Critical points and symmetries of a free energy function for biaxial nematic liquid crystals

We describe a general model for the free energy function for a homogeneous medium of mutually interacting molecules, based on the formalism for a biaxial nematic liquid crystal set out by Katriel {\em et al.} (1986) in an influential paper in {\em Liquid Crystals} {\bf 1} and subsequently called the KKLS formalism. The free energy is expressed as the sum of an entropy term and an interaction (Hamiltonian) term. Using the language of group representation theory we identify the order parameters as averaged components of a linear transformation, and characterise the full symmetry group of the entropy term in the liquid crystal context as a wreath product $SO(3)\wr Z_2$. The symmetry-breaking role of the Hamiltonian, pointed out by Katriel {\em et al.}, is here made explicit in terms of centre manifold reduction at bifurcation from isotropy. We use tools and methods of equivariant singularity theory to reduce the bifurcation study to that of a $D_3\,$-invariant function on ${\bf R}^2$, ubiquitous in liquid crystal theory, and to describe the 'universal' bifurcation geometry in terms of the superposition of a familiar swallowtail controlling uniaxial equilibria and another less familiar surface controlling biaxial equilibria. In principle this provides a template for {\em all} nematic liquid crystal phase transitions close to isotropy, although further work is needed to identify the absolute minima that are the critical points representing stable phases.Comment: 74 pages, 17 figures : submitted to Nonlinearit

Elementary catastrophes underlying bifurcations of vector fields and PDEs

A practical method was proposed recently for finding local bifurcation points in an n-dimensional vector field F by seeking their ‘underlying catastrophes’. Here we apply the idea to a partial differential equation as an example of the role that catastrophes can play in reaction diffusion. What are these ‘underlying’ catastrophes? We then show they essentially define a restricted class of ‘solvable’ rather than ‘all classifiable’ singularities, by identifying degenerate zeros of a vector field F without taking into account its vectorial character. As a result they are defined by a minimal set of r analytic conditions that providea practical means to solve for them, and a huge reduction from the calculations needed to classify a singularity, which we will also enumerate here. In this way, underlying catastrophes seem to allow us apply Thom’s elementary catastrophes in much broader contexts

Elementary catastrophes underlying bifurcations of vector fields and PDEs

A practical method was proposed recently for finding local bifurcation points in an $n$-dimensional vector field $F$ by seeking their `{\it underlying catastrophes}'. Here we apply the idea to a partial differential equation as an example of the role that catastrophes can play in reaction diffusion. What are these `underlying' catastrophes? We then show they essentially define a restricted class of `solvable' rather than `all classifiable' singularities, by identifying degenerate zeros of a vector field $F$ without taking into account its vectorial character. As a result they are defined by a minimal set of $r$ analytic conditions that provide a practical means to solve for them, and a huge reduction from the calculations needed to classify a singularity, which we will also enumerate here. In this way, {\it underlying catastrophes} seem to allow us apply Thom's {\it elementary catastrophes} in much broader contexts

Percolation, statistical topography, and transport in random-media

A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential psi(x). For rapidly decaying correlations of psi, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.Physic

Universality in Non-Equilibrium Quantum Systems

The phenomenon of universality is one of the most striking in many-body physics. Despite having sometimes wildly different microscopic constituents, systems can nonetheless behave in precisely the same way, with only the variable names interchanged. The canonical examples are those of liquid boiling into vapor and quantum spins aligning into a ferromagnet; despite their obvious differences, they nonetheless both obey quantitatively the same scaling laws, and are thus in the same universality class. Remarkable though this is, universality is generally a phenomenon limited to thermodynamic equilibrium, most commonly present at transitions between different equilibrium phases. Once out of equilibrium, the fate of universality is much less clear. Can strongly non-equilibrium systems behave universally, and are their universality classes different from those familiar from equilibrium? How is quantum mechanics important? This dissertation attempts to address these questions, at least in a small way, by showing and analyzing universal phenomena in several classes of non-equilibrium quantum systems.Comment: PhD thesis in physics submitted to the University of California, Berkeley, Summer 2020; 132 pages; based upon prior works arXiv:1909.05251 [Phys. Rev. Lett. 123, 246603 (2019)], arXiv:1906.11253 [Phys. Rev. Lett. 123, 230604 (2019)], arXiv:1807.09767 [Phys. Rev. B 98, 174203 (2018)], arXiv:1803.00019 [PNAS 115 (38) 9491-9496 (2018)], arXiv:1701.05899 [Phys. Rev. Lett. 118, 260602 (2017)

Approche liens de valence de la physique de basse énergie des systèmes antiferromagnétiques

L'objet de cette thèse est le traitement du modèle de Heisenberg antiferromagnétique dans la base de liens de valence, qui permet d'en décrire la physique de basse énergie. Le manuscrit est organisé en deux parties : dans la première nous utilisons le concept de fidélité afin de détecter les transitions de phases quantiques. Nous démontrons notamment que cette quantité est accessible dans un algorithme de Monte Carlo quantique, formulé dans la base de liens de valence, permettant ainsi de calculer la fidélité sur des systèmes de grande taille. La deuxième partie vise à développer l'idée initiale de Rokhsar et Kivelson, qui a pour but de transformer un modèle de Heisenberg en un modèle de dimères quantiques, généralement moins complexe d'un point de vue numérique. Après une dérivation rigoureuse, cette technique est appliquée au réseau kagomé et permet d'établir l'existence d'un point tricritique au voisinage du modèle initial. La même méthode est ensuite utilisée afin de traiter le modèle J1-J2-J3 sur le réseau hexagonal et démontre l'existence d'une phase plaquette dans un domaine de paramètres déterminé.The purpose of this thesis is the treatment of the antiferromagnetic Heisenberg model within the valence bond basis, allowing for the description of its low-energy physics. The manuscript is organized in two parts: In the first part we utilize the fidelity concept in order to detect quantum phase transitions. It is notably shown, that this quantity is accessible in a valence bond projector quantum Monte Carlo algorithm, making available the fidelity approach to large scale simulations. The second part is devoted to the generalization of an idea going back to Rokhsar and Kivelson, which aims to map a Heisenberg model onto a numerically less demanding quantum dimer model. Starting from a rigourous derivation, the method is then applied to the kagomé lattice and allows to establish the existence of a tricritical point in the vicinity of the original model. The same technique is also used to treat the J1-J2-J3 Heisenberg model on the honeycomb lattice, showing the existence of a plaquette phase in a determined parameter regime

Characterization of a metal-extracting water-poor microemulsion

Solvent extraction is the key separation method in hydrometallurgy. With proper conditioning of such a system, the transfer of metal-species from an aqueous phase into an organic phase can be triggered, either the enrich a desired metal species or to remove undesired impurities. On a laboratory scale, many powerful extractants have been synthesized that enable to selectively and efficiently select any desired metal species. However, application on an industrial scale is yet out of grasp, as fundamental principles of solvent extraction formulation are yet poorly understood. One key problem that is encountered by engineers is the formation of undesired phases (e.g. liquid crystals, emulsification and third-phase formation) and no model is available that allows to predict the occurrence of these phases. This work is dedicated to elaborate the macroscopic phase behaviour of solvent extraction systems by systematic screening of the macroscopic properties of a reference model. Therefore, we use HDEHP as extractant molecule and its sodium salt represents the extractant engaged in complex formation and how solvent penetration plays a crucial role. Establishing a phase prism, where the Z-axis represents ratio between HDEHP and NaDEHP, we are able to employ different cuts, known from surfactant science. These allow to deduce the properties of the complexes (also referred to as reverse micelles) on a nanoscopic level. The major revelation is that two different types of phase separation have been identified: an emulsification failure, where the interior solvent (water, that is co-solubilized inside the reverse micelles) is rejected from the extracting organic phase. Second, a liquid-gas type of phase separation, where the exterior solvent (the organic diluent) is repelled, resulting in two organic phases, where the heavy one carries all the extractants and the light one is purely the organic diluent. This second type needs to be avoided in solvent extraction formulation. In the second part of this work and based on the phase diagrams determined in the first part, the conductivity profile is extensively screened in monophasic regions. Three limits have been determined to play a major role on the conducting properties of a water-poor microemulsion: the critical aggregation concentration (CAC); the degree to which the head-groups are hydrated: transition from reverse micelles, where water is immobilized as it hydrates the extractant head-groups, towards swollen reverse micelles, exhibiting a liquid water-core; and the percolation threshold. In total, this gives 5 different regions which have dissimilar conducting properties