This research topic started with the Ph.D. thesis of S. Casado (my first graduate student), and currently is regarding to creation of defects when a breaking symmetry instability threshold is crossed over (for hydrodynamical and optical systems). We make comparison with cosmological phenomena.

The results presented here, have been mostly reported in some of the publications cited ⇒ here ⇐

Introduction

The etymology of the English word "defect" in the vernacular always carries a negative meaning. Either as an impairing worth or utility, or as a lack for completeness, adequacy, or perfection. From the point of view of physics, a defect could be defined as a localized state which has less symmetries than the linearly stable global state. Thus, the global state has a (locally) broken symmetry due to the existence of the defect. The topological properties of defects can be studied through their homotopy group [G. Tolouse et al. J. Physique Lett. 37 (1976) 149] [M. Kléman "Points, Lignes, Parois" (1977) Editions de Physique] [N.D. Mermin Rev. Mod. Phys. 51 (1979)] [L. Michel Rev. Mod. Phys. 52 (1980) 617] [M. Berry Proc. R. Soc. Lond. A 392 (1984) 45]

The appearance of topological defects is related to bifurcations between patterns with symmetry breaking. When crossing one of such bifurcations, phase singularities (the so called topological defect) appear as relics of the more symmetric state. Although the origin of the topological defects is of a higher symmetry, they are localized states stabilized by topological constraints, which diminish the symmetry of the defect, in such a way that it becomes even less symmetric that the globally stable state [A. Joets et al. J. Statis. Phys. 64 (1991) 981] [R. Ribotta et al. in Geometry and Topology of Caustics - Caustics '02 (2004) 223. Ed. by I. of Math., Polish Academy of Sciences].

Some important mechanisms in the appearance of defects in patterns are the following: geometrical constraints, thermal effects, nonlinear effects, and symmetry changing transitions.

In a symmetry breaking transition where two equivalent domains could grow in the more symmetric state, it may lead to a lack of "phase-matching" for the less symmetric domains. The phase is the parameter corresponding to the symmetry which is broken in the transition and could lead to the two (or more) equivalent domains. The localized state with even less symmetries appearing in the region where the lack of "phase-matching" occurs, is a topological defect.

The (topological) defects can be detected and characterized through image analysis using several methods including Voronoi analysis [G. Voronoi J. Reine Angew. Math. 134 (1908) 198] or complex demodulation [P. Bloomfield "Fourier Analysis of Time Series: An Introduction" (1976) Wiley] or by simple visual inspection.

Collaborators

(in alphabetical order, all levels of collaboration: in the lab, permanent collaboration, temporary collaboration)

• Stefano Boccaletti

• Javier Burguete

• Sergio Casado

• Héctor L. Mancini

• Montserrat A. Miranda

• Pier Luigi Ramazza

• Roland Ribotta

Results

The results reported here correspond to the experimental verification of the Zurek-Kibble mechanism [W.H. Zurek Phys. Rep. 276 (1996) 177] [T.W.B. Kibble J. Phys. A 9 (1976) 1387] [A. Rajantie Contemp. Phys. 44 (2003) 485] in bifurcations of out-of-equilibrium systems in continuum media [M.C. Cross et al. Rev. Mod. Phys. 65 (1993) 851]. The experimental verification of this mechanism in condensed matter phase transitions can be found in the references of our work. It is possible to find an extensive review of these results in our work

Here we will stress only showing some movies of a realization corresponding to the measurements done in a Kerr-like nonlinear optical medium [S. Residori Phys. Rep. 416 (2005) 201]:

• Case with a translation in the feedback loop. It gives stripes with number of defects depending on the rate of increase of the control parameter. Due to the translation, also there exists a drift of the structure (which has to be taken into account in the measurement process). The drift creates a "perfect" structure by coherent transport. In the movie you can see an example of the process.

• Case without the translation in the feedback loop. It gives an hexagonal structure with number of defects depending on the rate of increase of the control parameter. In the movie you can see an example of the process.

The experiments in convective systems (Rayleigh-Bénard and Bénard-Marangoni) not only comprise the study of the statistics of defects in the bifurcations to verify the Kibble-Zurek mechanism, but also the dynamics of the pattern and defects at the bifurcation itself.

Also, we report results on the equivalent experiments in secondary subcritical bifurcations which occur in convective systems under quasi-one-dimensional heating.