Two-dimensional Ginzburg-Landau Model Research Articles

Global Jacobian and Γ-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices

In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a 2D map u. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a 2D Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of u at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by Γ-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices.

From bulk descriptions to emergent interfaces: Connecting the Ginzburg-Landau and elastic-line models

Controlling interfaces is highly relevant from a technological point of view. However, their rich and complex behavior makes them very difficult to describe theoretically, and hence to predict. In this work, we establish a procedure to connect two levels of descriptions of interfaces: for a bulk description, we consider a two-dimensional Ginzburg-Landau model evolving with a Langevin equation, and boundary conditions imposing the formation of a rectilinear domain wall. At this level of description no assumptions need to be done over the interface, but analytical calculations are very difficult to handle, especially for disordered systems. On a different level of description, we consider a one-dimensional elastic line model evolving according to the Edwards-Wilkinson equation, which only allows one to study continuous and univalued interfaces, but which was up to now one of the most successful tools to treat interfaces analytically. To establish the connection between the bulk description and the interface description, we propose a simple method which has the advantage to be readily applicable to disordered systems. We probe the connection by numerical simulations at both levels for clean and disordered systems, and our simulations, in addition to making contact with experiments, allow us to test and provide insight to develop new analytical approaches to treat interfaces.

A universal thin film model for Ginzburg–Landau energy with dipolar interaction

We present an analytical treatment of a three-dimensional variational model of a system that exhibits a second-order phase transition in the presence of dipolar interactions. Within the framework of Ginzburg-Landau theory, we concentrate on the case in which the domain occupied by the sample has the shape of a flat thin film and obtain a reduced two-dimensional, non-local variational model that describes the energetics of the system in terms of the order parameter averages across the film thickness. Namely, we show that the reduced two-dimensional model is in a certain sense asymptotically equivalent to the original three-dimensional model for small film thicknesses. Using this asymptotic equivalence, we analyze two different thin film limits for the full three-dimensional model via the methods of $\Gamma$-convergence applied to the reduced two-dimensional model. In the first regime, in which the film thickness vanishes while all other parameters remain fixed, we recover the local two-dimensional Ginzburg-Landau model. On the other hand, when the film thickness vanishes while the sample's lateral dimensions diverge at the right rate, we show that the system exhibits a transition from homogeneous to spatially modulated global energy minimizers. We identify a sharp threshold for this transition.

Continuous generation of dissipative spatial solitons in two-dimensional Ginzburg–Landau models with elliptical shaped potentials

We report on the rich dynamics of two-dimensional fundamental solitons coupled and interacting on the top of an elliptical shaped potential in a two-dimensional Ginzburg–Landau model. Under the elliptical shaped potential, the solitons display unique and dynamic properties, such as the generation of straight-line arrays, emission of either one elliptical shaped soliton or several elliptical ring soliton arrays, and soliton decay. When changing the depth and sharpness of the external potential and fixing other parameters of the potential, various scenarios of soliton dynamics are also revealed. These results suggest some possible applications for all-optical data-processing schemes, such as the routing of light signals in optical communication devices.

Maze energetics revealed by a large-scale two-dimensional Ginzburg-Landau type simulation

To understand the magnetization process of permanent magnets, the structural analysis of a so-called maze pattern plays a key role, because demagnetized states consist of maze domains. Motivated by this viewpoint, we explore the phase space of a magnetic order parameter in a thin film and clarify the energetics of various nonlinear states including the mazes, based on a two-dimensional Ginzburg-Landau model augmented by long-range dipole-dipole interaction

Vortices and pinning effects for the Ginzburg-Landau model in multiply connected domains

We consider the two-dimensional Ginzburg-Landau model with magnetic fi eld for a superconductor with a multiply connected cross section. We study energy minimizers in the London limit as the Ginzburg-Landau parameter κ = 1/ǫ → ∞ to determine the number and asymptotic location of vortices. We show that the holes act as pinning sites, acquiring nonzero winding for bounded fields and attracting all vortices away from the interior for fields up to a critica l value hex = O(|ln ǫ|). At the critical level the pinning effect breaks down, and vortices appear in the interior of the superconductor at locations that we id entify explicitly in terms of the solutions of an elliptic boundary value problem. The method involves sharp upper and lower energy estimates, and a careful analysis of the limiting problem that captures the interaction between the vortices and the holes. c 2005 Wiley Periodicals, Inc.

ON THE SECOND CRITICAL FIELD FOR A GINZBURG–LANDAU MODEL WITH FERROMAGNETIC INTERACTIONS

We consider a two-dimensional Ginzburg–Landau model for superconductors which exhibit ferromagnetic ordering in the superconducting phase, introduced by physicists to describe unconventional p-wave superconductors. In this model the magnetic field is directly coupled to a vector-valued order parameter in the energy functional. We show that one effect of spin coupling is to increase the second critical field Hc2, the value of the applied magnetic field at which superconductivity is lost in the bulk. Indeed, when the spin coupling is strong we show that the upper critical field is no longer present, confirming predictions in the physics literature. We treat the energy density as a measure, and show that the order parameter converges (as the Ginzburg–Landau parameter κ→∞) in an average sense to a constant determined by the average energy.

Simulation of boundary condition influence in a second-order ferroelectric phase transition

Using a two-dimensional Ginzburg–Landau model and the finite-element computational method, we have calculated stable domain configurations resulting from a second-order ferroelectric phase transition for a finite-sized system. The boundary conditions applied here correspond to fully charge compensated situations, either by surface electrodes or by the injection of charges (or defects) near the sample surface. The domain wall thickness of a finite system without surface electrodes was found to become thinner as it approaches sample surfaces. This is distinctively different from that of an infinite system for which a planar wall assumption can be used. The orientation of the macroscopic polarization of a finite system without surface electrodes was found to be determined by its aspect ratio. A size effect was observed when all the dimensions were reduced simultaneously. The relaxation process in the formation of domains and the switching process have also been simulated for charge neutral boundary conditions using a time dependent Ginzburg–Landau model. The simulation results verified that the surfaces are the favored nucleation sites for domain switching.

Possible first-order transition in the two-dimensional Ginzburg-Landau model induced by thermally fluctuating vortex cores

We study the two-dimensional Ginzburg-Landau model of a neutral superfluid in the vicinity of the vortex unbinding transition. The model is mapped onto an effective interacting vortex gas by a systematic perturbative elimination of all fluctuating degrees of freedom (amplitudeand phase of the order parameter field) except the vortex positions. In the Coulomb gas descriptions derived previously in the literature, thermal amplitude fluctuations were neglected altogether. We argue that if one includes the latter, the vortices still form a two-dimensional Coulomb gas, but the vortex fugacity can be substantially raised. Under the assumption that Minnhagen's generic phase diagram of the two-dimensional Coulomb gas is correct, our results then point to a first-order transition rather than a Kosterlitz-Thouless transition, provided the Ginzburg-Landau correlation length is large enough in units of a microscopic cutoff lenght for fluctuations. The experimental relevance of these results is briefly discussed.