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arXiv:1005.2520 · Journal of Physics B Atomic Molecular and Optical Physics

Asymptotically exact trial wavefunctions for yrast states of rotating Bose gases

S. Viefers, Mathieu Taillefumier

Published Jul 19, 2010• 7 citations
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Abstract

We revisit the composite fermion (CF) construction of the lowest angular momentum yrast states of rotating Bose gases with weak short range interaction. For angular momenta at and below the single vortex, $L \leq N$, the overlaps between these trial wave functions and the corresponding exact solutions {\it increase} with increasing system size and appear to approach unity in the thermodynamic limit. In the special case $L=N$, this remarkable behaviour was previously observed numerically. Here we present methods to address this point analytically, and find strongly suggestive evidence in favour of similar behaviour for all $L \leq N$. While not constituting a fully conclusive proof of the converging overlaps, our results do demonstrate a striking similarity between the analytic structure of the exact ground state wave functions at $L \leq N$, and that of their CF counterparts. Results are given for two different projection methods commonly used in the CF approach.

DOI
10.1088/0953-4075/43/15/155302

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We revisit the composite fermion (CF) construction of the lowest angular momentum yrast states of rotating Bose gases with weak short range interaction. For angular momenta at and below the single vortex, L≤NL \leq NL≤N, the overlaps between these trial wave functions and the corresponding exact solutions {\it increase} with increasing system size and appear to approach unity in the thermodynamic limit. In the special case L=NL=NL=N, this remarkable behaviour was previously observed numerically. Here we present methods to address this point analytically, and find strongly suggestive evidence in favour of similar behaviour for all L≤NL \leq NL≤N. While not constituting a fully conclusive proof of the converging overlaps, our results do demonstrate a striking similarity between the analytic structure of the exact ground state wave functions at L≤NL \leq NL≤N, and that of their CF counterparts. Results are given for two different projection methods commonly used in the CF approach.
We study the composite-fermion (CF) construction at and below the single vortex (L=N)(L=N)(L=N) state of weakly interacting rotating Bose gases, presenting a new method for handling the large number of derivatives typically occurring via the Slater determinant. Remarkably, the CF wave function at L=NL=NL=N becomes asymptotically exact in the thermodynamic limit, even though this construction is not, a priori, expected to work in the low angular momentum regime. This implies an interesting mathematical identity which may be useful in other contexts.
We study the rotational properties of a two-component Bose gas trapped in a rotating harmonic potential, effectively in two dimensions. We first consider a completely homogeneous interaction between particles, and assume the bosons to be in the lowest Landau level. We derive analytical expressions for some wave functions and energies at low angular momenta, and use these results together with exact numerical diagonalization results to test the applicability of composite fermion (CF) trial wave functions to the system. We find that a general procedure based on CF wave functions reproduces the exact ground states and many of the excited states for sufficiently low angular momenta, and give very good approximate results at angular momenta up to the two-component analogy of a single vortex. Additionally, we produce very simple CF states that have remarkably high overlaps with the lowest-lying states. Finally, we briefly discuss the case of inhomogeneous interactions, pointing out the difficulties arising from this generalization.
Despite its success, the composite fermion (CF) construction possesses some mathematical features that have not been fully understood until recently. In particular, this construction is known to produce wave functions that are not necessarily orthogonal, or even linearly independent, after projection to the lowest Landau level. While this is usually not a problem in practice in the quantum Hall regime, we have previously shown that it presents a technical challenge for rotating Bose gases with low angular momenta. These are systems where the CF approach yields surprisingly good approximations to the exact eigenstates of weak short-range interactions, and so solving the problem of linearly dependent wave functions is of interest. It can also be useful for studying higher bands of fermionic quantum Hall states. Here we present several ways of constructing a basis for the space of so-called ``simple'' CF states for two-component rotating Bose gases in the lowest Landau level, and prove that they all give sets of linearly independent wave functions that span the space. Using this basis, we study the structure of the lowest-lying state using so-called restricted wave functions. We also examine the scaling of the overlap between the exact and CF wave functions at the maximal possible angular momentum for simple states.
We study the composite fermion construction at and below the single vortex (L=NL=NL=N) state of weakly interacting rotating Bose gases, presenting a new method for handling the large number of derivatives typically occurring via the Slater determinant. Remarkably, the CF wave function at L=NL=NL=N becomes {\em exact} in the large NNN limit, even though this construction is not, {\em a priori}, expected to work in the low angular momentum regime. This implies an interesting mathematical identity which may be useful in other contexts.
We study the rotational (yrast) spectra of dilute two-component atomic Bose gases in the low angular momentum regime, assuming equal interspecies and intraspecies interaction. Our analysis employs the composite fermion (CF) approach including a pseudospin degree of freedom. While the CF approach is not a priori expected to work well in this angular momentum regime, we show that composite fermion diagonalization gives remarkably accurate approximations to low-energy states in the spectra. For angular momenta 0<L<M0<L<M0<L<M (where NNN and MMM denote the numbers of particles of the two species, and $M\ensuremath{\ge}N$), we find that the CF states span the full Hilbert space and provide a convenient set of basis states which, by construction, are eigenstates of the symmetries of the Hamiltonian. Within this CF basis, we identify a subset of the basis states with the lowest $\ensuremath{\Lambda}$-level kinetic energy. Diagonalization within this significantly smaller subspace constitutes a major computational simplification and provides very close approximations to ground states and a number of low-lying states within each pseudospin and angular momentum channel.
For a weakly interacting Bose gas rotating in a harmonic trap we relate the yrast states of small systems (that can be treated exactly) to the thermodynamic limit (derived within the mean-field approximation). For a few dozens of atoms, the yrast line shows distinct quasiperiodic oscillations with increasing angular momentum that originate from the internal structure of the exact many-body states. These finite-size effects disappear in the thermodynamic limit, where the Gross-Pitaevskii approximation provides the exact energy to leading order in the number of particles NNN. However, the exact yrast states reveal significant structure not captured by the mean-field approximation: Even in the limit of large NNN, the corresponding mean-field solution accounts for only a fraction of the total weight of the exact quantum state.