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arXiv:cond-mat/9907498 · Physical review. B, Condensed matter

Composite fermion description of rotating Bose-Einstein condensates

Nigel R. Cooper, Nicola K. Wilkin

Published Dec 15, 1999• 130 citations
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Abstract

We study the properties of rotating Bose-Einstein condensates in parabolic traps, with coherence length large compared to the system size. In this limit, it has been shown that unusual ground states form which cannot be understood within a conventional many-vortex picture. Using comparisons with exact numerical results, we show that these ground states can be well-described by a model of noninteracting ``composite fermions.'' Our work emphasizes the similarities between the unusual states that appear in rotating Bose-Einstein condensates and incompressible fractional quantum Hall states.

DOI
10.1103/physrevb.60.r16279

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We study the properties of rotating Bose-Einstein condensates in parabolic traps, with coherence length large compared to the system size. In this limit, it has been shown that unusual ground states form which cannot be understood within a conventional many-vortex picture. Using comparisons with exact numerical results, we show that these ground states can be well-described by a model of noninteracting ``composite fermions.'' Our work emphasizes the similarities between the unusual states that appear in rotating Bose-Einstein condensates and incompressible fractional quantum Hall states.
The system of neutral bosons in rapidly rotating atomic trap is equivalent to charged bosons coupled to a magnetic field, which has opened up the possibility of fractional quantum Hall effect (FQHE) for bosons interacting with a short range interaction. We present numerical evidences for incompressible liquids at filling fractions ν = 1/2, 2/3, 3/4. Motivated by the composite fermion (CF) theory of the FQHE for electrons, we show how interacting bosons can be mapped into non-interacting spinless fermions carrying one vortex each. This provides an elegant explanation of the fractional quantum Hall states in this system. We use overlap calculations to test how the CF model fits with exact numerical results. At ν = 1, the fermionization appears to overcompensate for the repulsive interaction between bosons producing an attractive interaction between fermions, as evidence by a pairing of fermions. In that case, the Pfaffian state is a good candidate to describe the ground state of such system.
We provide evidence for several novel phases in the dilute limit of rotating Bose-Einstein condensates. By exact calculation of wave functions and energies for small numbers of particles, we show that the states near integer angular momentum per particle are best considered condensates of composite entities, involving vortices and atoms. We are led to this result by explicit comparison with a description purely in terms of vortices. Several parallels with the fractional quantum Hall effect emerge, including the presence of the Pfaffian state.
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.
Trapped Bose-Einstein condensates (BECs) differ considerably from the standard textbook example of a uniform Bose gas. In an isotropic harmonic potential V(r) = ½Mω2 r 2, the single-particle ground state introduces a new intrinsic scale of length [the ground-state size d = √ℏ/(Mω)] and energy [the ground-state energy E 0 = 32 \frac{3} {2} 23​ ℏω]. When the trap rotates at a low angular velocity, the behavior of a single vortex illustrates the crucial role of discrete quantized vorticity. For more rapid rotation, the condensate contains a vortex array. The resulting centrifugal forces expand the condensate radially and shrink it axially; thus, the condensate becomes effectively two dimensional. If the external rotation speed approaches the frequency of the radial harmonic confining potential, the condensate enters the “lowest-Landau-level” regime, and a simple description again becomes possible. Eventually, the system is predicted to make a quantum phase transition to a highly correlated state analogous to the fractional quantum Hall states of electrons in a strong magnetic field.
The nonperturbative effect of interaction can sometimes make interacting bosons behave as though they were free fermions. The system of neutral bosons in a rapidly rotating atomic trap is equivalent to charged bosons coupled to a magnetic field, which has opened up the possibility of a fractional quantum Hall effect for bosons interacting with a short-range interaction. Motivated by the composite fermion theory of a the fractional Hall effect of electrons, we test the idea that the interacting bosons map onto noninteracting spinless fermions carrying one vortex each, by comparing wave functions incorporating this physics with exact wave functions available for systems containing up to 12 bosons. We study here the analogy between interacting bosons at filling factors $\ensuremath{\nu}=n∕(n+1)$ with noninteracting fermions at ${\ensuremath{\nu}}^{*}=n$ for the ground state as well as the low-energy excited states and find that it provides a good account of the behavior for small nnn, but interactions between fermions become increasingly important with nnn. At $\ensuremath{\nu}=1$, which is obtained in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, the fermionization appears to overcompensate for the repulsive interaction between bosons, producing an attractive interaction between fermions, as evidenced by a pairing of fermions here.
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.