Harmonic and minimal unit vector fields on Riemannian symmetric spaces

We present new examples of harmonic and minimal unit vector fields on Riemannian symmetric spaces. These examples are constructed from cohomogeneity one actions with a reflective singular orbit. The radial unit vector field associated to such a reflective submanifold is harmonic and minimal

Efficient variational contraction of two dimensional tensor networks with a non trivial unit cell

Tensor network states provide an efficient class of states that faithfully capture strongly correlated quantum models and systems in classical statistical mechanics. While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state.Comment: 23 pages, 8 Figure

On Pythagoras' theorem for products of spectral triples

We discuss a version of Pythagoras theorem in noncommutative geometry. Usual Pythagoras theorem can be formulated in terms of Connes' distance, between pure states, in the product of commutative spectral triples. We investigate the generalization to both non pure states and arbitrary spectral triples. We show that Pythagoras theorem is replaced by some Pythagoras inequalities, that we prove for the product of arbitrary (i.e. non-necessarily commutative) spectral triples, assuming only some unitality condition. We show that these inequalities are optimal, and provide non-unital counter-examples inspired by K-homology.Comment: Paper slightly shortened to match the published version; Lett. Math. Phys. 201

Body size and dispersal mode as key traits determining metacommunity structure of aquatic organisms

Relationships between traits of organisms and the structure of their metacommunities have so far mainly been explored with meta-analyses. We compared metacommunities of a wide variety of aquatic organism groups (12 groups, ranging from bacteria to fish) in the same set of 99 ponds to minimise biases inherent to meta-analyses. In the category of passive dispersers, large-bodied groups showed stronger spatial patterning than small-bodied groups suggesting an increasing impact of dispersal limitation with increasing body size. Metacommunities of organisms with the ability to fly (i.e. insect groups) showed a weaker imprint of dispersal limitation than passive dispersers with similar body size. In contrast, dispersal movements of vertebrate groups (fish and amphibians) seemed to be mainly confined to local connectivity patterns. Our results reveal that body size and dispersal mode are important drivers of metacommunity structure and these traits should therefore be considered when developing a predictive framework for metacommunity dynamics