Abstract
Determining the statistics of elementary excitations supported by fractional quantum Hall states is crucial to understanding their properties and potential applications. In this paper, we use the topological entanglement entropy as an indicator of Abelian statistics to investigate the single-component $\nu=2/5$ and $3/7$ states for the Hofstadter model in the band mixing regime. We perform many-body simulations using the infinite cylinder density matrix renormalization group and present an efficient algorithm to construct the area law of entanglement, which accounts for both numerical and statistical errors. Using this algorithm, we show that the $\nu=2/5$ and $3/7$ states exhibit Abelian topological order in the case of two-body nearest-neighbor interactions. Moreover, we discuss the sensitivity of the proposed method and fractional quantum Hall states with respect to interaction range and strength.
Highlights
Throughout the extensive history of the fractional quantum Hall (FQH) effect, a lot has been learned about the elementary excitations above the ground state at FQH plateaus
In the Abelian case, exchange of quasiparticles in a given ground state yields a fractional phase shift of the wavefunction represented by a one-dimensional braid group, whereas in the non-Abelian case, the ground state is highly degenerate and an exchange of quasiparticles shifts between ground states, which is represented by a higherdimensional braid group
Having observed an increase in the topological entanglement entropy with interaction range, we investigate the effect of increasing the interaction strength, such that 10 ≤ V0 ≤ 50, at fixed interaction range κ = 1
Summary
Throughout the extensive history of the fractional quantum Hall (FQH) effect, a lot has been learned about the elementary excitations above the ground state at FQH plateaus. There are Abelian FQH states in lattice models stabilized by two-body interactions that have been shown to possess non-Abelian statistics when interactions are sufficiently long-range[13], which provides motivation for further study. We perform large-scale numerical calculations to investigate the Abelian nature of the singlecomponent ν = 2/5 and 3/7 FQH states in the Hofstadter model with a large interaction strength, chosen such that inter-band transitions are likely to occur. We build on previous studies in the field by presenting an efficient algorithm, which addresses both numerical and statistical errors Using this algorithm, we are able to compute the topological entanglement entropy to a high precision and demonstrate that these states are Abelian in the case of nearest-neighbor interactions.
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