Quantum Hall Systems on Toroidal Geometries

We present results of recent numerical calculations of second Landau level (LL) states on toroidal geometries. Calculations on the torus generally allow for smaller particle numbers than those on the sphere, due to less powerful symmetries. However, on the torus, different candidate states for particular quantum Hall plateaus appear at equal flux, in contrast to the situation on the sphere or plane. This means that working on the torus allows for more direct comparisons of trial states and reduces the problem of aliasing. Moreover, the torus brings interesting geometry, described by a modular parameter $\tau$. This potentially allows for a larger variety of phases as well as some interesting limits which can be treated analytically. It also allows for the calculation of the Hall viscosity, a quantity which corresponds to the shift on the sphere. Among fillings considered are $\nu=\frac{12}{5}$ and $\nu=\frac{5}{2}$ where states hosting non-Abelian anyons have been conjectured.