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Journal of Physics Condensed Matter

Liquid crystalline states in quantum Hall systems

Carlos Wexler, Orion Ciftja

Published Mar 28, 2002• 15 citations
Publisher

Abstract

We investigate liquid crystalline phases with nematic order at (1/3)-filling of the valence Landau level (LL). We generalize Laughlin's fractional quantum Hall (QH) effect wavefunction at ν = (1/3) to include anisotropic nodal distribution by modifying the Jastrow factors. Lengthy Monte Carlo simulations are then used to determine with unprecedented accuracy the (anisotropic) pair distribution function g ( r ) and static structure factor S ( q ) for various degrees of anisotropy. The determination of the correlation energies at (1/3)-filling of an arbitrary LL is then performed by using standard mappings of g ( r ) and S ( q ) to higher LLs. Our results indicate that while Laughlin's state is stable in the lowest LL, there are regions of instability towards nematic order in higher LLs. Possible connections to the recently discovered QH liquid crystals are discussed.

DOI
10.1088/0953-8984/14/14/303

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We investigate liquid crystalline phases with nematic order at (1/3)-filling of the valence Landau level (LL). We generalize Laughlin's fractional quantum Hall (QH) effect wavefunction at ν = (1/3) to include anisotropic nodal distribution by modifying the Jastrow factors. Lengthy Monte Carlo simulations are then used to determine with unprecedented accuracy the (anisotropic) pair distribution function g ( r ) and static structure factor S ( q ) for various degrees of anisotropy. The determination of the correlation energies at (1/3)-filling of an arbitrary LL is then performed by using standard mappings of g ( r ) and S ( q ) to higher LLs. Our results indicate that while Laughlin's state is stable in the lowest LL, there are regions of instability towards nematic order in higher LLs. Possible connections to the recently discovered QH liquid crystals are discussed.
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We investigate broken rotational symmetry (BRS) states for the fractional quantum Hall effect (FQHE) at 13\frac{1}{3}31​ filling of the valence Landau level (LL). Recent Monte Carlo calculations by Musaelian and Joynt [J. Phys.: Condens. Matter 8, L105 (1996)] suggest that Laughlin's state becomes unstable to a BRS state for some critical finite thickness value. We study in detail the properties of such state by performing a hypernetted-chain calculation that gives results in the thermodynamic limit, complementing other methods which are limited to a finite number of particles. Our results indicate that while Laughlin's state is stable in the lowest LL, in higher LL's a BRS instability occurs, perhaps indicating the absence of FQHE at partial fillings of higher LL's. Possible connections to the newly discovered liquid crystalline phases in higher LL's are also discussed.
The single mode approximation (SMA) has been a useful tool for the determination of the collective excitation spectra of various systems. For example, Feynman used it successfully for the calculation of the spectrum of sound waves in superfluid 4He [1]; and Girvin, MacDonald and Platzman employed a projected SMA to determine that of the fractional quantum Hall (QH) effect [2] for Laughlin‐like states [3]. In this paper we calculate the excitation spectrum for density fluctuations in a QH liquid crystal. We consider nematic, tetratic, and hexatic generalizations of Laughlin’s trial wave function [3] having two‐, four‐ and six‐fold broken rotational symmetry (BRS) respectively. We perform extensive Monte Carlo (MC) simulations to compute the density‐coupled spectrum of these liquid crystalline states in the lowest Landau levels. We find significant angular dependence of the spectrum with singular behavior for long wavelengths and considerable deepening of the magnetoroton as the anisotropy is increased.
We investigated the behavior of fractional quantum Hall (FQH) states in a two-dimensional electron system with layer thickness and an in-plane magnetic field. Our comparisons across various filling factors within the first Landau level revealed a crucial observation. A slight in-plane magnetic field specifically enhances the nematic order of the ν=7/3ν= 7/3ν=7/3 FQH state. For this particular filling, through calculating the energy gap, the Ising nematic order parameter, the pair-correlation function, and the static structure factor, we observed that as the in-plane magnetic field increases, the system first enters into an anisotropic FQH phase without closing the spectrum gap, then the FQH nematic (FQHN) phase after neutral gap closing. The system eventually enters a gapless one-dimensional charge density wave (CDW) phase for a large in-plane field. We thus provide a full phase diagram of the ν=7/3ν= 7/3ν=7/3 state in a tilted magnetic field, demonstrating the existence of the FQHN, which aligns with recent resonant inelastic light scattering (RILS) experimental observations.
We investigate broken rotational symmetry (BRS) states at half-filling of the valence Landau level (LL). We generalize Rezayi and Read's (RR) trial wave function, a special case of Jain's composite fermion (CF) wave functions, to include anisotropic coupling of the flux quanta to electrons, thus generating a nematic order in the underlying CF liquid. Using the Fermi hypernetted-chain method, which readily gives results in the thermodynamic limit, we determine the properties of these states in detail. By using the anisotropic pair distribution and static structure functions we determine the correlation energy and find that, as expected, RR's state is stable in the lowest LL, whereas BRS states may occur at half-filling of higher LL's, with a possible connection to the recently discovered quantum Hall liquid crystals.
We show the low-lying excitations at filling factor ν=n+1/3 with realistic interactions can be understood as quantum fluids with "Gaffnian quasiholes" as the proper elementary degrees of freedom. Each Laughlin quasihole can thus be understood as a bound state of two Gaffnian quasiholes, which in the lowest Landau level (LLL) behaves like "partons" with "asymptotic freedom" mediated by neutral excitations acting as "gluons." Near the experimentally observed nematic FQH phase in higher LLs, quasiholes become weakly bound and can fractionalize with rich dynamical properties. By studying the effective interactions between quasiholes, we predict a finite temperature phase transition of the Laughlin quasiholes even when the Laughlin ground state remains incompressible, and derive relevant experimental conditions for its possible observations.