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Topological phases are states of matter which are defined by an underlying topological structure, rather than by a broken symmetry. Many of these phases, in particular the "topological insulators" which were discovered recently, can be described in terms of models of non-interacting electrons. The effect of interactions on these phases remains, to a large extent, an open problem. Here, we discuss a general framework to classify gapped phases of one-dimensional systems, composed of either bosons or fermions with a given sets of symmetries, using concepts of entanglement. Each phase is identified according to the way its entanglement (Schmidt) eigenstates transform under the symmetry operations of the system, and exhibits a characteristic degeneracy of its entanglement spectrum. Examples include the symmetry protection of the Haldane phase of interacting spins or bosons, and topological superconductors in one dimension with time reversal symmetry. In the latter case, interactions are found to change the classification of distinct phases profoundly, relative to the non-interacting case.

In my talk, a coevolutionary model will be presented that incorporates the
logical structure of constitutional chemistry and its kinetics (i.e. the
dynamics) on the one hand and the topological evolution of the chemical
reaction network on the other hand (i.e. the metadynamics). The motivation
for designing this model is twofold. First, experiments that are to provide
insight into chemical problems should be expressed in a syntax that remains
as close as possible to real chemistry. Second, the study of physical
properties of the complex chemical reaction networks requires growing models
that incorporate features realistic from a biochemical perspective. In my
talk, the theory and algorithms underlying the coevolutionary model are
explained, and two illustrative examples, one in astrochemistry and another
one in prebiotic chemistry, are provided. The main lessons coming from these
simulations help to progress on our understanding of:

1) On the dynamics side:
a. The onset of homochirality as a symmetry breaking phenomenon

b. The definition of “autocatalysis” as a kinetic and not structural phenomenon

c. A call to better account for the energetic openness of the chemical systems and its thermodynamic properties

2) On the metadynamics side:
a. The need to be careful in making general claims concerning the structure of chemical reaction networks: random, scale-free, etc…

b. The benefits of relying on object-oriented principles for this kind of simulation

It is widely believed that symmetry-breaking electronic correlations are present in the pseudogap regime of high-Tc superconductors. If such correlations (e.g. stripes) are partially static, one would expect low-frequency noise from thermal switching among different configurations. We have found anomalous noise below a temperature of about 240 K in the normal state of underdoped YBCO and Ca-YBCO films. This noise regime, unlike the more typical noise above 240 K, has features expected for a symmetry-breaking collective electronic state. However, the onset temperature seems to be independent of doping, unlike other pseudogap effects. We speculate on a model involving coupling between stripes and impurities with thermally activated motion.