Topological defects on fluctuating surfaces: General properties and the Kosterlitz-Thouless transition

Abstract

We investigate the Kosterlitz-Thouless transition for hexatic order on a free fluctuating membrane and derive both a Coulomb gas and a sine-GordonHamiltonian to describe it. The Coulomb-gas Hamiltonian includes charge densities arising from disclinations and from Gaussian curvature. There is an interaction coupling the difference between these two densities, whose strength is determined by the hexatic rigidity, and an interaction coupling Gaussian curvature densities arising from the Liouville Hamiltonian resulting from the imposition of a covariant cutoff. In the sine-Gordon Hamiltonian, there is a linear coupling between a scalar field and the Gaussian curvature. We discuss gauge-invariant correlation function for hexatic order and the dielectric constant of the Coulomb gas. We also derive renormalization group recursion relations that predict a transition with decreasing bending rigidity κ.

Numerical insights into the phase diagram of p-atic membranes with spherical topology

“…For σ(H) the change is dramatic (see SI-6), depending both on ε and p for small κ and collapsing into a κ − 1 2 (see inset of SI-6 and SI-7) relationship for large κ, as expected from the equipartition theorem for semi-flexible membranes. This behavior is consistent with the crinkled phase, where the effective bending rigidity at short distances is given by κ, while the bending rigidity for the large-distance conformations tends to κ c (K A ) [32]. A closer analysis shows that σ(H) display approximative power law behavior in ε and p (see fig.…”

“…The shape of buckled p-atic membranes in the vicinity of the disclinations has also been studied in a mean-field setting [30] and more recently for tense p-atic membranes which deform into pseudo-spheres [12,31]. A more comprehensive RG analysis of the p-atic membrane, which includes these effects, was performed by Park and Lubensky [32][33][34] based on the descriptions given by Guitter and Kardar [29]. The crinkling line of fixed points became the solution of the equation κ c =K A (K A , κ c )/4 terminating in the melting/crumpling fixed point atK…”

“…The obtained phase diagram [32] is shown in fig. 1 (see also, Supplementary Material: SI-1 and SI-2).…”

Dual gauge field theory of quantum liquid crystals in two dimensions

“…These form the family of 'nematic-like' liquid crystals. There is actually an ambiguity in the vocabulary that is not settled: in the soft-matter community it is convention to reserve nematic for the uniaxial, D h∞ -symmetric variety (ordered states of 'rod-like' molecules), while for instance the nomenclature p-atics has been suggested for 2D nematics characterized by a p-fold axis [54]. In full generality, these substances are classified by their point group symmetries.…”

“…cannot change the energy of the system either, this rotation field has to drop out from Eq. (54). Nevertheless, at higher order in the expansion, derivatives ∂ m ω ab can appear, see Eq.…”