1D Cahn-Hilliard dynamics : coarsening and interrupted coarsening

Many systems exhibit a phase where the order parameter is spatially modulated. These patterns can be the result of a frustration caused by the competition between interaction forces with opposite effects. In all models with local interactions, these ordered phases disappear in the strong segregation regime (low temperature). It is expected however that these phases should persist in the case of long range interactions, which can't be correctly described by a Ginzburg-Landau type model with only a finite number of spatial derivatives of the order parameter. An alternative approach is to study the dynamics of the phase transition or pattern formation. While, in the usual process of Ostwald ripening, succession of doubling of the domain size leads to a total segregation, or macro-segregation, C. Misbah and P. Politi have shown that long-range interactions could cause an interruption of this coalescence process, stabilizing a pattern which then remains in a micro-structured state or super-crystal. We show that this is the case for a modified Cahn-Hilliard dynamics due to Oono which includes a non local term and which is particularly well suited to describe systems with a modulated phase

The Lamellar-Disorder Interface : One-Dimensional Modulated Profiles

We study interfacial behavior of a lamellar (stripe) phase coexisting with a disordered phase. Systematic analytical expansions are obtained for the interfacial profile in the vicinity of a tricritical point. They are characterized by a wide interfacial region involving a large number of lamellae. Our analytical results apply to systems with one dimensional symmetry in true thermodynamical equilibrium and are of relevance to metastable interfaces between lamellar and disordered phases in two and three dimensions. In addition, good agreement is found with numerical minimization schemes of the full free energy functional having the same one dimensional symmetry. The interfacial energy for the lamellar to disordered transition is obtained in accord with mean field scaling laws of tricritical points.Comment: 12 pages, 8 figure

Tails of Localized Density of States of Two-dimensional Dirac Fermions

The density of states of Dirac fermions with a random mass on a two-dimensional lattice is considered. We give the explicit asymptotic form of the single-electron density of states as a function of both energy and (average) Dirac mass, in the regime where all states are localized. We make use of a weak-disorder expansion in the parameter g/m^2, where g is the strength of disorder and m the average Dirac mass for the case in which the evaluation of the (supersymmetric) integrals corresponds to non-uniform solutions of the saddle point equation. The resulting density of states has tails which deviate from the typical pure Gaussian form by an analytic prefactor.Comment: 8 pages, REVTeX, 1 eps figure; to appear in Annalen der Physi

1D Cahn-Hilliard equation: Ostwald ripening and modulated phase systems

Using an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition, we compute the characteristic time for one step of period doubling in Langer's self similar scenario for Ostwald ripening. As an application, we compute the thermodynamically stable period of a 1D modulated phase pattern

Diluted planar ferromagnets: nonlinear excitations on a non-simply connected manifold

We study the behavior of magnetic vortices on a two-dimensional support manifold being not simply connected. It is done by considering the continuum approach of the XY-model on a plane with two disks removed from it. We argue that an effective attractive interaction between the two disks may exist due to the presence of a vortex. The results can be applied to diluted planar ferromagnets with easy-plane anisotropy, where the disks can be seen as nonmagnetic impurities. Simulations are also used to test the predictions of the continuum limit.Comment: 5 pages, 6 figure

Studying nonlinear effects on the early stage of phase ordering using a decomposition method

Nonlinear effects on the early stage of phase ordering are studied using Adomian's decomposition method for the Ginzburg-Landau equation for a nonconserved order parameter. While the long-time regime and the linear behavior at short times of the theory are well understood, the onset of nonlinearities at short times and the breaking of the linear theory at different length scales are less understood. In the Adomian's decomposition method, the solution is systematically calculated in the form of a polynomial expansion for the order parameter, with a time dependence given as a series expansion. The method is very accurate for short times, which allows to incorporate the short-time dynamics of the nonlinear terms in a analytical and controllable way.Comment: 11 pages, 1 figure, to appear in Phys Lett

Interfaces and Grain Boundaries of Lamellar Phases

Interfaces between lamellar and disordered phases, and grain boundaries within lamellar phases, are investigated employing a simple Landau free energy functional. The former are examined using analytic, approximate methods in the weak segregation limit, leading to density profiles which can extend over many wavelengths of the lamellar phase. The latter are studied numerically and exactly. We find a change from smooth chevron configurations typical of small tilt angles to distorted omega configurations at large tilt angles in agreement with experiment.Comment: 9 pages, 6 figures 9 pages, 6 figure

Coalescence in the 1D Cahn-Hilliard model

We present an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition. We have identified all the intermediate profiles, stationary solutions of the noiseless Cahn-Hilliard equation. Using properties of the soliton lattices, periodic solutions of the Ginzburg-Landau equation, we have construct a family of ansatz describing continuously the processus of destabilization and period doubling predicted in Langer's self similar scenario

Magnetic vortex-like excitations on a sphere

We study magnetic vortex-like solutions lying on the spherical surface. The simplest cylindrically symmetric vortex presents two cores (instead of one, like in open surfaces) with same charge, so repealing each other. However, the net vorticity is computed to vanish in accordance with Gauss theorem. We also address the problem of a flat plane in which a conical, a pseudospherical and a hemispherical segments were incorporated. In this case, if we allow the vortex to move without appreciable deformation in this support, then it is attracted by the conical apex and by the pseudosphere as well, while it is repealed by the hemisphere. This suggests that such surfaces could be viewed as pinning and depinning geometries for those excitations. Spherical harmonics coreless solutions are discussed within some details.Comment: 15 pages, 8 .eps figures, typed in tex. Version accepted in Physics Letters A (2007), please see DOI

A Uniform Approach to Antiferromagnetic Heisenberg Spins on Low Dimensional Lattices

Using group theoretical methods we show for both the triangular and square lattices that in the continuum limit the antiferromagnetic order parameter lives on SO3 without respect of the initial lattice. For the antiferromagnetic chain we recover the Haldane decomposition. This order parameter interacts with a local gauge field rather than with a global one as implicitly suggested in the literature which in our approach appears in a rather natural manner. In fact this merely corresponds to a novel extension of the spin group by a local gauge field. This analysis based on the real division algebras applies to low dimensional lattices.Comment: 5 pages; REVTeX