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Submitted by:** | MORAN, Niall
*National University of Ireland, Maynooth*
`niall.moran@gmail.com` |

*Quantum Hall Systems on Toroidal Geometries*

#### Niall Moran, Mikael Fremling, Joost Slingerland and Hans Hansson

We present and discuss results of recent numerical calculations of second Landau level 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, where the flux can be shifted. 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. This potentially allows for a larger
variety of phases as well as some interesting limits which can be treated analytically (notably the
thin torus limit). It also allows for the calculation of the Hall viscosity, a quantity which
corresponds to the shift on the sphere. Calculating this in principle allows for direct
identification of the correct number of flux quanta to consider in calculations on the sphere or
plane. We give an explanation of our methods and show results which include calculations of the
Coulomb spectra, Hall viscosity as well as excitation gaps and overlaps with various trial wave
functions. The fillings considered include nu=12/5
and nu=5/2
where states hosting non-Abelian anyons have been conjectured. We compare the situation to more
well-established first Landau level results and note that these second Landau level systems behave
quite differently away from the thin torus limit, in particular showing rather large finite size
effects.