Quantum Hall Ferroelectrics and Nematics in Multivalley Systems

We study broken symmetry states at integer Landau-level fillings in multivalley quantum Hall systems whose low-energy dispersions are anisotropic. When the Fermi surface of individual pockets lacks twofold rotational symmetry, like in bismuth (111) [Feldman et al. , Observation of a Nematic Quantum Hall Liquid on the Surface of Bismuth, Science 354, 316 (2016)] and in ${\mathrm{Sn}}_{1-x}{\mathrm{Pb}}_x\mathrm{Se}$ (001) [Dziawa et al., Topological Crystalline Insulator States in ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$, Nat. Mater. 11, 1023 (2012)] surfaces, interactions tend to drive the formation of quantum Hall ferroelectric states. We demonstrate that the dipole moment in these states has an intimate relation to the Fermi surface geometry of the parent metal. In quantum Hall nematic states, like those arising in AlAs quantum wells, we demonstrate the existence of unusually robust Skyrmion quasiparticles.

Popular Summary

When electrons are confined to move in a two-dimensional structure and are subjected to a strong perpendicular magnetic field, their orbital motion becomes quantized into Landau levels. Each of these levels contains a large number of states, which correspond to different locations of the center of the orbital motion of the electrons. When many electrons fill these Landau levels, their interactions are greatly enhanced, leading to a rich set of phenomena that can include various forms of spontaneous symmetry breaking and excitations that carry fractional quantum numbers. We studied a large class that spontaneously break space symmetries and that potentially arise when there are a few Landau levels that have nearly the same energy.

We found that when Landau levels have integer filling, the resulting states can break space symmetries like rotations, in which case they are known as quantum Hall nematic states. We also found new states that break space inversion, which we have dubbed “quantum Hall ferroelectric” states. This behavior arises because interactions favor states in which the electrons polarize into a single Landau level, resulting in a wave-function symmetry that is lower than the crystalline symmetry. Using a combination of analytical and numerical methods, we also found that some of these states can support exotic charged quasiparticles known as Skyrmions, which are hedgehoglike textures of the pseudospin magnetization.

Two open challenges are the experimental realization of our proposed quantum Hall ferroelectric state and the generalization of this physics to the case in which the Landau levels have fractional filling.

Figure 1

ARPES Fermi surfaces of (a) the Bi(111) surface from Ref. [11] and (b) the ${\mathrm{Sn}}_{1-x}{\mathrm{Pb}}_x\mathrm{Se}$ (001) surface from Ref. [12].

Figure 2

The green triangular region defines the allowed values of $p_I$, and the red dots at the corners represent the fully valley-polarized states in the case of $M=3$.

Figure 3

Probability amplitude contours for coherent states in the first Landau level of a Dirac cone with anisotropic velocities $v_x/v_y\approx 2.2$ (a) and with anisotropy and tilt $\delta v_x\approx 0.3$ (b). The wave function carries a dipole moment perpendicular to the tilt of the Dirac cone.

Figure 4

Number of minority valley electrons (left axis) for quasiparticles as a function of mass anisotropy in a two-valley system (AlAs) and their charge gap (right axis). The charge gap is in Coulomb energy units of $e^2/(\epsilon l)$.

Figure 5

The charge density distribution for the $N=24$ ($N_{\varphi }=N$) system with anisotropic ratio $m_x/m_y=1$, 5.0625, 16. All of the electrons tend to stay in the same flavor $\alpha$ orbitals ($j=0,2,\dots ,N_{\varphi }-1$) for the anisotropic case. Here, $\alpha =1$, 2 represents two components.

Figure 6

The charge density distribution for the $N=21$ ($N_{\varphi }=20$) system with anisotropic ratio $m_x/m_y=16$. The extra electron will stay in the orbital $K/(2\pi /N_{\varphi })$ if one targets the corresponding total momentum, and the ground-state energies are the same for different targeting momentum sectors. The total electron numbers for two flavors are $N_1=20$ and $N_2=1$.

Figure 7

The charge density distribution for the $N=21$ ($N_{\varphi }=20$) system with anisotropic ratio $m_x/m_y=2.0736$. The extra electron will stay near the orbital $K/(2\pi /N_{\varphi })$ if one targets the corresponding total momentum, and the ground-state energies are the same for different targeting momentum sectors. The electron numbers for two flavors are $N_1=18$ and $N_2=3$, depending on the system size.

Figure 8

The finite-size scaling of the charge gap in the quasiparticle regime for $m_x/m_y=6.5536$ and $m_x/m_y=10.4976$. The system size ranges from $N_{\varphi }=12$ to $N_{\varphi }=28$.

Figure 9

The left vertical axis shows the number of spin flips (blue) involved in the lowest-energy charged quasiparticle as a function of mass anisotropy for a system with two anisotropic pockets like AlAs. The right vertical axis shows different calculations of the charge gap including Hartree-Fock (purple) and DMRG for the largest system size (white) and the extrapolated values to infinite size (yellow).