Twist

Abstract: We explore the rotational degree of freedom between graphene layers via the simple prototype of the graphene twist bilayer, i.e., two layers rotated by some angle $\ensuremath{\theta}$. It is shown that, due to the weak interaction between graphene layers, many features of this system can be understood by interference conditions between the quantum states of the two layers, mathematically expressed as Diophantine problems. Based on this general analysis we demonstrate that while the Dirac cones from each layer are always effectively degenerate, the Fermi velocity ${v}_{F}$ of the Dirac cones decreases as $\ensuremath{\theta}\ensuremath{\rightarrow}0\ifmmode^\circ\else\textdegree\fi{}$; the form we derive for ${v}_{F}(\ensuremath{\theta})$ agrees with that found via a continuum approximation in [J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Phys. Rev. Lett. 99, 256802 (2007)]. From tight-binding calculations for structures with $1.47\ifmmode^\circ\else\textdegree\fi{}\ensuremath{\le}\ensuremath{\theta}l30\ifmmode^\circ\else\textdegree\fi{}$ we find agreement with this formula for $\ensuremath{\theta}\ensuremath{\gtrsim}5\ifmmode^\circ\else\textdegree\fi{}$. In contrast, for $\ensuremath{\theta}\ensuremath{\lesssim}5\ifmmode^\circ\else\textdegree\fi{}$ this formula breaks down and the Dirac bands become strongly warped as the limit $\ensuremath{\theta}\ensuremath{\rightarrow}0$ is approached. For an ideal system of twisted layers the limit as $\ensuremath{\theta}\ensuremath{\rightarrow}0\ifmmode^\circ\else\textdegree\fi{}$ is singular as for $\ensuremath{\theta}g0$ the Dirac point is fourfold degenerate, while at $\ensuremath{\theta}=0$ one has the twofold degeneracy of the $AB$ stacked bilayer. Interestingly, in this limit the electronic properties are in an essential way determined globally, in contrast to the ``nearsightedness'' [W. Kohn, Phys. Rev. Lett. 76, 3168 (1996)] of electronic structure generally found in condensed matter.

Abstract: The helicity of a localized solenoidal vector field (i.e. the integrated scalar product of the field and its vector potential) is known to be a conserved quantity under ‘frozen field’ distortion of the ambient medium. In this paper we present a number of results concerning the helicity of linked and knotted flux tubes, particularly as regards the topological interpretation of helicity in terms of the Gauss linking number and its limiting form (the Calugareanu invariant). The helicity of a single knotted flux tube is shown to be intimately related to the Calugareanu invariant and a new and direct derivation of this topological invariant from the invariance of helicity is given. Helicity is decomposed into writhe and twist contributions, the writhe contribution involving the Gauss integral (for definition, see equation (4.8)), which admits interpretation in terms of the sum of signed crossings of the knot, averaged over all projections. Part of the twist contribution is shown to be associated with the torsion of the knot and part with what may be described as ‘intrinsic twist’ of the field lines in the flux tube around the knot (see equations (5.13) and (5.15)). The generic behaviour associated with the deformation of the knot through a configuration with points of inflexion (points at which the curvature vanishes) is analysed and the role of the twist parameter is discussed. The derivation of the Calugareanu invariant from first principles of fluid mechanics provides a good demonstration of the relevance of fluid dynamical techniques to topological problems.