Licentiate Thesis: Non-abelian quantum Hall states on the thin torus

by Emma Wikberg (SU Fysikum)

Tuesday 28 Apr 2009, 13:15 → 15:15 Europe/Stockholm

FB52

We explore various quantum Hall systems on a torus. As torus circumference L1 becomes small, hopping terms in the interacting hamiltonian diminish and the problem is greatly simplified. Earlier studies of the gapless system of fermions at Landau Level filling ν = ½ showed that within a range of L1 on the very thin torus, the low-energy sector may be mapped onto 1D spin-1/2 chains. In this regime, the resulting spin hamiltonian is dominated by the nearest neighbor spin flip. This is the known XY-model, which is exactly solvable in terms of neutral dipoles with gapless excitations. The solution has high overlap with the Rezayi-Read wave function that describes this system. The same mapping is here applied to ν = 5/2, half-filling in the second Landau level, which as opposed to ν = ½ is a gapped quantum Hall system well described by the non-abelian Moore-Read (MR) wave function. In this case the spin hamiltonian is dominated by a ferromagnetic next-nearest neighbor Ising term, yielding sixfold degenerate ground states, in agreement with what is known about the MR wave function. We also find the expected gapped fractionally charged excitations to appear as domain walls between different ground states. Mathematically, there is a connection between fermions in a quantum Hall system and bosons in rapidly rotating Bose-Einstein condensates. Here, bosons at ν = 1 is the analog to fermions at half-filling. Led by this we find a way to map the bosonic system onto a spin-1/2 space, in analogy to the mapping for fermions above. As for ν = 5/2 we find the spin hamiltonian to be dominated by the negative next-nearest neighbor Ising interaction, which in this case yields a threefold degeneracy of the ground states. Furthermore, gapped fractionally charged excitations emerge as domain walls between the ground states. Again, this agrees well with what is expected from the non-abelian Moore-Read wave function that describes the ν = 1 phase. In addition to the above we explore a pseudopotential connection between ν = ½ and ν = 5/2, giving an idea of the stability of these phases. Exact diagonalization studies and overlap calculations support the theoretical conclusions drawn. Furthermore, we expand the picture of fractional charges as domain walls to more general (bosonic and fermionic) filling fractions, and sketch a way to count the excitation degeneracies.