Projective construction of theZkRead-Rezayi fractional quantum Hall states and their excitations on the torus geometry

Abstract

Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the Z k Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of Z k ground states on the torus. We develop an alternative projective construction of Z k Read-Rezayi states that complements the existing one. On the multi-layer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend the new symmetrization scheme to the plane and sphere geometries.

Phase diagram ofν=12+12bilayer bosons with interlayer couplings

“…The topological degeneracy is not that clear for small system sizes due to the strong finitesize effects, but it is greatly improved for N ≥ 12, suggesting a recovery of the exact degeneracy in the thermodynamic limit. We can see that the energy gap ∆ MR still has some finite size effects, much more than the equivalent quantity obtained with the model three-body interaction 9 . We then com- pute the ground-state overlap (defined as the square norm of the scalar product) with the exact MR state obtained by diagonalizing the three-body contact interaction or by projective construction on the torus 9 (especially for the largest systems).…”

“…Their respective dispersion relation should not depend on the particle number parity (as can be seen in Ref. 9 for the three-body model interaction). For the two-body contact interaction, our results are compatible with such a property up to more important finite-size effects.…”

“…It is expected that the energy gap for both even N and odd N will converge to the same value in the thermodynamic limit. Indeed each N sector should exhibit on the torus two types of neutral excitation modes: a magneto-roton mode 39,40 and a neutral fermion mode 9,40,41 . Their respective dispersion relation should not depend on the particle number parity (as can be seen in Ref.…”

“…When the two layers are decoupled, each of them is at filling factor ν = 1 2 leading to two copies of the ν = 1 2 Laughlin state. Applying the projective construction would allow to recover the MR state 9 . If all the bosons were located in the same layer thus having effectively a single layer at filling factor ν = 1, strong numerical evidence [15][16][17][18][19] has shown that the emerging phase would also be described by the MR state.…”

“…In this context, condensing some of the bosonic excitations of a given topological phase leads to a simpler (or equally rich) topological order. Conversely, the projective construction [3][4][5][6][7][8][9] starts from multiple copies of a simple topological phase hosting for example only Abelian excitations, to generate a new one that could potentially have a more complex topological order, involving non-Abelian excitations.…”