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arXiv:1507.00870 · Physical Review A

Density-functional theory for the crystalline phases of a two-dimensional dipolar Fermi gas

Brandon P. van Zyl, W. Kirkby, W. Ferguson

Published Aug 12, 2015• 14 citations
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Abstract

Density-functional theory is utilized to investigate the zero-temperature transition from a Fermi liquid to an inhomogeneous stripe, or Wigner crystal phase, predicted to occur in a one-component, spin-polarized, two-dimensional dipolar Fermi gas. Correlations are treated semiexactly within the local-density approximation using an empirical fit to quantum Monte Carlo data. We find that the inclusion of the nonlocal contribution to the Hartree-Fock energy is crucial for the onset of an instability to an inhomogeneous density distribution. Our density-functional theory supports a transition to both a one-dimensional stripe phase and a triangular Wigner crystal. However, we find that there is an instability first to the stripe phase, followed by a transition to the Wigner crystal at higher coupling.

DOI
10.1103/physreva.92.023614

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Density-functional theory is utilized to investigate the zero-temperature transition from a Fermi liquid to an inhomogeneous stripe, or Wigner crystal phase, predicted to occur in a one-component, spin-polarized, two-dimensional dipolar Fermi gas. Correlations are treated semiexactly within the local-density approximation using an empirical fit to quantum Monte Carlo data. We find that the inclusion of the nonlocal contribution to the Hartree-Fock energy is crucial for the onset of an instability to an inhomogeneous density distribution. Our density-functional theory supports a transition to both a one-dimensional stripe phase and a triangular Wigner crystal. However, we find that there is an instability first to the stripe phase, followed by a transition to the Wigner crystal at higher coupling.
The liquid-to-ordered phase transition in a bilayer system of fermions is studied within the context of our recently proposed density-functional theory [B. P. van Zyl, W. Kirkby, and W. Ferguson, Phys. Rev. A 92, 023614 (2015)]. In each two-dimensional layer, the fermions interact via a repulsive, isotropic dipolar interaction. The presence of a second layer introduces an attractive interlayer interaction, thereby allowing the presence of inhomogeneous density phases which would otherwise be energetically unfavorable. For any fixed layer separation, we find an instability to a commensurate one-dimensional stripe phase in each layer, which always precedes the formation of a triangular Wigner crystal. However, at a certain fixed coupling, tuning the separation can lead to the system favoring a commensurate triangular Wigner crystal, or one-dimensional stripe phase, completely bypassing the Fermi liquid state. While other crystalline symmetries, with energies lower than the liquid phase can be found, they are never allowed to form owing to their high energetic cost relative to the triangular Wigner crystal and stripe phase.
The liquid and crystal phases of a single-component Fermi gas with dipolar interactions are investigated using quantum Monte Carlo methods in two spatial dimensions and at zero temperature. The dipoles are oriented by an external field perpendicular to the plane of motion, resulting in a purely repulsive 1/r(3) interaction. In the liquid phase we calculate the equation of state as a function of the interaction strength and other relevant properties characterizing the Fermi-liquid behavior: effective mass, discontinuity at the Fermi surface, and pair correlation function. In the high density regime we calculate the equation of state of the Wigner crystal phase and the critical density of the liquid to solid quantum phase transition. Close to the freezing density we also search for the existence of a stripe phase, but such a phase is never found to be energetically favorable.
The rapid progress in the production and cooling of molecular gases indicates that experimental studies of quantum gases with a strong dipolar interaction is soon within reach. Dipolar gases are predicted to exhibit very rich physics including quantum liquid crystal phases such as density waves as well as superfluid phases, both of which play an important role for our understanding of strongly correlated systems. Here, we investigate the zero temperature properties of the density-wave phase of a two-dimensional (2D) system of fermionic dipoles using a conserving Hartree-Fock theory. We calculate the amplitude of the density waves as a function of the dipole moment and orientation with respect to the 2D plane. The stripes give rise to a 1D Brillouin zone structure, and the corresponding quasiparticle spectrum is shown to have gapped as well as gapless regions around the Fermi surface. As a result, the system remains compressible in the density-wave phase, and it collapses for strong attraction. We show that the density waves have clear signatures in the momentum distribution and in the momentum correlations. Both can be measured in time-of-flight experiments. Finally, we discuss how the striped phase can be realized with experimentally available systems.
We study the density instabilities of a two-dimensional gas of dipolar fermions with aligned dipole moments. The random phase approximation (RPA) for the density-density response function is never accurate for the dipolar gas, and so we incorporate correlations beyond RPA via an improved version of the Singwi-Tosi-Land-Sjölander scheme. In addition to density-wave instabilities, our formalism captures the collapse instability that is expected from Hartree-Fock calculations but is absent from RPA. Crucially, we find that when the dipoles are perpendicular to the layer, the system spontaneously breaks rotational symmetry and forms a stripe phase, in defiance of conventional wisdom.
We determine numerically the ground state of the two-dimensional fully polarized electron gas within the Hartree-Fock approximation without imposing any particular symmetries on the solutions. At low electronic densities, the Wigner crystal solution is stable, but for higher densities (rs{r}_{s}rs​ less than $\ensuremath{\sim}2.7$) we obtain a ground state of different symmetry: the charge density forms a triangular lattice with about 11% more sites than electrons. We prove analytically that this conducting state with broken translational symmetry has lower energy than the uniform Fermi-gas state in the high-density region giving rise to a metal to insulator transition.
We use quantum Monte Carlo methods to calculate the zero-temperature phase diagram of the two-dimensional homogeneous electron gas. We find a transition from a paramagnetic fluid to an antiferromagnetic triangular Wigner crystal at density parameter r(s)=31(1) a.u. and a transition to a ferromagnetic crystal at r(s)=38(5) a.u. The fully spin-polarized fluid is never stable. We search for, but do not find, the ferromagnetic "hybrid" phase proposed by H. Falakshahi and X. Waintal [Phys. Rev. Lett. 94, 046801 (2005)10.1103/PhysRevLett.94.046801].