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Variational Study of the Nematic State of the Two Dimensional Electron Gas in a Magnetic Field

Published Jan 1, 2008• 0 citations

Abstract

We have studied the nematic state of the two dimensional electron gas (2DEG) at halffilled Landau level (LL). Our motivation comes from experiments in which anisotropic transport in the 2DEG under high magnetic field and at low temperature was observed. Based on a model of the nematic state proposed by Oganesyan, Fradkin and Kivelson, we investigate this state and compare it with other competing states proposed by other groups. First, we investigate at what LL the nematic state becomes energetically favorable as compared to the isotropic state. Our studies indicate that this occurs at the second excited LL. Moreover, we compare the energy of the nematic state with that of the stripe state obtained within the Hartree-Fock approximation (HF) and we conclude that, for the samples studied experimentally, the nematic state might be more stable. In our study we have used two different methods which both have advantages and disadvantages. The first method used is the Fermi-hyper-netted chain (FHNC) which provides results valid in the thermodynamic limit (infinite size system), however, it is accurate only for a low density system. The second method used is the Monte Carlo method (MC) which can be used on a finite-size system and, thus, the issue arises of how to extrapolate in the thermodynamic limit (finite-size effect). The results obtained from both methods are in good agreement and indicate that the nematic state might be a viable candidate to explain the experimental findings. In order to compare our results for the nematic state to those obtained for the stripe state, we needed to include the kinetic energy contribution beyond the familiar hw_c/2 term in the case of the nematic state which comes from the deformed geometry of the Fermi sea. For the stripe state we have carried out a HF calculation for a more realistic potential for a 2DEG which includes the effects of the finite confinement of the electron wave function along the z-direction.

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We have studied the nematic state of the two dimensional electron gas (2DEG) at halffilled Landau level (LL). Our motivation comes from experiments in which anisotropic transport in the 2DEG under high magnetic field and at low temperature was observed. Based on a model of the nematic state proposed by Oganesyan, Fradkin and Kivelson, we investigate this state and compare it with other competing states proposed by other groups. First, we investigate at what LL the nematic state becomes energetically favorable as compared to the isotropic state. Our studies indicate that this occurs at the second excited LL. Moreover, we compare the energy of the nematic state with that of the stripe state obtained within the Hartree-Fock approximation (HF) and we conclude that, for the samples studied experimentally, the nematic state might be more stable. In our study we have used two different methods which both have advantages and disadvantages. The first method used is the Fermi-hyper-netted chain (FHNC) which provides results valid in the thermodynamic limit (infinite size system), however, it is accurate only for a low density system. The second method used is the Monte Carlo method (MC) which can be used on a finite-size system and, thus, the issue arises of how to extrapolate in the thermodynamic limit (finite-size effect). The results obtained from both methods are in good agreement and indicate that the nematic state might be a viable candidate to explain the experimental findings. In order to compare our results for the nematic state to those obtained for the stripe state, we needed to include the kinetic energy contribution beyond the familiar hw_c/2 term in the case of the nematic state which comes from the deformed geometry of the Fermi sea. For the stripe state we have carried out a HF calculation for a more realistic potential for a 2DEG which includes the effects of the finite confinement of the electron wave function along the z-direction.

We study the ground state of a nematic phase of the two-dimensional electron gas at filling fraction $\ensuremath{\nu}=1∕2$ using a variational wave function having Jastrow pair correlations of the form ${\ensuremath{\Pi}}_{i<j}{({z}_{i}\ensuremath{-}{z}_{j})}^{2}$ and an elliptical Fermi sea. Using the Fermi-hypernetted-chain approximation, we find that below a critical value of the broken-symmetry parameter, the nematic phase is energetically favorable as compared to the isotropic state for the second excited Landau level. We also find that below a critical value of the layer ``thickness'' parameter $\ensuremath{\lambda}$ (and in the actual materials), the quantum nematic is energetically favorable relative to the stripe ordered Wigner crystal phase.

The properties of a two-dimensional electron gas (2DEG) in a semiconductor host with two valleys related by an underlying

{C}_{4}

rotational symmetry are studied using Hartree-Fock and various other many-body approaches. A familiar artifact of the HF approach is a degeneracy between the valley-polarized---``Ising nematic''---and spin-polarized---ferromagnetic---phases, which is inconsistent with recent variational Monte Carlo (VMC) results. Correlation effects, computed either within the random phase approximation (RPA) or the T-matrix approximation, enhance the valley susceptibility relative to the spin susceptibility. Extrapolating the results to finite interaction strength, we find a direct first-order transition from a symmetry-unbroken state to a spin-unpolarized Ising nematic fluid with full valley polarization, in qualitative agreement with VMC. The RPA results are also reminiscent of experiments on the corresponding 2DEG in AlAs heterostructures.

At low temperatures, two-dimensional electron systems in a perpendicular magnetic field exhibit remarkable quantum phenomena. In these extreme conditions, strongly correlated electron systems stabilize in different quantum phases as the filling factor of Landau levels is varied. In this Brief Report we present finite-size Monte Carlo simulation results for anisotropic quantum Hall liquid states observed at certain even-denominator filling factors. The anisotropic electronic liquid phases are described by a broken rotational symmetry wave function. Our investigations of systems of few electrons in disk geometry indicate that an anisotropic liquid-crystalline quantum Hall phase with broken rotational symmetry is energetically favored relative to an isotropic liquid one.

Based on the Kosterlitz-Thouless-Halperin-Nelson-Young theory of two-dimensional melting and the analogy between Laughlin states and the two-dimensional one-component plasma, we investigate the possibility of liquid crystalline states in a single Landau level (LL). We introduce many-body trial wave functions that are translationally invariant but possess twofold (i.e., nematic), fourfold (tetratic), or sixfold (hexatic) broken rotational symmetry at respective filling factors $\ensuremath{\nu}=1/3,$ 1/5, and 1/7 of the valence LL. We find that the above liquid crystalline states exhibit a soft charge-density wave (CDW) which underlies the translationally invariant state but which is destroyed by quantum fluctuations. By means of Monte Carlo simulations, we determine that, for a considerable variety of interaction potentials, the anisotropic states are energetically unfavorable for the lowest and first excited LL's (with index

L=0,1),

whereas the nematic is favorable at the second excited LL

(L=2).

We have used the variational and diffusion quantum Monte Carlo methods to calculate the energy, pair-correlation function, static structure factor, and momentum density of the ground state of the two-dimensional (2D) homogeneous electron gas. We have used highly accurate Slater-Jastrow-backflow trial wave functions and twist averaging to reduce finite-size effects where applicable. We compare our results with others in the literature and construct a local-density-approximation exchange-correlation functional for 2D systems.

A transport study of two-dimensional (2D) holes confined to wide GaAs quantum wells provides a glimpse of a subtle competition between different many-body phases at a Landau level filling $\ensuremath{\nu}=3/2$ in tilted magnetic fields. At large tilt angles ($\ensuremath{\theta}$), an anisotropic, stripe (or nematic) phase replaces the isotropic compressible Fermi sea at $\ensuremath{\nu}=3/2$ if the quantum well has a symmetric charge distribution. When the charge distribution is made asymmetric, instead of the stripe phase, an even-denominator fractional quantum state appears at $\ensuremath{\nu}=3/2$ in a range of large $\ensuremath{\theta}$, and reverts back to a compressible state at even higher $\ensuremath{\theta}$. We attribute this remarkable evolution to the significant mixing of the excited and ground-state Landau levels of 2D hole systems in tilted fields.