Simple Critical Point Research Articles

Uniaxial and biaxial structures in the elastic Maier-Saupe model.

We perform statistical mechanics calculations to analyze the global phase diagram of a fully connected version of a Maier-Saupe-Zwanzig lattice model with the inclusion of couplings to an elastic strain field. We point out the presence of uniaxial and biaxial nematic structures, depending on temperature T and on the applied stress σ. Under uniaxial extensive tension, applied stress favors uniaxial orientation, and we obtain a first-order boundary along which there is a coexistence of two uniaxial paranematic phases, and which ends at a simple critical point. Under uniaxial compressive tension, stress favors biaxial orientation; for small values of the coupling parameters, the first-order boundary ends at a tricritical point, beyond which there is a continuous transition between a paranematic and a biaxially ordered structure. For some representative choices of the model parameters, we obtain a number of analytic results, including the location of critical and tricritical points and the line of stability of the biaxial phase.

Field-induced uniaxial and biaxial nematic phases in the Maier–Saupe–Zwanzig (MSZ) lattice model

ABSTRACTWe report detailed statistical mechanics calculations to analyse the field-temperature phase diagram of a fully connected Maier–Saupe–Zwanzig (MSZ) lattice model for the nematic transitions in liquid-crystalline systems. If the field is applied along the uniaxial direction, with positive anisotropy, there is a first-order boundary between distinct uniaxial paranematic phases, which ends at a simple critical point. For negative values of the anisotropy, the first-order boundary becomes continuous at a tricritical point, and the transition is between field-induced uniaxial and biaxial structures. This elementary MSZ lattice model provides a way to systematically regain all of the main features of earlier phenomenological and model calculations. Due to the simplicity of the calculations, critical and tricritical points, as well as critical lines, can be located analytically, which leads to a possibility of comparisons with values for real experimental systems. Also, we formulate a generalised six-state MSZ model, which takes into account intrinsic molecular biaxiality, and perform calculations to determine the location of the critical point in the field-temperature phase diagram.

Thermodynamic curvature and phase transitions in Kerr-Newman black holes

Singularities in the thermodynamics of Kerr-Newman black holes are commonly associated with phase transitions. However, such interpretations are complicated by a lack of stability and, more significantly, by a lack of conclusive insight from microscopic models. Here, I focus on the later problem. I use the thermodynamic Riemannian curvature scalar $R$ as a try to get microscopic information from the known thermodynamics. The hope is that this could facilitate matching black hole thermodynamics to known models of statistical mechanics. For the Kerr-Newman black hole, the sign of $R$ is mostly positive, in contrast to that for ordinary thermodynamic models, where $R$ is mostly negative. Cases with negative $R$ include most of the simple critical point models. An exception is the Fermi gas, which has positive $R$. I demonstrate several exact correspondences between the two-dimensional Fermi gas and the extremal Kerr-Newman black hole. Away from the extremal case, $R$ diverges to $+\ensuremath{\infty}$ along curves of diverging heat capacities ${C}_{J,\ensuremath{\Phi}}$ and ${C}_{\ensuremath{\Omega},Q}$, but not along the Davies curve of diverging ${C}_{J,Q}$. Finding statistical mechanical models with like behavior might yield additional insight into the microscopic properties of black holes. I also discuss a possible physical interpretation of $|R|$.

The period function of hyperelliptic Hamiltonians of degree 5 with real critical points

We provide a criterion to determine the convexity of the period function for a class of planar Hamiltonian systems. As an application we prove that the period function of all hyperelliptic Hamiltonians of degree 5 with real critical points has at most one simple critical point. More precisely, if the period annulus surrounds only one non-degenerate singularity, then the period function is monotone; otherwise (i.e. the period annulus surrounds three singularities, taking into account the multiplicities), the period function has exactly one critical point.

Dynamics, stability, and bifurcations

Dynamics, stability, and bifurcations

Critical behavior at superconductor-insulator phase transitions near one dimension

I argue that the system of interacting bosons at zero temperature and in random external potential possesses a simple critical point which describes the proliferation of disorder-induced topological defects in the superfluid ground state, and which is located at weak disorder close to and above one dimension. This makes it possible to address the critical behavior at the superfluid-Bose glass transition in dirty boson systems by expanding around the lower critical dimension d=1. Within the formulated renormalization procedure near d=1 the dynamical critical exponent is obtained exactly and the correlation length exponent is calculated as a Laurent series in the parameter \sqrt{\epsilon}, with \epsilon=d-1: z=d, \nu=1/\sqrt{3\epsilon} for the short range, and z=1, \nu=\sqrt{2/3\epsilon}, for the long-range Coulomb interaction between bosons. The identified critical point should be stable against the residual perturbations in the effective action for the superfluid, at least in dimensions 1\leq d \leq 2, for both short-range and Coulomb interactions. For the superfluid-Mott insulator transition in the system in a periodic potential and at a commensurate density of bosons I find \nu=(1/2\sqrt{\epsilon})+ 1/4+O(\sqrt{\epsilon}), which yields a result reasonably close to the known XY critical exponent in d=2+1. The critical behavior of the superfluid density, phonon velocity and the compressibility in the system with the short-range interactions is discussed.

A quantitative study of the piezooptical activity in GaAs

The optical activity induced by uniaxial stress in GaAs is shown to be an order of magnitude smaller than previously reported. The former over-estimation is explained by birefringence effects related to tiny misalignments in the stress, which we properly separated from the true optical activity using phase-sensitive transmission experiments. The new experimental results are compared to a simple critical point analysis and to full band structure empirical pseudopotential calculations. We find good agreement for the sign, the absolute value and the trends in the dispersion of the optical activity. Attention is paid to the role of the interband deformation potentials which determine the sign of the gyrotropic effects. A sign conflict, present in former calculations, is solved.

Element behavior in post-critical plane frame analysis

Depending on the type of uni-modal (cuspoid) catastrophe which governs the local behavior of a structure at a simple critical point, correct prediction of the post-critical path requires an accurate estimation of a number of terms in the Taylor series expansion of the total potential function at the critical point. This imposes certain conditions on the accuracy of the adopted non-linear elements. These conditions are mainly related to the quality of the kinematic assumptions which underlie the strain definition used in the model. If these assumptions are not accurate enough, the fourth and higher order terms in the Taylor expansion of the total potential function will not be correctly represented. As a consequence, the model will fail to predict correctly even the initial post-buckling behavior whenever the criticality at hand is more complex than the fold (limit point, asymmetric bifurcation). This proves to be the case for the so-called ‘technical’ beam models. This inability is inherited by any constructed finite element model, regardless of the interpolation function used in its definition and of the number of elements used in the discretization. An improved non-linear model based on the treatment by Antman (Bifurcation problems in non-linearly elastic structures, in: P.H. Rabinowitz, ed., Application of Bifurcation Theory, Academic Press, NY, 1977), is adopted, and several finite elements are developed on the basis of this model. These elements are tested for a number of problems for which the critical behavior is governed by fold, cusp and butterfly singularities. The numerical results outline the importance of the ‘small’ kinematic terms, especially in conjunction with the occurrence of higher-order uni-modal singularities. Some risks for numerical locking are pointed out and remedied. The recommended element leads to a computational cost, which is comparable to an implemented ‘shallow arch’ element.