Potts Transition Research Articles

Over Half a Century of Research in Oxides

An attempt will be made to recall some of the highlights of the research in oxides achieved by the author and various collaborators over a period of 58 years. These were the observation of the paramagnetic resonance of transition metal ions of high valency like Fe3+, Fe4+, Fe5+ and Co4+ and their photochromic behaviour; then the detection of Mn4+ in BaTiO3 and the proof for the presence of order–disorder character in these ferroelectric transitions, and then the temperature dependence of the rotational order parameter of the structural phase transitions in LaAlO3 and SrTiO3. Near the transition, the observation of critical behaviour compatible with renormalisation group theory and the presence of a Potts transition line, a critical end point and a Lifshitz point under uniaxial stress in SrTiO3 and CaGdF3. This will be followed by the determination of the stabilisation energy of the Jahn–Teller effect with octahedral oxygen neighbours to various ions; leading over to the concept of Jahn–Teller polarons as quasi particles for the generation of superconductivity, and the experimental observation of Jahn–Teller bipolarons and their stabilisation energy determined in one of the HTS copper oxides.

Quantum phase transition between theZ2spin liquid and valence bond crystals on a triangular lattice

We study the quantum phase transition between the $Z_2$ spin liquid and valence bond solid (VBS) orders on a triangular lattice. With a fully isotropic triangular lattice, the transition from a columnar or resonating-plaquette VBS order can be either first order or there could be two transitions with an intermediate phase. If the transition splits into two, then the $Z_2$ spin liquid will first experience a first order $q=3$ Potts transition to a new nematic $Z_2$ spin liquid that breaks the $2\pi/3$ lattice rotation symmetry (but retain translation symmetry unlike the VBS states). The second transition will then take this new nematic $Z_2$ spin liquid to a columnar or resonating-plaquette VBS state through a second order $3d$ XY$^*$ transition. On a distorted triangular lattice, the degeneracy between some of the different columnar VBS orders is lifted, and the phase transition can reduce to a single $3d$ XY$^*$ transition.

Swendsen--Wang Is Faster than Single-Bond Dynamics

We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model is larger than the spectral gap of a single-bond dynamics, that updates only a single edge per step. For this we give a representation of the algorithms on the joint (Potts/random-cluster) model. Furthermore we obtain upper and lower bounds on the mixing time of the single-bond dynamics on the discrete $d$-dimensional torus of side length $L$ at the Potts transition temperature for $q$ large enough that are exponential in $L^{d-1}$, complementing a result of Borgs, Chayes and Tetali.

Equation of state from the Potts-percolation model of a solid

We expand the Potts-percolation model of a solid to include stress and strain. Neighboring atoms are connected by bonds. We set the energy of a bond to be given by the Lennard-Jones potential. If the energy is larger than a threshold the bond is more likely to fail, whereas if the energy is lower than the threshold, the bond is more likely to be alive. In two dimensions we compute the equation of state: stress as a function of interatomic distance and temperature by using renormalization-group and Monte Carlo simulations. The phase diagram, the equation of state, and the isothermal modulus are determined. When the Potts heat capacity is divergent the continuous transition is replaced by a weak first-order transition through the van der Waals loop mechanism. When the Potts transition is first order the stress exhibits a large discontinuity as a function of the interatomic distance.

Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point

We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice \({\mathbb{Z}^{d}}\)—heat bath dynamics and the Swendsen–Wang algorithm—and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen–Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L d-1. This work provides the first upper bound of this form for the Swendsen–Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)2.

Potts fully frustrated model: thermodynamics, percolation, and dynamics in two dimensions

We consider a Potts model diluted by fully frustrated Ising spins. The model corresponds to a fully frustrated Potts model with variables having an integer absolute value and a sign. This model presents precursor phenomena of a glass transition in the high-temperature region. We show that the onset of these phenomena can be related to a thermodynamic transition. Furthermore, this transition can be mapped onto a percolation transition. We numerically study the phase diagram in two dimensions (2D) for this model with frustration and without disorder and we compare it to the phase diagram of (i) the model with frustration and disorder and (ii) the ferromagnetic model. Introducing a parameter that connects the three models, we generalize the exact expression of the ferromagnetic Potts transition temperature in 2D to the other cases. Finally, we estimate the dynamic critical exponents related to the Potts order parameter and to the energy.

The Potts frustrated model: relations with glasses

Similarities between fragile glasses and spin glasses (SG) suggest the study of frustrated spin model to understand the complex dynamics of glasses above the glass transition. We consider a frustrated spin model with Ising spins and s-state Potts spins both with and without disorder. We study the two models by Monte Carlo simulations in two dimensions. The Potts spins mimic orientational degrees of freedom and the coupled frustrated Ising spins take into account for frustrating effects like the geometrical hindrance. We show that in this model dynamical transitions and crossovers are related to static transitions. In particular, when disorder is present, as predicted and verified in SG, a dynamical transition between high-temperatures exponential to low-temperatures non-exponential correlation functions numerically coincides with the ordering temperature of the ferromagnetic regions, i.e. the Griffiths temperature $T_c(s)$, while a crossover between power law growth of correlation times to Arrhenius law occurs near a Potts transition at $T_p(s)<T_c(s)$. In the model without disorder, where $T_c(s)$ is not defined, both dynamical transition and crossover occur at $T_p(s)$. Furthermore the static susceptibility and the autocorrelation times of quantities depending on Potts spins diverge at $T_p(s)$. This is reminescent of recent experimental results on glass-forming liquids.

Precursor phenomena in frustrated systems.

To understand the origin of the dynamical transition, between high-temperature exponential relaxation and low-temperature nonexponential relaxation, that occurs well above the static transition in glassy systems, a frustrated spin model, with and without disorder, is considered. The model has two phase transitions, the lower being a standard spin glass transition (in the presence of disorder) or fully frustrated Ising (in the absence of disorder), and the higher being a Potts transition. Monte Carlo results clarify that in the model with (or without) disorder the precursor phenomena are related to the Griffiths (or Potts) transition. The Griffiths transition is a vanishing transition which occurs above the Potts transition and is present only when disorder is present, while the Potts transition which signals the effect due to frustration is always present. These results suggest that precursor phenomena in frustrated systems are due either to disorder and/or to frustration, giving a consistent interpretation also for the limiting cases of Ising spin glass and of Ising fully frustrated model, where also the Potts transition is vanishing. This interpretation could play a relevant role in glassy systems beyond the spin systems case.

Helicity modulus near a three-state Potts-like transition

By employing Monte Carlo simulations, both the helicity modulus and the heat capacity of a coupled classical XY model in two dimensions have been calculated. The model is based on the Hamiltonian proposed by Bruinsma and Aeppli [Phys. Rev. Lett. 48, 1625 (1982)]. For the first time, the model enables us to study the temperature dependence of the helicity modulus and obtain the associated critical exponent near a three-state Potts transition.

New ordered phases and multicriticality of Te/Ni(111): Model calculations and comparisons with experiment

The chemisorbed system tellurium on Ni(111) has been studied using a lattice gas model with pair interactions as long range as sixth neighbor. We discover three distinct (2√3×2√3)R30 phases: low coverage (θ= (1)/(12) ), centered (θ= (1)/(6) ), and high coverage (θ= (1)/(4) ), when interactions are adjusted to match qualitatively the other features of the experimental phase diagram. Each pair of neighboring ordered phases is separated by a coexistence region at low temperatures. The low-coverage 2√3 phase disorders via a first-order transition at higher temperature. At higher coverage, a (√3×√3)R30 phase occurs at intermediate temperature. This phase melts by a first-order transition at low coverages and a continuous, three-state Potts transition at higher coverages. Thus, the simulations suggest that this chemisorbed system may provide an experimental realization of Potts tricritical behavior. Alternate sets of interactions in this long-range lattice gas model produce the (√7×√7)R19, (3×3), and (2×2) phases, which have been reported for other chemisorbed systems on triangular lattices.

Phase diagram and phase transitions of Krypton on graphite in the extended monolayer regime

We report high resolution x-ray diffraction studies of the structures and phase transitions of monolayer krypton, adsorbed on both powder and single crystal graphite substrates. A comprehensive series of powder diffraction profiles is used to construct the two dimensional phase diagram. The melting of the fix fi commensurate solid is shown to be strongly first order throughout the region where tricritical behavior was previously thought to occur; fluid solid coexistence extends up to the termination of the commensurate phase at 130 K. A disordered weakly incommensurate phase is shown to be a reentrant fluid, a system which may be described as a disordered network of domain walls and which evolves continuously into a more conventional 2D fluid. This evolution is marked by the disappearance of satellite peaks which are caused by the modulation of the overlayer by the substrate. The freezing of the reentrant fluid into the commensurate phase is shown to be consistent with a chiral Potts transition, its freezing into the incommensurate solid consistent with a dislocation binding transition. Single crystal experiments reveal the orientation of the weakly incommensurate phase. The reentrant fluid is found to have no visible orientational fluctuations, manifesting isotropic diffraction peaks. This is attributed to the strong epitaxy of domain walls. The incommensurate solid is shown to undergo an aligned-rotated transition which is well described by zerotemperature calculations.

Q-state Potts model by Wilson's exact renormalization-group equation

Critical properties of the $Q$-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of $Q$ that annihilate as $Q$ approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For $Q&gt;{Q}_{c}$, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients $a$ and $b$ cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation, $a=6.52$ and $b=0.065$. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, $Q=1$, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.

Melting and wetting transitions in the three-state chiral clock model

The melting transition of the two-dimensional, three-state, asymmetric or chiral clock model is examined. Evidence from scaling arguments and analysis of perturbation series is presented, indicating that the chiral symmetry-breaking operator is relevant at the symmetric (or pure Potts) critical point with a crossover exponent of ø ≈ 0.2. The remainder of the commensurate-disordered phase boundary therefore appears to be in a new universality class, distinct from the pure three-state Potts transition. An interfacial wetting transition that plays an important role in the crossover between the two types of critical behavior is discussed. The location and exponents of this wetting transition are obtained both in a low-temperature limit using generating function techniques and in a systematic low-temperature expansion of the transfer matrix.

Evidence for Ising and Potts Transitions in theε−ζTransformation of Two-DimensionalO2

Two-dimensional ${\mathrm{O}}_{2}$ physisorbed on graphite shows a structural and magnetic phase transition from the paramagnetic to the antiferromagnetic state. A high-resolution specific-heat measurement is presented which reveals that there are two successive phase transitions: first a three-state Potts transition, producing a distorted lattice being still paramagnetic, and second an elastically driven Ising transition from the paramagnetic to the antiferromagnetic state.

Three-state Potts transition in sodium azide: experimental study of an order of the transition by means of a heat capacity measurement

It is pointed out that the alpha - beta phase transition in sodium azide, NaN3, is a clear-cut realisation of the phase transition of the three-state Potts model. With the use of AC calorimetry, the temperature dependence of the heat capacity was studied in the close vicinity of the transition point (Tc=19.80+or-0.05 degrees C). Thermal hysteresis of the anomalous heat capacity, if any, quite small, i.e. smaller than 0.3 degrees C. The present result suggests that this phase transition is continuous, or quite close to being continuous, and gives rise to the possibility that this transition violates one of the Landau conditions for second-order phase transitions (i.e. the third-order-invariant condition).

Universality, Tricriticality, and the Potts Transition in First-Stage Lithium-Intercalated Graphite

With use of ac calorimetry and neutron diffraction, the lithium sublattice melting transition in ${\mathrm{C}}_{6}$Li, which belongs to the universality class of the discrete three-dimensional, three-state Potts model, has been discovered and studied. Strong precursive effects surround the (weak) first-order transition at 715\ifmmode\pm\else\textpm\fi{}2 K. Fitting power laws away from the discontinuous region yields $\ensuremath{\alpha}=0.50\ifmmode\pm\else\textpm\fi{}0.05$ and $\ensuremath{\beta}=0.22\ifmmode\pm\else\textpm\fi{}0.03$. These values are consistent with three-dimensional tricritical exponents and may reflect conjectured tricritical behavior.

Q-state Potts model in general dimension

Critical properties of the $q$-state Potts and Potts-lattice-gas models are calculated by means of the Kadanoff variational renormalization group for dimensions 1.58, 2, and 2.32, and the Migdal approximation for arbitrary dimensions. Accurate results for the critical and tricritical exponents are obtained as well as for ${q}_{c}$, the largest value of $q$ for which the Potts transition is continuous. It is concluded unambiguously that the three-state Potts model in three dimensions undergoes a first-order phase transition.