A theorem regarding families of topologically non-trivial fermionic systems

We introduce a Hamiltonian for fermions on a lattice and prove a theorem regarding its topological properties. We identify the topological criterion as a $\mathbb{Z}_2-$ topological invariant $p(\textbf{k})$ (the Pfaffian polynomial). The topological invariant is not only the first Chern number, but also the sign of the Pfaffian polynomial coming from a notion of duality. Such Hamiltonian can describe non-trivial Chern insulators, single band superconductors or multiorbital superconductors. The topological features of these families are completely determined as a consequence of our theorem. Some specific model examples are explicitly worked out, with the computation of different possible topological invariants.Comment: 6 page

WKB Approximation to the Power Wall

We present a semiclassical analysis of the quantum propagator of a particle confined on one side by a steeply, monotonically rising potential. The models studied in detail have potentials proportional to $x^{\alpha}$ for $x>0$; the limit $\alpha\to\infty$ would reproduce a perfectly reflecting boundary, but at present we concentrate on the cases $\alpha =1$ and 2, for which exact solutions in terms of well known functions are available for comparison. We classify the classical paths in this system by their qualitative nature and calculate the contributions of the various classes to the leading-order semiclassical approximation: For each classical path we find the action $S$, the amplitude function $A$ and the Laplacian of $A$. (The Laplacian is of interest because it gives an estimate of the error in the approximation and is needed for computing higher-order approximations.) The resulting semiclassical propagator can be used to rewrite the exact problem as a Volterra integral equation, whose formal solution by iteration (Neumann series) is a semiclassical, not perturbative, expansion. We thereby test, in the context of a concrete problem, the validity of the two technical hypotheses in a previous proof of the convergence of such a Neumann series in the more abstract setting of an arbitrary smooth potential. Not surprisingly, we find that the hypotheses are violated when caustics develop in the classical dynamics; this opens up the interesting future project of extending the methods to momentum space.Comment: 30 pages, 8 figures. Minor corrections in v.

Scene representation and matching for visual localization in hybrid camera scenarios

Scene representation and matching are crucial steps in a variety of tasks ranging from 3D reconstruction to virtual/augmented/mixed reality applications, to robotics, and others. While approaches exist that tackle these tasks, they mostly overlook the issue of efficiency in the scene representation, which is fundamental in resource-constrained systems and for increasing computing speed. Also, they normally assume the use of projective cameras, while performance on systems based on other camera geometries remains suboptimal. This dissertation contributes with a new efficient scene representation method that dramatically reduces the number of 3D points. The approach sets up an optimization problem for the automated selection of the most relevant points to retain. This leads to a constrained quadratic program, which is solved optimally with a newly introduced variant of the sequential minimal optimization method. In addition, a new initialization approach is introduced for the fast convergence of the method. Extensive experimentation on public benchmark datasets demonstrates that the approach produces a compressed scene representation quickly while delivering accurate pose estimates. The dissertation also contributes with new methods for scene matching that go beyond the use of projective cameras. Alternative camera geometries, like fisheye cameras, produce images with very high distortion, making current image feature point detectors and descriptors less efficient, since designed for projective cameras. New methods based on deep learning are introduced to address this problem, where feature detectors and descriptors can overcome distortion effects and more effectively perform feature matching between pairs of fisheye images, and also between hybrid pairs of fisheye and perspective images. Due to the limited availability of fisheye-perspective image datasets, three datasets were collected for training and testing the methods. The results demonstrate an increase of the detection and matching rates which outperform the current state-of-the-art methods