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arXiv:0903.4093 · Journal of Statistical Mechanics Theory and Experiment

Spin chain description of rotating bosons at ν = 1

Emma Wikberg, Emil J. Bergholtz, A. Karlhede

Published Jul 23, 2009• 10 citations
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Abstract

We consider bosons at Landau level filling ν = 1 on a thin torus. In analogy with previous work on fermions at filling ν = 1/2, we map the low-energy sector onto a spin-1/2 chain. While the fermionic system may realize the gapless XY phase, we show that typically this does not happen for the bosonic system. Instead, both delta function and Coulomb interaction lead to gapped phases in the bosonic system, and in particular we identify a phase corresponding to the non-Abelian Moore–Read state. In the spin language, the Hamiltonian is dominated by a ferromagnetic next-nearest-neighbor interaction, which leads to a description consistent with the non-trivial degeneracies of the ground and excited states of this phase of matter. In addition we comment on the similarities and differences of the two systems mentioned above and fermions at ν = 5/2.

DOI
10.1088/1742-5468/2009/07/p07038

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Abstract Fractional quantum Hall states have been observed at filling factor ν = 3/4 in GaAs hole system and bilayer graphene. In theoretical bootstrap analysis, it was revealed that non-Abelian topological orders with Ising anyons can be realized at ν = 3/4, which exhibit 12 fold ground state degeneracy on the torus. The properties of ν = 3/4 states can be analyzed using two complementary approaches. In the first one, they are treated as particle–hole conjugate of ν = 1/4 Moore–Read types states. In the second one, they are mapped to composite fermions with reverse flux attachment at effective filling factor 3/2, whose integral part realizes an integer quantum Hall state and the fractional part realizes ν = 1/2 Moore–Read type states. For bilayer graphene with appropriate Landau level mixing, numerical calculations have found 12 quasi-degenerate ground states on the torus at ν = 3/4. Chiral graviton spectral functions of these states have one low energy peak with negative chirality and one high energy peak with positive chirality. This points to a specific member of the Moore–Read type states and agrees with the deduction based on daughter states.