Long-range interacting classical particle systems on curved manifolds: the helix, torus, and Möbius strip

Abstract

This thesis investigates the behavior of classical ions and dipoles that are confined to curved geometries. The combination of confining forces and the interactions between particles gives rise to geometry-dependent effective interactions. The latter exhibit different characteristics compared to interactions in flat geometries. For instance, Coulomb-interacting ions confined to a helical path exhibit oscillating effective interactions, allowing two or more ions to form a bound state. One example of such a bound state are two ions trapping each other on opposite sides of a helix winding. The diverse phenomenology induced by such geometry-dependent effective interactions is studied in detail throughout this thesis by considering a wide range of geometries, and both isotropic and anisotropic interparticle interactions. We first investigate the impact of external electric fields on the properties of ions confined to a helix. We consider many-body equilibrium configurations on a toroidal helix, focusing on the evolution of these configurations in the presence of a static external electric field. We are able to characterize the statistical properties of these equilibria. Additionally, we find that by varying the field strength a crossover between staggered and ordered equilibrium configurations occurs. This crossover persists for a wide range of system parameters. Next, we explore time-dependent fields. We analyze the dynamics of a single particle confined to a toroidal helix, driven by an either oscillating or rotating external field. Using phase space analysis, we identify a mechanism responsible for effectively inducing directed transport of the particle, with the transport direction being determined by the initial conditions. Remarkably, this directed transport occurs even without any bias or asymmetries in the driving potential. In the case of the oscillating external field, adding a static potential along the helical path will change the systems behavior to that of a generalized Kapitza pendulum. We also investigate what happens when considering anisotropic dipole-dipole interactions instead of isotropic Coulomb interactions. We begin with a system of freely rotating dipoles at fixed equidistant positions along a helical path. Our analysis of the ground-state equilibrium configurations reveals a complex behavior and shows a dependence on geometrical parameters, such as the helix radius and the (parametric) distance between two dipoles along the helix. In particular, the equilibrium configurations can be uniquely described by integer tuples that can be mapped to fractions of the number-theoretical Farey sequence, while in the parameter space, a self-similar bifurcation tree akin to the Stern-Brocot tree is identified. Beyond helical geometries, we explore dipole arrays on curved two-dimensional surfaces. For this setup, the geometric curvature can lead to a ground state exhibiting domain walls that separate regions of different dipole alignments. These curvatureinduced domain walls behave differently from typical (degeneracy-induced) domain walls. We highlight these differences by examining the domain-walls response to an external field, as well as the impact of the domain-wall on the dispersion of excitations. Notably, for the latter example, low-energy dynamics are confined within the domains, without being able to cross the boundary of the domain wall. Finally, we show that the emergence and annihilation of these curvature-induced domains are accompanied by structural crossovers that are indicated by a dip in the 2D compression modulus.