We discuss a generalization of the standard Bloch sphere representation for single qubit to two qubits, in framework Hopf fibrations high-dimensional spheres by lower dimensional spheres. The single-qubit Hilbert space is three-dimensional S 3. 2 base suitably oriented 3 fibration nothing but sphere, while circular fibres represent overall phase degree freedom. For two-qubits case, seven-dimensional 7, which also allows fibration, with and 4 base. most striking result that 7 are entanglement...

We show that the time evolution of a space curve is associated with geometric phase. This phase arises from path dependence rotation natural Frenet-Serret triad respect to nonrotating (Fermi-Walker) frame. derive general expression in 1+1 dimension for and gauge potential, discuss application this formalism classical, continuous, antiferromagnetic Heisenberg spin chain.

The subject of space curves finds many applications in physics such as optical fibers, magnetic spin chains, and vortex filaments a fluid. We show that the time evolution curve is associated with geometric phase. Using concept Fermi-Walker parallel transport, we this phase arises because path dependence rotation natural Frenet-Serret triad one moves along curve. employ Lamb's formalism for space-curve dynamics to derive an expression anholonomy density general evolution. This manifests...

The authors investigate geometrical properties of a space curve and its spherical images. They show that is characterised by two 'phase-like' quantities comment on the relation these to Berry phase.

We present an exact calculation of the effective geometry-induced quantum potential for a particle confined on helicoidal ribbon. This leads to appearance localized states at rim helicoid. In this geometry twist ribbon plays role transverse electric field surface and thus is reminiscent Hall effect.

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

We discuss the dynamics of a classical spinless quantum particle carrying electric charge and constrained to move on non singular static surface in ordinary three dimensional space presence arbitrary configurations time independent currents. Starting from canonical action embedding we show that charged with $q$ couples term linear $qA^3M$, where $A^3$ is transverse component electromagnetic vector potential $M$ mean curvature surface. This cancels exactly contribution orbital magnetic moment...

We study classical Heisenberg spins on an infinite elastic cylinder. In the continuum limit Hamiltonian of system is given by nonlinear $\ensuremath{\sigma}$ model. investigate periodic, cylindrically symmetric solution sine-Gordon equation (the Euler-Lagrange for this Hamiltonian). The does not satisfy self-dual equations Bogomol'nyi [Sov. J. Nucl. Phys. 24, 449 (1976)] which give minimum energy configuration in each homotopy class. This leads to a novel geometric effect: periodic shrinking

We study classical Heisenberg spins coupled by an isotropic or anisotropic spin-spin interaction on infinite elastic cylinder. In the continuum limit, Hamiltonian of system is given a nonlinear \ensuremath{\sigma} model. investigate cylindrically symmetric solutions sine-Gordon equation (the Euler-Lagrange for this Hamiltonian). The periodic solution as well one-soliton do not satisfy self-dual equations Bogomol'nyi [Sov. J. Nucl. Phys. 24, 449 (1976)] which are necessary condition to reach...

Blue phases are frustrated systems characterized by a local constraint of double twist described mathematically in terms connection. We give rigourous construction such connection on any Riemannian 3-manifold using the Levi-Civita apply it to S 3 sphere which frustration is relieved. Using isomorphism between and unit quaternion group, we show that blue phase can be considered as nematic Lie algebra this group. This description enables an easy classification defects homotopy theory...

We study the effect of conformations on charge transport in a thin elastic tube. Using Kirchhoff model for tube with any given Poisson ratio, cross-sectional shape and intrinsic twist, we obtain class exact solutions its conformation. The tube's torsion is found terms twist while curvature satisfies nonlinear differential equation which supports periodic form Jacobi elliptic functions, call conformon lattice solutions. These typically describe loops. Each solution induces corresponding...

We show that the metastable skyrmion on a plane can be stabilized by removing disk of radius ${\ensuremath{\rho}}_{0}$ at origin. This renders non-simply connected but provides characteristic length for stabilizing half-skyrmion which is centered ${\ensuremath{\rho}}_{0}(1+\sqrt{2}).$ obtain exact solutions and lattice demonstrate these are limiting case Heisenberg spins (i) truncated cone with half-angle $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\pi}/2$ (ii) an...

We study classical Heisenberg spins in the continuum limit (i.e., nonlinear \ensuremath{\sigma} model) on an elastic two-dimensional manifold with at least one nonconstant principal curvature. If corresponding Euler-Lagrange equations support a soliton solution, curvature of geometry induces geometrical frustration region which is relieved by deformation soliton. illustrate these results elliptic cylinder where we find terms variable ellipticity along axis cylinder.