Phase Transition Of Continuous Order Research Articles

Limits of the phonon quasi-particle picture at the cubic-to-tetragonal phase transition in halide perovskites

The soft modes associated with continuous-order phase transitions are associated with strong anharmonicity. This leads to the overdamped limit where the phonon quasi-particle picture can break down. However, this limit is commonly restricted to a narrow temperature range, making it difficult to observe its signature feature, namely the breakdown of the inverse relationship between the relaxation time and damping. Here we present a physically intuitive picture based on the relaxation times of the mode coordinate and its conjugate momentum, which at the instability approach infinity and the inverse damping factor, respectively. We demonstrate this behavior for the cubic-to-tetragonal phase transition of the inorganic halide perovskite CsPbBr3 via molecular dynamics simulations, and show that the overdamped region extends almost 200 K above the transition temperature. Further, we investigate how the dynamics of these soft phonon modes change when crossing the phase transition.

Limiting Measure of Lee–Yang Zeros for the Cayley Tree

This paper is devoted to an in-depth study of the limiting measure of Lee--Yang zeroes for the Ising Model on the Cayley Tree. We build on previous works of M\"uller-Hartmann-Zittartz (1974 and 1977), Barata--Marchetti (1997), and Barata--Goldbaum (2001), to determine the support of the limiting measure, prove that the limiting measure is not absolutely continuous with respect to Lebesgue measure, and determine the pointwise dimension of the measure at Lebesgue a.e. point on the unit circle and every temperature. The latter is related to the critical exponents for the phase transitions in the model as one crosses the unit circle at Lebesgue a.e. point, providing a global version of the "phase transition of continuous order" discovered by M\"uller-Hartmann-Zittartz. The key techniques are from dynamical systems because there is an explicit formula for the Lee-Yang zeros of the finite Cayley Tree of level $n$ in terms of the $n$-th iterate of an expanding Blaschke Product. A subtlety arises because the conjugacies between Blaschke Products at different parameter values are not absolutely continuous.

Absence and presence of phase transitions for Potts models on pseudo-lattices

By considering a Potts model on a pseudo-lattice such as a tree or a Husimi tree, it is shown that the analyticity of the free energy in the absence of a field for such systems is due to a graph-theoretical property of these lattices called locality. It is proved that, if this property is destroyed by introducing next-nearest neighbour interactions, then the free energy in the absence of a field is nonanalytic as a function of the nearest-neighbour interaction energy below a critical temperature. An exact relation shows that this nonanalyticity is of the same type (phase transition of continuous order) as that found for the Ising model on a Cayley tree with external field.

S-State potts model on a Cayley tree

The Potts model is shown to exhibit a phase transition of continuous order on a Cayley tree. The leading nonanalytic part of the free energy ∣ L∣ K s involves a critical exponent K s going from one to infinity as the coupling goes f rom infinity to K BP, the Bethe-Peierls critical coupling.

Spherical model on the Cayley tree

We investigate the phase transition of the spherical model on the Cayley tree. There is only one transition temperature T,; the specific heat shows a finite discontinuity at Tc; there is no spontaneous magnetisation; the susceptibility remains finite, but shows a cusp at Tc with infinite slope on the high temperature side. These results differ significantly from spherical model properties on a lattice and from the recent Ising model calculations on the Cayley tree with its phase transition of continuous order.

Monte Carlo investigation of magnetic properties of a two-dimensional plane-rotator model

The equilibrium properties of a simple quadratic lattice of plane rotators with nearest-neighbor isotropic interactions have been examined by the Monte Carlo method. It was found that the 2- d plane-rotator model shows much the same magnetic properties as those of the 2- d Heisenberg model if temperature is scaled in units of T M , the transition temperature predicted by the mean-field theory. It is pointed out that the present results are closely related to the prediction of Zittartz of a phase transition of continuous order.

Theory of the Ising model on a Cayley tree

The Ising model on a Cayley tree is known to exhibit a phase transition of continuous order. In this paper we present a complete and quantitative analysis of the leading singular term in the free energy which is associated with this phase transition. We have been able to solve this problem by considering the distribution of zeros of the partition function. The most interesting new feature in our results is a contribution to the free energy which performs singular oscillations as the magnetic field approaches zero.

Infrared singularities in one-dimensional fermi systems

We study the infrared problem in one-dimensional Fermi systems with general interactions including backward scattering. The singular structure of the ground state energy which is a function of two coupling parametersu andv is determined exactly. For attractiveu-coupling one has power law singularities inv with a characteristic singular exponentk varying continuously withu. This behavior is reminiscent of a phase transition of continuous order and is obtained here by exploiting the analogy of the 1-d Fermi system and a 2-d classical plasma with a long range logarithmic Coulomb potential and a short range cut off. We also discuss other recent approaches to the problem.

Continuous order phase transitions on a decorated Cayley tree of connectivity two

The treatment of the Ising model on a Cayley tree given by Muller-Hartmann and Zittartz is extended in the case of connectivity two to a decorated tree containing additional bonds with an arbitrary coupling constant. The possibility of phase transitions is investigated and discussed. The positions of the singular surfaces, on which continuous order phase transitions take place, are examined as functions of coupling constants and external fields.

Phase transitions of continuous order: Ising model on a Cayley tree

We introduce the notion of a phase transition of continuous order. As a model which displays such a phase transition an Ising model on a Cayley tree is solved exactly and discussed in detail.