The topological origin of the Peierls–Nabarro barrier

Abstract

Crystals and other condensed matter systems described by density waves often exhibit dislocations. Here we show, by considering the topology of the ground state manifolds (GSMs) of such systems, that dislocations in the density phase field always split into disclinations, and that the disclinations themselves are constrained to sit at particular phase values in the GSM. This constraint on the location of the disclinations results in an energy barrier to dislocation glide. Consequently, the topology of the GSM alone gives rise to a Peierls–Nabarro barrier.

Complex-tensor theory of simple smectics

“…This implies an energy barrier associated with the sliding of dislocations with respect to the lamellar structure. This indicates the possibility of non-zero Peierls-Nabarro energy barriers 26 , 48 , further validating of the E formulation.…”

“…Though elegant and economical, the traditional formalism has known shortcomings 26 . These can be stated in a number of equivalent ways: (i) Only the variation of the density is physical, and so the wave function is not truly a single-valued function of position 23 , 48 but rather both Ψ and Ψ * appear simultaneously 27 . (ii) This can be stated as Φ is not an element of the unit circle S 1 but rather of the orbifold 25 , 49 , 50 .…”

Network topology of interlocked chiral particles

“…The algorithm for the extraction of these layers is designed to create an output analogous to the topological picture of conventional uniaxial smectics. There, topological charge conservation is guaranteed by explicitly considering smectic layers (density peaks) and so-called half-layers in between (density minima) as topological entities, that carry topological charge [22][23][24][25]. In that way, the indices of the vertices become topological, i.e., adhere to topological charge conservation in analogy to conventional electrodynamics, where the total charge, consisting of inside and boundary charge, adjusts to the topology of the confining container.…”

“…Those vertices are connected via a set of edge lines [33,34]. As known from the treatment of the topology of layers in conventional smectics [22][23][24][25], a charge conservation follows, if the species of networks alternate. In other words, between any two smectic layers has to be a density minimum, i.e., a half-layer.…”

Continuum Limit of 2D Fractional Nonlinear Schrödinger Equation

“…In contrast to the (continuum) nonlinear Schrödinger equation that admits the Galilean boost from which traveling wave solutions emerge, many lattice systems lack translational invariance. It is known that highly localized solutions in a lattice system do not propagate due to the presence of the Peirels-Nabarro potential [26,10]; for a recent work on FNLSE in this context, see [22]. All this is to point out the challenges and open problems that need to be studied by a combination of analytical and numerical tools.…”