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arXiv:cond-mat/0108119 · Physical review. B, Condensed matter

Hypernetted-chain study of broken rotational symmetry states for the ν=13 fractional quantum Hall effect and other fractionally filled Landau levels

Orion Ciftja, Carlos Wexler

Published Dec 28, 2001• 16 citations
arXivHTMLPDF

Abstract

We investigate broken rotational symmetry (BRS) states for the fractional quantum Hall effect (FQHE) at $\frac{1}{3}$ 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.

DOI
10.1103/physrevb.65.045306

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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.
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 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.
The nature of the state at low Landau-level filling factors has been a long-standing puzzle in the field of the fractional quantum Hall effect (FQHE). While theoretical calculations suggest that a crystal is favored at filling factors $\ensuremath{\nu}\ensuremath{\lesssim}\frac{1}{6}$, experiments show, at somewhat elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect at $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{11}, \frac{2}{13}, \frac{3}{17}, \frac{3}{19}, \frac{1}{9}, \frac{2}{15}$, and 217\frac{2}{17}172​, which belong to the standard sequences $\ensuremath{\nu}=n/(6n\ifmmode\pm\else\textpm\fi{}1)$ and $n/(8n\ifmmode\pm\else\textpm\fi{}1)$. To address this paradox, we investigate the nature of some of the low-$\ensuremath{\nu}$ states, specifically $\ensuremath{\nu}=\frac{1}{7}, \frac{2}{13}$, and 19\frac{1}{9}91​, by variational Monte Carlo, density matrix renormalization group, and exact diagonalization methods. We conclude that in the thermodynamic limit, these are likely to be incompressible fractional quantum Hall liquids, albeit with strong short-range crystalline correlations. This suggests a natural explanation for the experimentally observed behavior and a rich phase diagram that admits, in the low-disorder limit, a multitude of crystal-FQHE liquid transitions as the filling factor is reduced.
The Fermi hypernetted-chain theory is applied to study the half-filled state of the fractional quantum Hall effect in the thermodynamic limit. We study in detail the radial distribution function, the correlation energy, and the quasiparticle-quasihole excitation spectrum of an unprojected Fermi wave function of the form ${\ensuremath{\psi}}_{\ensuremath{\nu}=1/2}^{\mathrm{Fermi}}={\ensuremath{\Pi}}_{j<k}^{N}{(z}_{j}\ensuremath{-}{z}_{k}{)}^{2}\mathrm{Det}{{\ensuremath{\varphi}}_{\stackrel{\ensuremath{\rightarrow}}{k}}(\stackrel{\ensuremath{\rightarrow}}{r})},$ a possible candidate to describe the half-filled state. Adopting a technique originating from nuclear physics, we compute the effective mass of the fermion excitations near the Fermi surface for this wave function. We find it to be exactly the bare mass of the electron, in accordance with the mean field approximation of not imposing the lowest Landau level constraint. Similar calculations were performed on other related wave functions, which, based on the composite fermion picture, describe the half-filled state of the electrons as a limit of infinite-filled composite fermion Landau levels.
While it is generally believed that a crystal state is favored at filling factors ν<1/6\nu< 1/6ν<1/6 in the lowest Landau level, experiments show, at elevated temperatures, minima in the longitudinal resistance that are associated with fractional quantum Hall effect (FQHE) at the standard sequences ν=n/(6n±1)\nu=n/(6n\pm 1)ν=n/(6n±1) and ν=n/(8n±1)\nu=n/(8n\pm 1)ν=n/(8n±1). To address this paradox, we investigate the nature of the state at filling factors ν=1/7\nu=1/7ν=1/7, 1/9, 1/11 and 2/13 by several methods and find that the competition between the crystal and liquid states is more subtle than previously thought. Exact diagonalization study in the spherical geometry shows behavior consistent with an FQHE liquid for system sizes available. Density matrix renormalization group study in the cylindrical geometry shows that the nature of the state depends on the particle number, with crystal favored for certain particle numbers but not all; these studies do not allow us to make conclusive statements regarding the thermodynamic nature of the state. A variational study shows that the composite-fermion crystal and the renormalized FQHE liquid are very close in energy. Based on these studies, it is not possible to rule out, definitively, an FQHE liquid with strong short-range crystalline correlations at these filling factors.
The authors discuss the existence of fractional quantised Hall states in other than the lowest (n=0) Landau level. They present the results of numerical studies of systems with up to nine particles in the n=1 Landau level. They find that a fractional quantised Hall state is likely for filling fractions of the n=1 Landau level, nu 1 , with nu 1 -1 =5. At nu 1 -1 =3 they find that the Laughlin trial wavefunctions is not a good candidate ground-state wavefunction. If there is a gap at nu 1 -1 it is predicted to be small. At nu 1 -1 = 7 / 2 they predict the existence of a fractional quantised Hall state despite the small or non-existent gap at nu 1 -1 =3. They also show that introducing truncated pseudopotential interactions is not always a physical procedure outside of the lowest Landau level.