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arXiv:1710.05597 · Physical review. B./Physical review. B

Phase transition and intrinsic metric of dipolar fermions in the quantum Hall regime

Zi-Xiang Hu, Qi Li, Lin-Peng Yang, Wu-Qing Yang, Na Jiang, Ruizhi Qiu, Bo Yang

Published Jan 17, 2018• 10 citations
arXivHTMLPDF

Abstract

For the fast rotating quasi-two-dimensional dipolar fermions in the quantum Hall regime, the rotational symmetry in the two-body interaction breaks when the dipole moment has an in-plane component that can be tuned by an external field. Assuming that all the dipoles are polarized in the same direction, we perform the numerical diagonalization for finite size systems on a torus. We find that while $\ensuremath{\nu}=1/3$ Laughlin state is stable in the lowest Landau level (LLL), it is not stable in the first Landau level (1LL); instead, the most stable Laughlin state in the 1LL is the $\ensuremath{\nu}=2+1/5$ Laughlin state. These FQH states are robust against moderate introduction of anisotropy, but large anisotropy induces a transition into a compressible phase in which all the particles are attracted and form a bound state. We show that such phase transitions can be detected by the intrinsic geometrical properties of the ground states alone. The anisotropy and the phase transition are systematically studied with the generalized pseudopotentials and characterized by the intrinsic metric, the wave function overlap, and the nematic order parameter. We also propose simple model Hamiltonians for this physical system in the LLL and 1LL, respectively.

DOI
10.1103/physrevb.97.035140

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For the fast rotating quasi-two-dimensional dipolar fermions in the quantum Hall regime, the rotational symmetry in the two-body interaction breaks when the dipole moment has an in-plane component that can be tuned by an external field. Assuming that all the dipoles are polarized in the same direction, we perform the numerical diagonalization for finite size systems on a torus. We find that while \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 Laughlin state is stable in the lowest Landau level (LLL), it is not stable in the first Landau level (1LL); instead, the most stable Laughlin state in the 1LL is the \ensuremathν=2+1/5\ensuremath{\nu}=2+1/5\ensuremathν=2+1/5 Laughlin state. These FQH states are robust against moderate introduction of anisotropy, but large anisotropy induces a transition into a compressible phase in which all the particles are attracted and form a bound state. We show that such phase transitions can be detected by the intrinsic geometrical properties of the ground states alone. The anisotropy and the phase transition are systematically studied with the generalized pseudopotentials and characterized by the intrinsic metric, the wave function overlap, and the nematic order parameter. We also propose simple model Hamiltonians for this physical system in the LLL and 1LL, respectively.
We investigate fast rotating quasi-two-dimensional dipolar Fermi gases in the quantum Hall regime. By tuning the direction of the dipole moments with respect to the zzz axis, the dipole-dipole interaction becomes anisotropic in the xxx-yyy plane. For a soft confining potential we find that, as we tilt the angle of the dipole moments, the system evolves from a $\ensuremath{\nu}=1/3$ Laughlin state with dipoles being polarized along the zzz axis to a series of ground states characterized by distinct mean total angular momentum, and finally to an anisotropic integer quantum Hall state. During the transition from the fractional regime to the integer regime, we find that the density profile of the system exhibits crystal-like structures. We map out the ground states as a function of the tilt angle and the confining potential, revealing the competition of the isotropic confining potential and both the isotropic and anisotropic components of the dipole-dipole interaction.
The fractional quantum Hall (FQH) nematic state is a conjectured state of matter in which FQH topological order coexists with spontaneously broken rotation symmetry. Recent experiments on GaAs quantum wells suggest this state, as well as the concomitant isotropic-nematic phase transition, may be realized in a two-dimensional electron gas at certain rational filling factors. Although effective field theories of the FQH nematic state have been constructed, there has been so far no explicit realization of this state as the ground state of a microscopic model of interacting electrons. The authors show via a variational approach and numerical exact diagonalization that tuning Haldane pseudopotentials in a model of interacting electrons in the lowest Landau level induces a quantum phase transition out of the isotropic Laughlin FQH liquid and into the FQH nematic state.
Geometric fluctuations of the density mode in a fractional quantum Hall (FQH) state can give rise to a nematic FQH phase, a topological state with a spontaneously broken rotational symmetry. While experiments on FQH states in the second Landau level have reported signatures of putative FQH nematics in anisotropic transport, a realistic model for this state has been lacking. We show that the standard model of particles in the lowest Landau level interacting via the Coulomb potential realizes the FQH nematic transition, which is reached by a progressive reduction of the strength of the shortest-range Haldane pseudopotential. Using exact diagonalization and variational wave functions, we demonstrate that the FQH nematic transition occurs when the system's neutral gap closes in the long-wavelength limit while the charge gap remains open. We confirm the symmetry-breaking nature of the transition by demonstrating the existence of a "circular moat" potential in the manifold of states with broken rotational symmetry, while its geometric character is revealed through the strong fluctuations of the nematic susceptibility and Hall viscosity.
We examine the superfluid and collapse instabilities of a quasi-two-dimensional gas of dipolar fermions aligned by an orientable external field. It is shown that the interplay between the anisotropy of the dipole-dipole interaction, the geometry of the system, and the p-wave symmetry of the superfluid order parameter means that the effective interaction for pairing can be made very large without the system collapsing. This leads to a broad region in the phase diagram where the system forms a stable superfluid. Analyzing the superfluid transition at finite temperatures, we calculate the Berezinskii-Kosterlitz-Thouless temperature as a function of the dipole angle.
We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where L1{L}_{1}L1​ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd qqq, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd qqq, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L1{L}_{1}L1​ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1{L}_{1}L1​; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.
Motivated by the search for quantum liquid crystal phases in a gas of ultracold atoms and molecules, we study the density-wave and nematic instabilities of dipolar fermions on the two-dimensional square lattice (in the $x\text{\ensuremath{-}}y$ plane) with dipoles pointing to the zzz direction. We determine the phase diagram using two complementary methods, the Hartree-Fock mean-field theory and the linear-response analysis of compressibility. Both give consistent results. In addition to the staggered $(\ensuremath{\pi},\ensuremath{\pi})$ density wave, over a finite range of densities and hopping parameters, the ground state of the system first becomes nematic and then smectic, when the dipolar interaction strength is increased. Both phases are characterized by the same broken fourfold (C4)({C}_{4})(C4​) rotational symmetry. The difference is that the nematic phase has a closed Fermi surface but the smectic does not. The transition from the nematic to the smectic phase is associated with a jump in the nematic order parameter. This jump is closely related to the Van Hove singularities. We derive the kinetic equation for collective excitations in the normal isotropic phase and find that the zero sound mode is strongly Landau damped and thus is not a well-defined excitation. Experimental implications of our results are discussed.