Numerical benchmarking of fluid-rigid body interactions

We propose a fluid-rigid body interaction benchmark problem, consisting of a solid spherical obstacle in a Newtonian fluid, whose centre of mass is fixed but is free to rotate. A number of different problems are defined for both two and three spatial dimensions. The geometry is chosen specifically, such that the fluid-solid partition does not change over time and classical fluid solvers are able to solve the fluid-structure interaction problem. We summarise the different approaches used to handle the fluid-solid coupling and numerical methods used to solve the arising problems. The results obtained by the described methods are presented and we give reference intervals for the relevant quantities of interest

Quantization of systems with internal degrees of freedom in two-dimensional manifolds

Presented is a primary step towards quantization of infinitesimal rigid body moving in a two-dimensional manifold. The special stress is laid on spaces of constant curvature like the two-dimensional sphere and pseudosphere (Lobatschevski space). Also two-dimensional torus is briefly discussed as an interesting algebraic manifold.Comment: 19 page

Competing states for the fractional quantum Hall effect in the 1/3-filled second Landau level

In this work, we investigate the nature of the fractional quantum Hall state in the 1/3-filled second Landau level (SLL) at filling factor $\nu=7/3$ (and 8/3 in the presence of the particle-hole symmetry) via exact diagonalization in both torus and spherical geometries. Specifically, we compute the overlap between the exact 7/3 ground state and various competing states including (i) the Laughlin state, (ii) the fermionic Haffnian state, (iii) the antisymmetrized product state of two composite fermion seas at 1/6 filling, and (iv) the particle-hole (PH) conjugate of the $Z_4$ parafermion state. All these trial states are constructed according to a guiding principle called the bilayer mapping approach, where a trial state is obtained as the antisymmetrized projection of a bilayer quantum Hall state with interlayer distance $d$ as a variational parameter. Under the proper understanding of the ground-state degeneracy in the torus geometry, the $Z_4$ parafermion state can be obtained as the antisymmetrized projection of the Halperin (330) state. Similarly, it is proved in this work that the fermionic Haffnian state can be obtained as the antisymmetrized projection of the Halperin (551) state. It is shown that, while extremely accurate at sufficiently large positive Haldane pseudopotential variation $\delta V_1^{(1)}$, the Laughlin state loses its overlap with the exact 7/3 ground state significantly at $\delta V_1^{(1)} \simeq 0$. At slightly negative $\delta V_1^{(1)}$, it is shown that the PH-conjugated $Z_4$ parafermion state has a substantial overlap with the exact 7/3 ground state, which is the highest among the above four trial states.Comment: 22 pages, 5 figure