Triangle and Star-Star Relations in Statistical Mechanics

The homogeneous three-layer Zamolodchikov model is equivalent to a four-state model on the checkerboard lattice which closely resembles the four-state critical Potts model, but with some of its Boltzmann weights negated. Here we show that it satisfies a "star-to-reverse-star" (or simply star-star) relation, even though we know of no star-triangle relation for this model. For any nearest-neighbour checkerboard model, we show that this star-star relation is sufficient to ensure that the decimated model (where half the spins have been summed over) satisfies a "twisted" Yang-Baxter relation. This ensures that the transfer matrices of the original model commute in pairs, which is an adequate condition for "solvability".

Lack of self-averaging in critical disordered systems.

We consider the sample to sample fluctuations that occur in the value of a thermodynamic quantity $P$ in an ensemble of finite systems with quenched disorder, at equilibrium. The variance of $P$, $V_{P}$, which characterizes these fluctuations is calculated as a function of the systems' linear size $l$, focusing on the behavior at the critical point. The specific model considered is the bond-disordered Ashkin-Teller model on a square lattice. Using Monte Carlo simulations, several bond-disordered Ashkin-Teller models were examined, including the bond-disordered Ising model and the bond-disordered four-state Potts model. It was found that far from criticality the energy, magnetization, specific heat and susceptibility are strongly self averaging, that is $V_{P}\sim l^{-d}$ (where $d=2$ is the dimension). At criticality though, the results indicate that the magnetization $M$ and the susceptibility $\chi$ are non self averaging, i.e. $\frac{V_{\chi}}{\chi^{2}}, \frac{V_{M}}{M^{2}}\not \rightarrow 0$. The energy $E$ at criticality is weakly self averaging, that is $V_{E}\sim l^{-y_{v}}$ with $0<y_{v}<d$. Less conclusively, and possibly only as a transient behavior, the specific heat too is found to be weakly self averaging. A phenomenological theory of finite size scaling for disordered systems is developed. Its main prediction is that when the specific heat exponent $\alpha<0$ ($\alpha$ of the disordered model) then, for a quantity $P$ which scales as $l^{\rho}$ at criticality, its variance $V_{P}$ will scale asymptotically as $l^{2\rho+\frac{\alpha}{\nu}}$. we found very good agreement between the theory and the data for $V_{\chi}$ and $V_{E}$.

First-order phase transition with global Z3 and Z4 symmetry

Mechanism behind the weakness of first-order phase transition with global Z 3 symmetry in three spatial dimensions is discussed based on large-scale numerical simulations of first-order phase transitions in the three- and four-state Potts models.

Logarithmic corrections for the percolative properties of the four-state Potts model

Clusters in the Potts model are connected sets of nearest neighbour sites for which the spin variables are in the same state. Droplets can be obtained by adding bonds with probability p=1-exp(-K) between the sites in a cluster (where K is the Potts model's inverse temperature). When q to 4, renormalization group (RG) fixed points describing clusters and droplets will coalesce, leading to logarithmic corrections. We calculate the precise form of these corrections used in a differential RG method. Our predictions are then tested using extensive Monte Carlo calculations. Theory and the simulations are found to be in excellent agreement.

Buckling instabilities of a confined colloid crystal layer.

A model predicting the structure of repulsive, spherically symmetric, monodisperse particles confined between two walls is presented. We study the buckling transition of a single flat layer as the double layer state develops. Experimental realizations of this model are suspensions of stabilized colloidal particles squeezed between glass plates. By expanding the thermodynamic potential about a flat state of \( N \) confined colloidal particles, we derive a free energy as a functional of in-plane and out-of-plane displacements. The wavevectors of these first buckling instabilities correspond to three different ordered structures. Landau theory predicts that the symmetry of these phases allows for second order phase transitions. This possibility exists even in the presence of gravity or plate asymmetry. These transitions lead to critical behavior and phases with the symmetry of the three-state and four-state Potts models, the X-Y model with 6-fold anisotropy, and the Heisenberg model with cubic interactions. Experimental detection of these structures is discussed.

A first-order phase transition in the three-dimensional four-state antiferromagnetic Potts model

Studies the four-state antiferromagnetic Potts model with next-nearest-neighbour ferromagnetic interactions on the simple cubic lattice by the Monte Carlo method. The ratio J2/J1 of the strength of the next-nearest-neighbour interactions to that of the nearest-neighbour interactions is to be 10. The author analyses the finite-size effects of the energy, the specific heat, the fourth-order cumulant, the magnetization, the susceptibility and the effective transition temperatures. With the help of the finite-size scaling theory the author concludes that the transition of the model is assigned to be of first order.

ALGEBRAIC VARIETIES FOR THE CHIRAL POTTS MODEL

We describe the symmetries of the chiral checkerboard Potts model (duality, inversion relation, …) and write down the algebraic variety corresponding to the integrable case advocated by Baxter, Perk, Au-Yang. We examine some of its subvarieties, in different limits and for various lattices, with a special emphasis on q=3. This yields for q=3, a new algebraic variety where the standard scalar checkerboard Potts model is solvable. By a comparative analysis of the parametrization of the integrable four-state chiral Potts model and the one of the symmetric Ashkin-Teller model, we bring to light algebraic subvarieties for the q-state chiral Potts model which are invariant under the symmetries of the model. We recover in this manner the Fateev-Zamolodchikov points.

Diagonalisation of a colouring problem (on a strip)

A bond colouring problem on a honeycomb lattice, equivalent to evaluating the residual entropy of a four-state antiferromagnetic Potts model, is considered. The transfer matrix for strips with three or four hexagons in each row and any number of rows is diagonalised.

MONTE CARLO RENORMALIZATION GROUP STUDY OF THREE-STATE AND FOUR-STATE POTTS MODEL ON TWO-DIMENSIONAL RANDOM TRIANGLE LATTICE

An improved Monte Carlo renormalization group method is applied to three-state and four-state Potts models on two-dimensional random triangle lattice. Their fixed points and critical exponents are analysed. The critical exponents obtained are consistent with the theoretical analytical values.

Critical behaviors of melting and high-temperature structures in stage-1 rubidium-graphite intercalation compounds.

The phase diagram, the high-temperature (T) structures of liquid and solids, and the critical behaviors of melting in stage-1 Rb-graphite intercalation compounds have been investigated with use of x-ray diffraction. The (${P}_{\mathrm{Rb}}$,T) phase diagram shows a narrow region of a high-T \ensuremath{\alpha}\ensuremath{\beta} stacking solid with the in-plane (2\ifmmode\times\else\texttimes\fi{}2) structure between the low-T \ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\gamma}\ensuremath{\delta} solid and the high-T liquid. At high temperatures Rb atoms behave as a quasi-two-dimensional liquid with in-plane correlations, and as T approaches ${T}_{m}$ (melting temperature) they have strong in-plane and out-of-plane correlations with the same symmetry as the \ensuremath{\alpha}\ensuremath{\beta} solid; the two-dimensional (2D) to 3D crossover effect is observed. Remarkable features of the continuous melting are observed at ${T}_{m}$: (1) the peak wave number of the in-plane short-range order in the liquid changes continuously to the Bragg peak wave number of the \ensuremath{\alpha}\ensuremath{\beta} solid, (2) the order parameter of the \ensuremath{\alpha}\ensuremath{\beta} solid grows continuously with \ensuremath{\beta}=0.30\ifmmode\pm\else\textpm\fi{}0.06, and (3) the in-plane positional and orientational correlation lengths \ensuremath{\xi} increase in the solid and the liquid and have critical exponents ${\ensuremath{\nu}}_{a}$=0.24\ifmmode\pm\else\textpm\fi{}0.04 and ${\ensuremath{\nu}}_{\ensuremath{\varphi}}$=0.34\ifmmode\pm\else\textpm\fi{}0.04 in the liquid, respectively. All the critical parameters show no appreciable hysteresis at ${T}_{m}$. Contrary to these continuous behaviors, the correlation lengths remain finite at ${T}_{m}$, suggesting the presence of a small but finite first-order gap. These experimental results are discussed in terms of the 3D four-state Potts model. In the low-${P}_{\mathrm{Rb}}$ case, the transition is of first order in contrast to the high-${P}_{\mathrm{Rb}}$ case mentioned above. In some samples, the maximum temperature in \ensuremath{\xi}(T) which is obtained in the off-Bragg scan shifts a few degrees lower than ${T}_{m}$. Some discussions are given. The \ensuremath{\alpha}\ensuremath{\beta} solid at high T shows remarkable in-plane anisotropic Debye-Waller factors.

Logarithmic corrections to finite-size scaling in the four-state Potts model

The leading corrections to finite-size scaling predictions for eigenvalues of the quantum Hamiltonian limit of the critical four-state Potts model are calculated analytically from the Bethe ansatz equations for equivalent eigenstates of a modifiedXXZ chain. Scaled gaps are found to behave for large chain lengthL asx+d⦵L+0[(lnL)−1], wherex is the anomalous dimension of the associated primary scaling operator. For the gaps associated with the energy and magnetic operators, the values of the amplitudesd are in agreement with predictions of conformai invariance. The implications of these analytical results for the extrapolation of finite lattice data are discussed. Accurate estimates of x andd are found to be extremely difficult even with data available from large lattices,L∼500.

On the critical dynamics of the diluted Q-state Potts models

The results of Monte Carlo simulations of the four-state Potts model on a square lattice at the bond percolation threshold are presented. Estimates are given for the new dynamical exponents A and B. The result for A is in clear contradiction with a recent conjecture of Nunes da Silva and Lage (1987).

Evidence for universality of the potts model on the two-dimensional penrose lattice

A microcanonical simulation of the four-state ferromagnetic Potts model on the two-dimensional Penrose lattice indicates that the phase transition is second order and is in the same universality class as the periodic system. The five-state ferromagnetic Potts model on the two-dimensional Penrose lattice is found to be first order, as in the periodic system.

Two-dimensional structural phase transitions in the stage-1 OsF6-graphite intercalated compound.

Neutron scattering has been used to study the temperature dependence of the in-plane structure of the stage-1 graphite intercalation compound ${\mathrm{C}}_{9}$${\mathrm{OsF}}_{6}$. The low-temperature structure has a two-dimensional character and can be described as a nearly hexagonal domain-wall lattice in which sharp domain walls separate regions of (2\ifmmode\times\else\texttimes\fi{}2)R0\ifmmode^\circ\else\textdegree\fi{} structure on a 60-A\r{} scale. The domain-wall lattice is imperfect with a correlation length that only extends over a few such regions. The in-plane intercalant structure is disordered above 280 K and resistivity measurements show a clear hysteresis in the transition region above 250 K, indicating a first-order transition. The structural data are consistent with coexistence of a solidlike and fluidlike phase in the transition region, as expected for constant volume melting. The results suggest that this class of intercalation compounds is a good realization of a four-state Potts model with effective dimensionalities less than 3.

Commuting transfer matrices for the four-state self-dual chiral Potts model with a genus-three uniformizing fermat curve

We consider the diagonal-to-diagonal transfer matrix of the four-state self-dual chiral Potts model on a square lattice. We show that there exists a family of models parametrized by a spectral variable u and an auxiliary variable b such that the commutation requirement [ T( u, b), T( u′, b)] = 0 needed for the existence of an infinite number of commuting constants of the motion holds. This family is uniformized by the curve x 4+ y 4 = 1 for all values of b except 0, 1 and ∞, where b = 1 corresponds to the critical self-dual model of Fateev and Zamolodchikov.

Critical behaviour and logarithmic corrections of a quantum model with three-spin interaction

Extended finite-size scaling and series expansion methods are used to determine the critical properties of the one-dimensional quantum Ising model with three-spin interaction. By supposing the presence of logarithmic corrections the universality class of the model is investigated. The results obtained by different methods support the conjecture that the model belongs to the same universality class as the four-state Potts model.

Experimental verification of critical exponents in the two-dimensional four-state Potts universality class: Oxygen on Ru(0001).

Order-disorder critical behavior of the adsorption system O/Ru(0001) at (1/4 monolayer, ordered in a p(2\ifmmode\times\else\texttimes\fi{}2) structure, is studied by use of LEED. By measuring the profiles---and the peak and integrated intensities---of the (1/2-order oxygen LEED spots as function of temperature for 0.004\ensuremath{\Vert}(T-${\mathrm{T}}_{\mathrm{c}}$)/${\mathrm{T}}_{\mathrm{c}}$\ensuremath{\Vert}0.2, we determine the critical exponents \ensuremath{\alpha}, \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\nu} independently. They agree to within 3% to 10% of the theoretical values for the two-dimensional four-state Potts model universality class.

Critical dynamics of the four-state Potts model

The critical relaxation of the four-state Potts model on L*L lattices with L >tc(L), where tc(L) is some crossover time. This crossover is discussed within the context of dynamic finite-size scaling theory. As a result, the dynamic critical exponent is estimated to be Z=2.94+or-0.29.

Equivalence of an Ising model with two- and three-spin interactions with the four-state Potts model

The Ising model model on the square lattice with two-spin interactions in the y direction and three-spin interactions in the x direction is related to the four-state Potts model on the square lattice. This mapping becomes exact in the limit of strong three-spin couplings.