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arXiv:0704.3859 · Physical Review Letters

Density Waves and Supersolidity in Rapidly Rotating Atomic Fermi Gases

Gunnar Möller, Nigel R. Cooper

Published Nov 9, 2007• 21 citations
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Abstract

We study theoretically the low-temperature phases of a two-component atomic Fermi gas with attractive s-wave interactions under conditions of rapid rotation. We find that, in the extreme quantum limit, when all particles occupy the lowest Landau level, the normal state is unstable to the formation of charge density wave (CDW) order. At lower rotation rates, when many Landau levels are occupied, we show that the low-temperature phases can be supersolids, involving both CDW and superconducting order.

DOI
10.1103/physrevlett.99.190409

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We study theoretically the low-temperature phases of a two-component atomic Fermi gas with attractive s-wave interactions under conditions of rapid rotation. We find that, in the extreme quantum limit, when all particles occupy the lowest Landau level, the normal state is unstable to the formation of charge density wave (CDW) order. At lower rotation rates, when many Landau levels are occupied, we show that the low-temperature phases can be supersolids, involving both CDW and superconducting order.
In this article, we review developments in the theory of rapidly rotating degenerate atomic gases. The main focus is on the equilibrium properties of a single-component atomic Bose gas, which (at least at rest) forms a Bose–Einstein condensate. Rotation leads to the formation of quantized vortices which order into a vortex array, in close analogy with the behaviour of superfluid helium. Under conditions of rapid rotation, when the vortex density becomes large, atomic Bose gases offer the possibility to explore the physics of quantized vortices in novel parameter regimes. First, there is an interesting regime in which the vortices become sufficiently dense that their cores, as set by the healing length, start to overlap. In this regime, the theoretical description simplifies, allowing a reduction to single-particle states in the lowest Landau level. Second, one can envisage entering a regime of very high vortex density, when the number of vortices becomes comparable to the number of particles in the gas. In this regime, theory predicts the appearance of a series of strongly correlated phases, which can be viewed as bosonic versions of fractional quantum Hall states. In this article, we describe the equilibrium properties of rapidly rotating atomic Bose gases in both the mean-field and the strongly correlated regimes, and related theoretical developments for Bose gases in lattices, for multi-component Bose gases and for atomic Fermi gases. The current experimental situation and outlook for the future are discussed in light of these theoretical developments.
We study the phase diagram of a dipolar Fermi gas at half-filling in a cubic optical lattice with dipole moments aligned along the zzz axis. The anisotropic dipole-dipole interaction leads to the competition between pz{p}_{z}pz​-wave superfluid and nematic charge-density-wave (CDW) orders at low temperatures. We find that the superfluid phase survives with weak interactions and the CDW phase dominates with strong interactions. In-between, the supersolid phase appears as a balance between superfluid and CDW orders. The superfluid density is anisotropic in the supersolid and superfluid phases. In the CDW phase, there is a semimetal-to-insulator transition with increase of the interaction strength. Experimental implications are discussed.
Experiments on quantum degenerate Fermi gases of magnetic atoms and dipolar molecules begin to probe their broken-symmetry phases dominated by the long-range, anisotropic dipole-dipole interaction. Several candidate phases including the ppp-wave superfluid, the stripe density wave, and a supersolid have been proposed theoretically for two-dimensional spinless dipolar Fermi gases. Yet the phase boundaries predicted by different approximations vary greatly and a definitive phase diagram is still lacking. Here we present a theory that treats all competing many-body instabilities in the particle-particle and particle-hole channel on equal footing. We obtain the low-temperature phase diagram by numerically solving the functional renormalization-group flow equations and find a nontrivial density wave phase at small dipolar tilting angles and strong interactions, but no evidence of the supersolid phase. We also estimate the critical temperatures of the ordered phases.
We use the Gutzwiller ansatz and analyze the phase diagram of the extended Bose-Hubbard Hamiltonian with on-site (UUU) and nearest-neighbor (VVV) repulsions. For ddd-dimensional hypercubic lattices, when 2dV<U2\mathit{dV}<U2dV<U, it is well known that the ground state alternates between the charge-density-wave (CDW) and Mott insulators, and the supersolid (SS) phase occupies small regions around the CDW insulators. However, when 2dV>U2\mathit{dV}>U2dV>U, in this Rapid Communication, we show that the ground state has only CDW insulators, and more importantly, the SS phase occupies a much larger region in the phase diagram, existing up to very large hopping values which could be orders of magnitude higher than that of the well-known case. In particular, the SS-superfluid phase boundary increases linearly as a function of hopping when $2\mathit{dV}\ensuremath{\gtrsim}1.5U$, for which the prospects of observing the SS phase with dipolar Bose gases loaded into optical lattices is much higher.
We study a rapidly rotating gas of unpolarized spin-1/2 ultracold fermions in the two-dimensional regime when all atoms reside in the lowest Landau level. Due to the presence of the spin degree of freedom both sss-wave and ppp-wave interactions are allowed at ultralow temperatures. We investigate the phase diagram of this system as a function of the filling factor in the lowest Landau level and in terms of the ratio between sss- and ppp-wave interaction strengths. We show that the presence of attractive interactions induces a wide regime of phase separation with formation of maximally compact droplets that are either fully polarized or composed of spin-singlets. In the regime with no phase separation, we give evidence for fractional quantum Hall states. Most notably, we find two distinct singlet states at the filling $\ensuremath{\nu}=2/3$ for different interactions. One of these states is accounted for by the composite fermion theory, while the other one is a paired state for which we identify two competing descriptions with different topological structures. This paired state may be an Abelian liquid of composite spin-singlet Bose molecules with Laughlin correlations. Alternatively, it may be a known non-Abelian paired state, indicated by good overlaps with the corresponding trial wave function. By fine tuning of the scattering lengths it is possible to create the non-Abelian critical Haldane-Rezayi state for $\ensuremath{\nu}=1/2$ and the permanent state of Moore and Read for $\ensuremath{\nu}=1$. For purely repulsive interactions, we also find evidence for a gapped Halperin state at $\ensuremath{\nu}=2/5$.
In the limit of very fast rotation atomic Bose-Einstein condensates may entirely reside in the lowest two-dimensional Landau level (LLL). For small enough filling factor of the LLL, one may have formation of fractional quantum Hall states. We investigate the case of bosons with dipolar interactions as may be realized with chromium-52 atoms. We show that at a filling factor equal to unity the ground state is a Moore-Read (or Pfaffian) paired state as is the case of bosons with purely sss-wave scattering interactions. This Pfaffian state is destabilized when the interaction in the sss-wave channel is small enough and the ground state is a stripe phase with unidimensional density modulation. For filling factor 1/3, we show that there is formation of a Fermi sea of ``composite fermions.'' These composites are made of one boson bound with three vortices. This phase has a wide range of stability and the effective mass of the fermions essentially depends only on the scattering amplitude in momentum channels larger or equal to 2. The formation of such a Fermi sea opens up a new possible route to detection of the quantum Hall correlations.