Degeneracy and inversion of band structure for Wigner crystals on a closed helix

Abstract

Constraining long-range interacting particles to move on a curved manifold can drastically alter their effective interactions. As a prototype we explore the structure and vibrational dynamics of crystalline configurations formed on a closed helix. We show that the ground state undergoes a pitchfork bifurcation from a symmetric polygonic to a zigzag-like configuration with increasing radius of the helix. Remarkably, we find that, for a specific value of the helix radius, below the bifurcation point, the vibrational frequency spectrum collapses to a single frequency. This allows for an essentially independent small-amplitude motion of the individual particles and, consequently, localized excitations can propagate in time without significant spreading. Upon increasing the radius beyond the degeneracy point, the band structure is inverted, with the out-of-phase oscillation mode becoming lower in frequency than the mode corresponding to the center-of-mass motion

Electrostatic bending response of a charged helix

“…Returning to the case of charged helical structures it has been shown that, even in the absence of an electrolyte solution, charges confined on a helical trap can display interesting properties [22][23][24][25][26][27][28] such as effective oscillatory interactions [22,23], band structure inversions [26] and unconventional pinned-to-sliding transitions [28]. The helical confining manifold essential for the emergence of such features can have different shapes.…”

“…The helical confining manifold essential for the emergence of such features can have different shapes. An infinite homogeneous [22][23][24] or inhomogeneous helix [25] as well as a closed helix [26][27][28] have all been used as prerequisites in order to highlight different aspects of the helical confinement. The question, however, of what is the energetically preferable conformation of the helical filament given the Coulomb interactions between the charges confined on it still remains open.…”

Dynamics of nonlinear excitations of helically confined charges

“…As discussed in Ref. [23] the spreading velocity decreases as r is increased, following the width of the linear spectrum [ Fig. 1(b)].…”

“…However, the subsequent drop in P is much stronger in Fig In Ref. [23] it was demonstrated that a small localized initial excitation does not spread significantly for short times at the degeneracy point, in contrast to the behavior for other geometries. This fact was understood solely by an inspection of the linearization spectrum.…”

“…In a previous work [23], we have shown that in such a system, a tuning of the geometry controlled by the helix radius, leads to an unconventional deformation of the phononic band structure including a regime of strong degeneracy. As a consequence, the propagation of small amplitude localized excitations is affected significantly and a specific geometry exists at which the excitations remain localized up to long times.…”