Klein paradox between transmitted and reflected Dirac waves on Bour surfaces

It is supposed the existence of a curved graphene sheet with the geometry of a Bour surface $B_{n}$, such as the catenoid (or helicoid), $B_{0}$, and the classical Enneper surface, $B_{2}$, among others. In particular, in this work, the propagation of the electronic degrees of freedom on these surfaces is studied based on the Dirac equation. As a consequence of the polar geometry of $B_{n}$, it is found that the geometry of the surface causes the Dirac fermions to move as if they would be subjected to an external potential coupled to a spin-orbit term. The geometry-induced potential is interpreted as a barrier potential, which is asymptotically zero. Furthermore, the behaviour of asymptotic Dirac states and scattering states are studied through the Lippmann-Schwinger formalism. It is found that for surfaces $B_{0}$ and $B_{1}$, the total transmission phenomenon is found for sufficiently large values of energy, while for surfaces $B_{n}$, with $n\geq 2$, it is shown that there is an energy point $E_{K}$ where Klein's paradox is realized, while for energy values $E\gg E_{K}$ it is found that the conductance of the hypothetical material is completely suppressed, $\mathcal{G}(E)\to 0$.Comment: 31 pages, 9 figure

Brownian motion meets Riemann curvature

The general covariance of the diffusion equation is exploited in order to explore the curvature effects appearing on brownian motion over a d-dimensional curved manifold. We use the local frame defined by the so called Riemann normal coordinates to derive a general formula for the mean-square geodesic distance (MSD) at the short-time regime. This formula is written in terms of $O(d)$ invariants that depend on the Riemann curvature tensor. We study the n-dimensional sphere case to validate these results. We also show that the diffusion for positive constant curvature is slower than the diffusion in a plane space, while the diffusion for negative constant curvature turns out to be faster. Finally the two-dimensional case is emphasized, as it is relevant for the single particle diffusion on biomembranes.Comment: 16 pages and 3 figure

Magnetized granular particles running and tumbling on S1S^{1}

It has been shown that a nonvibrated magnetic granular system, when it is feeded by means an altenating magnetic field, behaves with most of the distinctive physical features of active matter systems. In this work we focus our attention on the simplest granular system composed by a single magnetized spherical particle allocated in a quasi one-dimensional circular channel that receives energy from a magnetic field reservoir and transduces it into a running and tumbling motion. The theoretical analysis based on the run and tumble model on a circle of radius R forecasts the existence of a dynamical phase transition between an erratic motion (disordered phase) when the characteristic persistence length of the run and tumble motion, $\ell_{c} < R/2$, to a persistent motion (ordered phase) when $\ell_{c}> R/2$. It is found that the limiting behaviours of these phases correspond to a Brownian motion on the circle and a simple uniform circular motion, respectively. It is qualitatively shown that the lower magnetization of a particle, the larger persistence lenght is. It is so at least within the experimental limit of validity of our experiments. Our results show a very good agreement between theory and experiment.Comment: 32 pages, 10 figure

Covariant description of the colloidal dynamics on curved manifolds

Brownian motion is a universal characteristic of colloidal particles embedded in a host medium, and it is the fingerprint of molecular transport or diffusion, a generic feature of relevance not only in physics but also in several branches of science and engineering. Since its discovery, Brownian motion, also known as colloidal dynamics, has been important in elucidating the connection between the molecular details of the diffusing macromolecule and the macroscopic information on the host medium. However, colloidal dynamics is far from being completely understood. For instance, the diffusion of non-spherical colloids and the effects of the underlying geometry of the host medium on the dynamics of either passive or active particles are a few representative cases that are part of the current challenges in soft matter physics. In this contribution, we take a step forward to introduce a covariant description of the colloidal dynamics in curved spaces. Without the loss of generality, we consider the case where hydrodynamic interactions are neglected. This formalism will allow us to understand several phenomena, for instance, the curvature effects on the kinetics during spinodal decomposition and the thermodynamic properties of colloidal dispersion, to mention a few examples. This theoretical framework will also serve as the starting point to highlight the role of geometry on colloidal dynamics, an aspect that is of paramount importance to understanding more complex transport phenomena, such as the diffusive mechanisms of proteins embedded in cell membranes

Axially symmetric membranes with polar tethers

Axially symmetric equilibrium configurations of the conformally invariant Willmore energy are shown to satisfy an equation that is two orders lower in derivatives of the embedding functions than the equilibrium shape equation, not one as would be expected on the basis of axial symmetry. Modulo a translation along the axis, this equation involves a single free parameter c.If c\ne 0, a geometry with spherical topology will possess curvature singularities at its poles. The physical origin of the singularity is identified by examining the Noether charge associated with the translational invariance of the energy; it is consistent with an external axial force acting at the poles. A one-parameter family of exact solutions displaying a discocyte to stomatocyte transition is described.Comment: 13 pages, extended and revised version of Non-local sine-Gordon equation for the shape of axi-symmetric membrane