Related papers: Geodesics in Information Geometry : Classical and …
We study the behaviour of geodesics on a Riemannian manifold near a generalized conical or cuspidal singularity. We show that geodesics entering a small neighbourhood of the singularity either hit the singularity or approach it to a…
Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry.…
We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of…
We analyze the geometry of the ground-state manifold in parameter-dependent many-body systems with quantum phase transitions (QPTs) and describe finite-size precursors of the singular geometry emerging at the QPT boundary in the…
Several physical problems such as the `twin paradox' in curved spacetimes have purely geometrical nature and may be reduced to studying properties of bundles of timelike geodesics. The paper is a general introduction to systematic…
The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at…
This study targets quantum phases which are characterized by topological properties and no associated with the symmetry breaking. We concern ourselves primarily with the transitions among these quantum phases. This type of quantum phase…
The state of a quantum system, adiabatically driven in a cycle, may acquire a measurable phase depending only on the closed trajectory in parameter space. Such geometric phases are ubiquitous, and also underline the physics of robust…
This paper surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical…
Geometric phases of simple harmonic oscillator system are studied. Complete sets of "eigenfunctions" are constructed, which depend on the way of choosing classical solutions. For an eigenfunction, two different motions of the probability…
This article first gives a concise introduction to quantum phase transitions, emphasizing similarities with and differences to classical thermal transitions. After pointing out the computational challenges posed by quantum phase…
We introduce a quantitative condition on orbits of dynamical systems which measures their aperiodicity. We show the existence of sequences in the Bernoulli-shift and geodesics on closed hyperbolic manifolds which are as aperiodic as…
We study the geometric phase of an open two-level quantum system under the influence of a squeezed, thermal environment for both non-dissipative as well as dissipative system-environment interactions. In the non-dissipative case, squeezing…
This paper is a review on recently found connection between geodesically equivalent metrics and integrable geodesic flows. Suppose two different metrics on one manifold have the same geodesics. We show that then the geodesic flows of these…
We use the system of p-adic numbers for the description of information processes. Basic objects of our models are so called transformers of information, basic processes are information processes, the statistics are information statistics…
The relation between thermodynamic phase transitions in classical systems and topology changes in their state space is discussed for systems in which equivalence of statistical ensembles does not hold. As an example, the spherical model…
The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant…
Quantum phase transitions encompass a variety of phenomena that occur in quantum systems exhibiting several possible symmetries. Traditionally, these transitions are explored by continuously varying a control parameter that connects two…
The possibility of realization of quantum gates by means of the non-adiabatic geometric phase is considered. It is shown that the non-adiabatic phase can be used for quantum gates realization as well as the adiabatic one.
In this paper we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum frame work. We apply this formulation to the…