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arXiv:0808.1293 · Physical Review A

Topological entropy of quantum Hall states in rotating Bose gases

Alexis G. Morris, David L. Feder

Published Jan 23, 2009• 8 citations
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Abstract

Through exact numerical diagonalization, the von Neumann entropy is calculated for the Laughlin and Pfaffian quantum Hall states in rotating interacting Bose gases at zero temperature in the lowest Landau level limit. The particles comprising the states are indistinguishable, so the required spatial bipartitioning is effected by tracing over a subset of single-particle orbitals. The topological entropy is then extracted through a finite-size scaling analysis. The results for the Laughlin and the Pfaffian states are consistent with the expected values of $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\sqrt{2}$ and $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\sqrt{4}$, respectively.

DOI
10.1103/physreva.79.013619

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Through exact numerical diagonalization, the von Neumann entropy is calculated for the Laughlin and Pfaffian quantum Hall states in rotating interacting Bose gases at zero temperature in the lowest Landau level limit. The particles comprising the states are indistinguishable, so the required spatial bipartitioning is effected by tracing over a subset of single-particle orbitals. The topological entropy is then extracted through a finite-size scaling analysis. The results for the Laughlin and the Pfaffian states are consistent with the expected values of ln2\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\sqrt{2}ln2​ and ln4\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\sqrt{4}ln4​, respectively.
We present analytic and numerical calculations on the bipartite entanglement entropy in fractional quantum Hall states of the fermionic Laughlin sequence. The partitioning of the system is done both by dividing Landau-level orbitals and by grouping the fermions themselves. For the case of orbital partitioning, our results can be related to spatial partitioning, enabling us to extract a topological quantity (the "total quantum dimension") characterizing the Laughlin states. For particle partitioning we prove a very close upper bound for the entanglement entropy of a subset of the particles with the rest, and provide an interpretation in terms of exclusion statistics.
The energy spectrum for an ultracold rotating Bose gas in a harmonic trap is calculated exactly for small systems, allowing the atoms to occupy several Landau levels. Two vortexlike states and two strongly correlated states (the Pfaffian and Laughlin) are considered in detail. In particular, their critical rotation frequencies and energy gaps are determined as a function of particle number, interaction strength, and the number of Landau levels occupied (up to three). For the vortexlike states, the lowest-Landau-level (LLL) approximation is justified only if the interaction strength decreases with the number of particles; nevertheless, the constant of proportionality increases rapidly with the angular momentum per particle. For the strongly correlated states, however, the interaction strength can increase with particle number without violating the LLL condition. The results suggest that, in large systems, the Pfaffian and Laughlin states might be stabilized at rotation frequencies below the centrifugal limit for sufficiently large interaction strengths, with energy gaps a significant fraction of the trap energy.
We study the quantum Hall phases that appear in the dilute limit of rotating Bose-Einstein condensates. By exact diagonalization in a spherical geometry we obtain the ground state and low-lying excited states of a small number of bosons as a function of the filling fraction nu, the ratio of the number of bosons to the number of vortices. We show the occurrence of the Jain principal sequence of incompressible liquids for nu=2/3,3/4,4/3,5/4 in addition to the Laughlin state nu=1/2 as well as the Pfaffian state for nu=1. We give gap estimates by finite-size scaling of both charged and neutral excitations.
We present a detailed analysis of bipartite entanglement entropies in fractional quantum Hall (FQH) states, considering both Abelian (Laughlin) and non-Abelian (Moore-Read) states. We derive upper bounds for the entanglement between two subsets of the particles making up the state. We also consider the entanglement between spatial regions supporting a FQH state. Using the latter, we show how the so-called topological entanglement entropy $\ensuremath{\gamma}$ of a FQH state can be extracted from wave functions for a limited number of particles.
The Moore-Read Pfaffian and anti-Pfaffian states have been under scrupulous review as candidates which describe the fractional quantum Hall effect at filling factor 5/2. Quantum states in the universality class of the Moore-Read Pfaffian/anti-Pfaffian have non-trivial intrinsic topological order and support low-energy non-Abelian excitations that have applications in fault-tolerant topological quantum computing schemes. Both states are exact ground states of three-body Hamiltonians that explicitly break particle-hole symmetry. We study the topological order of a competing ansatz state &PSgr;2 that is the exact ground state of a two-body Hamiltonian that preserves particle-hole symmetry. In particular, we calculate the bipartite entanglement entropy and spectra in the lowest Landau level in the spherical geometry for &PSgr; 2. We perform such calculations for a finite number of electrons up to 14. We then extrapolate to the thermodynamic limit the topological entanglement entropy γ as a measure of the topological order of the ansatz and compare to the known value of the Moore-Read Pfaffian/anti-Pfaffian state. We also study the orbital entanglement spectra for &PSgr;2 and compare with the Moore-Read Pfaffian and two-body Coulomb ground states. We show that our extrapolation of γ lies within the uncertainty of the known value of γ for the Moore-Read Pfaffian state, and that the orbital entanglement spectra of &PSgr;2 assumes a similar structure to that of the two-body Coulomb interaction.
We study the quantum Hall phases that appear in the fast-rotation limit for Bose-Einstein condensates of spinless bosonic atoms. We use exact diagonalization in a spherical geometry to obtain low-lying states of a small number of bosons as a function of the angular momentum. This allows us to understand or guess the physics at a given filling fraction $\ensuremath{\nu}$, the ratio of the number of bosons to the number of vortices. This is also the filling factor of the lowest Landau level. In addition to the well-known Bose-Laughlin state at $\ensuremath{\nu}=1∕2$, we give evidence for the Jain principal sequence of incompressible states at $\ensuremath{\nu}=p∕(p\ifmmode\pm\else\textpm\fi{}1)$ for a few values of ppp. There is a collective mode in these states whose phenomenology is in agreement with standard arguments coming, e.g., from the composite-fermion picture. At filling factor 1, the potential Fermi sea of composite fermions is replaced by a paired state, the Moore-Read state. This is most clearly seen from the half-flux nature of elementary excitations. We find that the hierarchy picture does not extend up to the point of transition towards a vortex lattice. While we cannot come to any definite conclusions, we investigate the clustered Read-Rezayi states and show evidence for incompressible states at the expected ratio of flux versus number of Bose particles.