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arXiv:1003.2239

Quantum Melting of a Wigner crystal of Rotating Dipolar Fermions in the Lowest Landau Level

Szu-Cheng Cheng, Shih-Da Jheng, Tsin-Fu Jiang

Published Mar 11, 2010• 0 citations
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Abstract

We have investigated the behavior and stability of a Wigner crystal of rotating dipolar fermions in two dimensions. Using an ansatz wave function for the ground state of rotating two-dimensional dipolar fermions, which occupy only partially the lowest Landau level, we study the correlation energy, elastic moduli and collective modes of Wigner crystals in the lowest Landau level. We then calculate the mean square of the displacement vector of Wigner crystals. The critical filling factor, below which the crystalline state is expected, is evaluated at absolute zero by use of the Lindeman's criterion. We find that the particle (hole) crystal is locally stable for filling factor is less than 1/15 (between filling factors 14/15 and 1), where the stable regime of the crystal is much narrower than the result from Baranov, Fehrmann and Lewenstein, [Phys. Rev. Lett. 100, 200402 (2008)].

DOI
10.48550/arxiv.1003.2239

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We have investigated the behavior and stability of a Wigner crystal of rotating dipolar fermions in two dimensions. Using an ansatz wave function for the ground state of rotating two-dimensional dipolar fermions, which occupy only partially the lowest Landau level, we study the correlation energy, elastic moduli and collective modes of Wigner crystals in the lowest Landau level. We then calculate the mean square of the displacement vector of Wigner crystals. The critical filling factor, below which the crystalline state is expected, is evaluated at absolute zero by use of the Lindeman's criterion. We find that the particle (hole) crystal is locally stable for filling factor is less than 1/15 (between filling factors 14/15 and 1), where the stable regime of the crystal is much narrower than the result from Baranov, Fehrmann and Lewenstein, [Phys. Rev. Lett. 100, 200402 (2008)].
Using an ansatz wave function for the ground state of rotating two-dimensional dipolar fermions, which occupy only partially the lowest Landau level, we study the correlation energy and elastic properties of the Wigner crystal of rotating dipolar Fermi gases. From a simple Hartree-Fock approach, we show that the correlation energy of a particle crystal is lower and higher than the correlation energy of a hole crystal for filling factors v<1/2 and v>1/2, respectively. Furthermore we find that the shear moduli of these dipolar crystals have a nonmonotonic behavior as a function of the filling factor. We also examine the stability of a Wigner crystal. The Wigner crystal with the sample width being zero is locally stable for 0< v <1/2, while the corresponding hole crystal is locally stable for 1/2< v <1. Due to the WC being unstable around v=1/2, we also conclude that a new liquid state, not a quantum Hall state, can exist at v=1/2.
We show the possible existence of the Wigner crystal (WC) in the Fractional Quantum Hall (FQH) regime. We find that the Landau-level mixing (LLM) will lower the energy of the WC significantly in the high-density regime. The WC is lower in energy than the FQH liquid in the high-density regime. We conclude that the crystal phase is expected at high density for rotating dipolar gases, which is consistent with non-rotating dipolar gases, but is inconsistent with the low-density conclusion from Baranov et al. [Phys. Rev. Lett. 100, 200402 (2008)], where the effect of LLM is ignored.
We point out that the Wigner crystal (WC) state of rotating dipolar fermions in the lowest Landau level is strongly correlated. From intra-Landau-level excitations of the fractional quantum Hall effect (FQHE) liquid, we find that the roton minimum of the excitation spectrum at $\ensuremath{\nu}=1/7$ becomes negative, indicating an absence of the FQHE state, which is supposed to be the ground state of the system. Therefore, we show that the Hartree-Fock theory of the WC does not predict the correct ground-state energy in the lowest Landau level. There exists a strong correlation effect on the Wigner crystallization of rotating dipolar gases.
We study the competition between the Wigner crystal and the Laughlin liquid states in an ultracold quasi-two-dimensional rapidly rotating polarized fermionic dipolar gas, and find that the Wigner crystal has a lower energy below a critical filling factor. We examine the quantum crystal to liquid transition for different confinements in the third direction. Our analysis of the phonon spectra of the Wigner crystal taking into account the phonon-phonon interactions also shows the stability of the Wigner crystal for sufficiently low filling factors (nu < 1/7).
A two-dimensional dipolar Fermi gas in a harmonic trap under rotation is studied by solving ab initio Kohn-Sham equations. The physical parameters used match those of an ultracold gas of fermionic 23Na40K^{23}\text{Na}\phantom{\rule{0.16em}{0ex}}^{40}\text{K}23Na40K molecules, a prototypical system of strongly interacting dipolar quantum matter, which was created very recently. We find that, as the critical rotational frequency is approached and the system collapses into the lowest Landau level, an array of tightly packed quantum vortices develops, in spite of the nonsuperfluid character of the system. In this state the system loses axial symmetry and the fermionic cloud boundaries assume an almost perfect square shape. At higher values of the filling factor the vortex lattice disappears, while the system still exhibits square-shaped boundaries. At lower values of the filling factor the fermions become instead localized in a Wigner cluster structure.
We report on results of finite-size numerical studies of partially filled lowest Landau level, at low electron filling factors. We find convincing evidence suggesting that electrons form Wigner crystals at sufficiently low filling factors, and the critical filling factor is ${\ensuremath{\nu}}_{c}\ensuremath{\approx}1/7.$ At $\ensuremath{\nu}=1/7$ we find the system undergoes a phase transition from the Wigner crystal to the incompressible Laughlin state when the short-range part of the Coulomb interaction is modified slightly. This transition is either continuous or very weakly first order.