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arXiv:cond-mat/9711160 · Physical review. B, Condensed matter

Fermi-fluid description of the half-filled Landau level

Tapash Chakraborty

Published Apr 15, 1998• 8 citations
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Abstract

We present a many-body approach to calculate the ground-state properties of a system of electrons in a half-filled Landau level. Our starting point is a simplified version of the recently proposed trial wave function where one includes the antisymmetrization operator to the bosonic Laughlin state. Using the classical plasma analogy, we calculate the pair-correlation function, the static structure function, and the ground-state energy in the thermodynamic limit. These results are in good agreement with the expected behavior at $\ensuremath{\nu}=$$\frac{1}{2}$.

DOI
10.1103/physrevb.57.8812

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We present a many-body approach to calculate the ground-state properties of a system of electrons in a half-filled Landau level. Our starting point is a simplified version of the recently proposed trial wave function where one includes the antisymmetrization operator to the bosonic Laughlin state. Using the classical plasma analogy, we calculate the pair-correlation function, the static structure function, and the ground-state energy in the thermodynamic limit. These results are in good agreement with the expected behavior at \ensuremath{\nu}=$$\frac{1}{2}.
New trial wave functions corresponding to half filling quantum Hall states are proposed. These wave functions are constructed by first pairing up the quasielectrons of the 1/3 Laughlin quantum Hall state, with the same relative angular momentum for each pair, and then making the paired quasielectrons condense into a 1/4 Laughlin state. The quasiparticle excitations of the proposed wave functions carry ±1/4\pm1/4±1/4 of electron charge, and obey Abelian fractional statistics. In the spherical geometry, the total flux quanta NϕN_ϕNϕ​ is shown to be related to the number of electrons NNN by Nϕ=2N−(5−q)N_ϕ = 2N-(5-q)Nϕ​=2N−(5−q) with qqq being the relative angular momentum between the quasielectrons in each pair which takes values of non-negative even integers. The overlaps are calculated between the proposed trial wave functions, including the ground state, quasiexciton states, and quasihole states, and the exact states of the finite size systems at Nϕ=2N−3N_ϕ = 2N-3Nϕ​=2N−3. The near unity overlaps are obtained in the lowest Landau level, while the moderate overlaps are obtained in the second Landau level. The relevance of the wave functions to the yet to be discovered fractional quantum Hall effect in the lowest Landau level, as well as the 5/2 quantum Hall effect is also discussed.
A system of electrons in a half-filled Landau level is investigated in spherical geometry. For systems of size from N=1 to 14 electrons with flux ${\mathit{N}}_{\mathrm{\ensuremath{\varphi}}}$=2(N-1) the angular momentum of the ground state is as predicted by Hund's second rule for composite fermions of one electron and two vortices at zero magnetic field. Low lying excitations also fit this interpretation and trial wave functions give excellent overlaps. The two-particle correlation function shows a significant correlation hole at short distances and suggests an asymptotically oscillating form at long distances, as in a Fermi liquid.
Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at 13\frac{1}{3}31​ filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.
Strongly correlated two-dimensional electronic systems subject to a perpendicular magnetic field at lowest Landau level (LLL) filling factors: 1/2, 1/4 and 1/6, are believed to be composite-fermion (CF) Fermi liquid phases. Even though a Bose Laughlin wave function cannot describe these filling factors, we investigate whether such a wave function provides a lower energy bound to the true CF Fermi liquid energies. By using Monte Carlo simulations in disk geometry we compute the Bose Laughlin energies and compare them to corresponding results for the spin-polarized LLL CF Fermi liquid state and avalable data from the literature. We find the unexpected result that, for filling factors ν = 1/4 and 1/6, the Bose Laughlin ground-state energy is practically identical to the true CF liquid energy while this is not the case at ν = 1/2 where the Bose Laughlin ground-state energy is sizeably lower than the energy of the CF Fermi liquid state.
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.
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.