Search Results (9)

We briefly review the early development of the mean-field dynamics for cooperatively interacting quantum many-body systems, mapped to pseudo-spin (Ising-like) systems. We start with (Anderson, 1958) pseudo-spin mapping the BCS (1957) Hamiltonian of superconductivity, reducing it to a mean-field Hamiltonian of the XY (or effectively Ising) model in a transverse field. Then, we obtain the mean-field estimate for the equilibrium gap in the ground-state energy at different temperatures (gap disappearing at the transition temperature), which fits Landau’s (1949) phenomenological theory of superfluidity. We then present in detail a general dynamical extension (for non-equilibrium cases) of the mean-field theory of quantum Ising systems (in a transverse field), following de Gennes’ (1963) decomposition of the mean field into the orthogonal classical cooperative (longitudinal) component and the quantum (transverse) component, with each of the component following Suzuki–Kubo (1968) mean-field dynamics. Next, we discuss its applications to quantum hysteresis in Ising magnets (in the presence of an oscillating transverse field), to quantum heat engines (employing the transverse Ising model as a working fluid), and to the quantum annealing of the Sherrington–Kirkpatrick (1975) spin glass by tuning down (to zero) the transverse field, which provides us with a very fast computational algorithm, leading to ground-state energy values converging to the best-known analytic estimate for the model. Finally, we summarize the main results obtained and draw conclusions about the effectiveness of the de Gennes–Suzuki–Kubo mean-field equations for the study of various dynamical aspects of quantum condensed matter systems. Full article

The switching properties of nematic liquid crystals under electrical and mechanical stresses play a fundamental role in the design and fabrication of electro-optical devices. Depending on the stress applied to a nematic texture confined in a pi-cell, different nematic configurations are allowed inside the cell, while the induced distortion is relaxed by means of growing biaxial domains which can end with the order reconstruction phenomenon, a transition connecting two topologically different nematic textures which can occur in different regions of the pi-cell. Due to the different space and time scales involved, modelling in the frame of the Landau–de Gennes order tensor theory is mandatory to correctly describe the fast-switching mechanisms involved, while from a computational point of view, sophisticated numerical techniques are required to grasp tiny and fast features which can be predicted by the mathematical modelling. In this paper, we review the results obtained from the mathematical and numerical modelling of a 5CB liquid crystal confined in a pi-cell performed by using a numerical technique based on the equidistribution principle, tailored for the description of a complex physical system in which fast switching phenomena are coupled with strong distortions. After a recap on the underneath theory and on the numerical method, we focus on the switching properties of the nematic material when subjected to variable mechanical and electrical stresses in both symmetric and asymmetric conditions. Full article

Based on two-band Bogoliubov–de Gennes theory, we study the boundary effect of an interface between a two-gap superconductor FeSe and insulator (or vacuum). New boundary terms are introduced into two-band Ginzburg–Landau formalism, which modifies the boundary conditions for the corresponding order parameters of superconductor. The theory allows for a mean-field calculation of the critical temperature suppression with the decrease in FeSe film thickness. Our numerical results are in good agreement with the experimental data observed in this material. Full article

Intense electric fields applied to an asymmetric π-cell containing a nematic liquid crystal subjected to strong mechanical stresses induce distortions that are relaxed through a fast-switching mechanism: the order reconstruction transition. Topologically different nematic textures are connected by such a mechanism that is spatially driven by the intensity of the applied electric fields and by the anchoring angles of the nematic molecules on the confining plates of the cell. Using the finite element method, we implemented the moving mesh partial differential equation numerical technique, and we simulated the nematic evolution inside the cell in the context of the Landau–de Gennes order tensor theory. The order dynamics have been well captured, putting in evidence the possible existence of a metastable biaxial state, and a phase diagram of the nematic texture has been built, therefore confirming the appropriateness of the used technique for the study of this type of problem. Full article

We consider temperature-driven phase transitions occurring in a liquid crystal suspension of ferromagnetic particles within the Landau–de Gennes theory. The temperature dependences of the order parameters in the uniaxial model with a vector order parameter for the magnetic subsystem are obtained. The dimensionless expression for the free energy density of the suspension has been used for the study of the phase behavior general regularities of the system. Phase state diagrams of the suspension and temperature dependences of the order parameters of the liquid crystal and the ensemble of magnetic particles for different values of the phenomenological expansion coefficients are constructed. It is shown that the considered model admits the existence of a cascade of temperature phase transitions: isotropic phase–superparamagnetic nematic phase–ferromagnetic nematic phase. We have shown that in the mesomorphic state of the liquid crystal, the spontaneous magnetization can appear in a continuous way or by a jump with decreasing temperature, which corresponds to the tricritical behavior. The values of temperature and expansion coefficients corresponding to the tricritical and triple points are numerically found. Full article

The dynamics of a disclination loop (s = ±1/2) in nematic liquid crystals constrained between two coaxial cylinders were investigated based on two-dimensional Landau–de Gennes tensorial formalism by using a finite-difference iterative method. The effect of thickness (d = R₂ − R₁, where R₁ and R₂ represent the internal and external radii of the cylindrical cavity, respectively) on the director distribution of the defect was simulated using different R₁ values. The results show that the order reconstruction occurs at a critical value of d_c, which decreases with increasing inner ratio R₁. The loop also shrinks, and the defect center deviates from the middle of the system, which is a non-planar structure. The deviation decreases with decreasing d or increasing R₁, implying that the system tends to be a planar cell. Two models were then established to analyze the combined effect of non-planar geometry and electric field. The common action of these parameters facilitates order reconstruction, whereas their opposite action complicates the process. Full article

At the wall in a hybrid nematic cell with strong anchoring, the nematic director is parallel to one wall and perpendicular to the other. Within the Landau-de Gennes theory, we have investigated the dynamics of s = ±1/2 wedge disclinations in such a cell, using the two-dimensional finite-difference iterative method. Our results show that with the cell gap decreasing, the core of the defect explodes, and the biaxiality propagates inside the cell. At a critical value of d_c* ≈ 9ξ (where ξ is the characteristic length for order-parameter changes), the exchange solution is stable, while the defect core solution becomes metastable. Comparing to the case with no initial disclination, the value at which the exchange solution becomes stable increases relatively. At a critical separation of d_c ≈ 6ξ, the system undergoes a structural transition, and the defect core merges into a biaxial layer with large biaxiality. For weak anchoring boundary conditions, a similar structural transition takes place at a relative lower critical value. Because of the weakened frustration, the asymmetric boundary conditions repel the defect to the weak anchoring boundary and have a relatively lower critical value of d_a, where the shape of the defect deforms. Further, the response time between two very close cell gaps is about tens of microseconds, and the response becomes slower as the defect explodes. Full article

The Landau–de Gennes theory provides a successful macroscopic description of nematics. Cornerstone of this theory is a phenomenological expression for the effective free energy as a function of the orientational order parameter. Here, we show how such a macroscopic Landau–de Gennes free energy can systematically be constructed for a microscopic model of liquid crystals formed by interacting mesogens. For the specific example of the Gay–Berne model, we obtain an enhanced free energy that reduces to the familiar Landau–de Gennes expression in the limit of weak ordering. By carefully separating energetic and entropic contributions to the free energy, our approach reconciles the two traditional views on the isotropic–nematic transition of Maier–Saupe and Onsager, attributing the driving mechanism to attractive interactions and entropic effects, respectively. Full article

Biological liquid crystalline polymers are found in cellulosic, chitin, and DNA based natural materials. Chiral nematic liquid crystalline orientational order is observed frozen-in in the solid state in plant cell walls and is known as a liquid crystal analogue characterized by a helicoidal plywood architecture. The emergence of the plywood architecture by directed chiral nematic liquid crystalline self assembly has been postulated as the mechanism that leads to optimal cellulose fibril organization. In natural systems, tissue growth and development takes place in the presence of inclusions and secondary phases leaving behind characteristic defects and textures, which provide a unique testing ground for the validity of the liquid crystal self-assembly postulate. In this work, a mathematical model, based on the Landau-de Gennes theory of liquid crystals, is used to simulate defect textures arising in the domain of self assembly, due to presence of secondary phases representing plant cells, lumens and pit canals. It is shown that the obtained defect patterns observed in some plant cell walls are those expected from a truly liquid crystalline phase. The analysis reveals the nature and magnitude of the viscoelastic material parameters that lead to observed patterns in plant-based helicoids through directed self-assembly. In addition, the results provide new guidance to develop biomimetic plywoods for structural and functional applications. Full article