Rossen Dandoloff

Geometry-induced potential on a two-dimensional section of a wormhole: Catenoid

We show that a two-dimensional wormhole geometry is equivalent to a catenoid, a minimal surface. We then obtain the curvature-induced geometric potential and show that the ground state with zero energy corresponds to a reflectionless potential. By introducing an appropriate coordinate system we also obtain bound states for different angular momentum channels. Our findings can be realized in suitably bent bilayer graphene sheets with a neck, in a honeycomb lattice with an array of dislocations, or in nanoscale waveguides in the shape of a catenoid.

Curvature-induced quantum behaviour on a helical nanotube

We investigate the effect of curvature on the behaviour of a quantum particle bound to move on a surface shaped as a helical tube. We derive and discuss the governing Schrodinger equation and the corresponding quantum effective potential which is periodic and points to the helical configuration as more energetically favorable as compared to the straight tube. The exhibited periodicity also leads to energy band structure of pure geometrical origin.

Curvature induced quantum potential on deformed surfaces

We investigate the effect of curvature on the behaviour of a quantum particle bound to move on a surface. For the Gaussian bump we derive and discuss the quantum potential which results in the appearance of a bound state for particles with vanishing angular momentum. The Gaussian bump provides a characteristic length for the problem. For completeness we solve the inverse problem and show the way to derive a surface with prescribed quantum properties. We also show that there exist surfaces in the form of a circular strip around the axis of symmetry which allow particles with generic angular momentum to bind.

Effect of conformations on charge transport in a thin elastic tube

We study the effect of conformations on charge transport in a thin elastic tube. Using the Kirchhoff model for a tube with any given Poisson ratio, cross-sectional shape and intrinsic twist, we obtain a class of exact solutions for its conformation. The tubes torsion is found in terms of its intrinsic twist and its Poisson ratio, while its curvature satisfies a nonlinear differential equation which supports exact periodic solutions in the form of Jacobi elliptic functions, which we call conformon lattice solutions. These solutions typically describe conformations with loops. Each solution induces a corresponding quantum effective periodic potential in the Schrodinger equation for an electron in the tube. The wave function describes the delocalization of the electron along the central axis of the tube. We discuss some possible applications of this novel mechanism of charge transport.

Berry's phase and Fermi-Walker parallel transport

Abstract The usual condition for parallel transport of the eigenstates is identified with the Fermi-Walker parallel transport law known from general relativity. Other possible parallel transport laws are discussed.

Coupling between magnetic field and curvature in Heisenberg spins on surfaces with rotational symmetry

Article history: We study the nonlinear σ -model in an external magnetic field applied on curved surfaces with rotational symmetry. The Euler-Lagrange equations derived from the Hamiltonian yield the double sine-Gordon equation (DSG) provided the magnetic field is tuned with the curvature of the surface. A 2π skyrmion appears like a solution for this model and surface deformations are predicted at the sector where the spins point in the opposite direction to the magnetic field. We also study some specific examples by applying the model on three rotationally symmetric surfaces: the cylinder, the catenoid and the hyperboloid.

Heisenberg spins on a bilayer connected by a neck and other geometries with a characteristic length scale

We obtain a half-skyrmion solution in the orientation of Heisenberg spins on a neck joining two planes with a semi-circular region. In addition, we consider several geometries, topologically equivalent to either a plane with a hole or a truncated circular cone or a cylinder due to the presence of an intrinsic length scale, for which we obtain skyrmion solutions. We also consider two minimal surfaces, namely a catenoid and a helicoid. Finally, we consider Heisenberg spins on single-sheet paraboloid and hyperboloid geometries. These spin textures may possibly be realized in elastically soft, curved magnetic thin films.

Quantum effective potential, electron transport and conformons in biopolymers

In the Kirchhoff model of a biopolymer, conformation dynamics can be described in terms of solitary waves, for certain special cross-section asymmetries. Applying this to the problem of electron transport, we show that the quantum effective potential arising due to the bends and twists of the polymer enables us to formalize and quantify the concept of a conformon that has been hypothesized in biology. Its connection to the soliton solution of the cubic nonlinear Schrodinger equation emerges in a natural fashion.

Heisenberg spins on an elastic torus section

Abstract Classical Heisenberg spins in the continuum limit (i.e., the nonlinear σ-model) are studied on an elastic torus section with homogeneous boundary conditions. The corresponding rigid model exhibits topological soliton configurations with geometrical frustration due to the torus eccentricity. Assuming small and smooth deformations allows us to find shapes of the elastic support by relaxing the rigidity constraint: an inhomogeneous Lame equation arises. Finally, this leads to a novel geometric effect: a global shrinking with swellings.

Topological Spin Excitations Induced by an External Magnetic Field Coupled to a Surface with Rotational Symmetry

We study the Heisenberg model in an external magnetic field on curved surfaces with rotational symmetry. The Euler–Lagrange static equations, derived from the Hamiltonian, lead to the inhomogeneous double sine-Gordon equation. Nonetheless, if the magnetic field is coupled to the metric elements of the surface, and consequently to its curvature, the homogeneous double sine-Gordon equation emerges and a