Flow and Dirac particles on curved and topological manifolds

Abstract

A compelling challenge for simulations is the situation when fluid flows threw confining surfaces strongly deform. Here we present a novel method for fluid structure interaction (FSI) simulations where an original 2$^{nd}$-order curved space lattice Boltzmann fluid solver (LBM) is coupled to a finite element method (FEM) for thin shells. The LBM can work independently on a standard lattice in curved coordinates without the need for interpolation, re-meshing or an immersed boundary. The LBM distribution functions are transformed dynamically under coordinate change. In addition, force and momentum can be calculated on the nodes exactly in any geometry. Furthermore, the FEM shell is a complete numerical tool with implementations such as growth, self-contact and strong external forces. We show resolution convergent error for standard tests under metric deformation. Mass and volume conservation, momentum transfer, boundary-slip and pressure maintenance are verified through specific examples. Additionally, a brief deformation stability analysis is carried out. Next, we study the interaction of a square fluid flow channel to a deformable shell. Finally, we simulate a flag at moderate Reynolds number, air flow channel. The scheme is limited to small deformations of $\mathcal{O}(10\%)$ relative to domain size, by improving its stability the method can be naturally extended to multiple applications without further implementations. Furthermore, it is common knowledge that Dirac particles have been notoriously difficult to confine, an important property in the context of quantum computing and waveguides. Implementing a curved space Dirac equation solver based on the quantum Lattice Boltzmann method, we show that curvature in a 2-D space can confine a portion of a charged, mass-less Dirac fermion wave-packet. This is equivalent to a finite probability of confining the Dirac fermion within a curved space region. We propose a general power law expression for the probability of confinement with respect to average spatial curvature for the studied geometry. Additionally, the characteristic 2D honeycomb carbon atom lattice of graphene makes it a perfect flat electronic material which can be stacked and reshaped resulting in spectacular electronic properties. Here we study the discrete energy spectrum of curved graphene sheets in the presence of a magnetic field. The shifting of the Landau levels is determined for complex and realistic geometries of curved graphene sheets. The energy levels follow a similar square root dependence on the energy quantum number as for rippled and flat graphene sheets. The Landau levels are shifted towards lower energies proportionally to the average deformation and the effect is larger compared to a simple uni-axially rippled geometry. The resistivity of wrinkled graphene sheets is calculated for different average space curvatures and shown to obey a linear relation. Moreover, we propose a periodic quantized alternating current device with a curved graphene sheet. The study is carried out with a quantum lattice Boltzmann method, solving the Dirac equation on curved manifolds. Finally, it is currently evident that topological phases of matter have revolutionized quantum engineering. Implementing a curved space Dirac equation solver based on the quantum Lattice Boltzmann method, we study the topological and geometrical transport properties of a M\"obius graphene ribbon. We measure a quantum Spin-Hall current on the graphene strip, in the absence of a magnetic field, originating from topology and curvature, whereas a quantum Hall current is not observed. In the torus geometry a Hall current is measured. Additionally, a specific illustration of the equivalence between the Berry and Ricci curvature is presented through a travelling wave-packet around the M\"obius band.