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arXiv:cond-mat/0609649

Ground-state of fractional and integral quantum Hall systems at $ν\leq 1$ and it excitations

O. G. Balev

Published Sep 25, 2006• 0 citations
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Abstract

Many-body variational ground-state wave function of two-dimensional electron system (2DES), localized in the main strip (MS)$L_{x}^{\square} \times L_{y}$ of the finite width $L_{x}^{\square}=\sqrt{2 πm} \ell_{0}$ (and the periodic boundary condition (PBC) imposed along $x-$direction), is presented at the fractional and the integral filling factors $ν=1/m$ for two different ion backgrounds: microscopical uniform ion background (UIB) and classical ion jellium background (IJB); $\ell_{0}$ is the magnetic length, $m=2\ell+1$ and $\ell=0, 1, 2,...$. It is shown that the ground-state and the lowest excited-state can correspond to partial crystal-like correlation order among $N$ electrons of the main region (MR) $L_{x} \times L_{y}$; then the study of 2DES of $N$ electrons within MR is exactly reduced to the treatment of 2DES of $\tilde{N}=N L_{x}^{\square}/L_{x}$ electrons localized within MS, with PBC along $x$. The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the $y-$ direction, with the period $L_{x}^{\square}/m$, and independent of $x$. For IJB, at $m=3, 5$, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state (the Laughlin model uses IJB); the same holds at $m=1$. Obtained compound exciton and compound spin-exciton states show finite excitation gaps. The excited compound electron (hole) is composed, within MS, from $m$ strongly correlated quasielectrons (quasiholes) of the charge $e/m$ ($-e/m$. Quantized Hall conductance $σ_{H}=e^{2}/(2 m π\hbar)$ is obtained. The theory is in good agreement with experiments.

DOI
10.48550/arxiv.cond-mat/0609649

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Ground-state of fractional and integral quantum Hall systems at ν≤1ν\leq 1ν≤1 and it excitations
Many-body variational ground-state wave function of two-dimensional electron system (2DES), localized in the main strip (MS)Lx□×LyL_{x}^{\square} \times L_{y}Lx□​×Ly​ of the finite width Lx□=2πmℓ0L_{x}^{\square}=\sqrt{2 πm} \ell_{0}Lx□​=2πm​ℓ0​ (and the periodic boundary condition (PBC) imposed along x−x-x−direction), is presented at the fractional and the integral filling factors ν=1/mν=1/mν=1/m for two different ion backgrounds: microscopical uniform ion background (UIB) and classical ion jellium background (IJB); ℓ0\ell_{0}ℓ0​ is the magnetic length, m=2ℓ+1m=2\ell+1m=2ℓ+1 and ℓ=0,1,2,...\ell=0, 1, 2,...ℓ=0,1,2,.... It is shown that the ground-state and the lowest excited-state can correspond to partial crystal-like correlation order among NNN electrons of the main region (MR) Lx×LyL_{x} \times L_{y}Lx​×Ly​; then the study of 2DES of NNN electrons within MR is exactly reduced to the treatment of 2DES of N~=NLx□/Lx\tilde{N}=N L_{x}^{\square}/L_{x}N~=NLx□​/Lx​ electrons localized within MS, with PBC along xxx. The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the y−y-y− direction, with the period Lx□/mL_{x}^{\square}/mLx□​/m, and independent of xxx. For IJB, at m=3,5m=3, 5m=3,5, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state (the Laughlin model uses IJB); the same holds at m=1m=1m=1. Obtained compound exciton and compound spin-exciton states show finite excitation gaps. The excited compound electron (hole) is composed, within MS, from mmm strongly correlated quasielectrons (quasiholes) of the charge e/me/me/m (−e/m-e/m−e/m. Quantized Hall conductance σH=e2/(2mπℏ)σ_{H}=e^{2}/(2 m π\hbar)σH​=e2/(2mπℏ) is obtained. The theory is in good agreement with experiments.
It is shown that the ground-state and the lowest excited-states of two-dimensional electron system (2DES), with ion jellium background, correspond to partial crystal-like (with the period Lx□=2mπℓ0L_{x}^{\square}=\sqrt{2 m π} \ell_{0}Lx□​=2mπ​ℓ0​) correlation order among NNN electrons of the main region (MR; Lx×LyL_{x} \times L_{y}Lx​×Ly​). Many-body variational ground-state wave function of 2DES is presented at the fractional and the integral filling factors ν=1/mν=1/mν=1/m; m=2ℓ+1m=2\ell+1m=2ℓ+1 and ℓ=0,1,2,...\ell=0, 1, 2,...ℓ=0,1,2,.... The ground-state manifests the broken symmetry liquid-crystal state with 2DES density that is periodic along the y−y-y− direction, with the period Lx□/mL_{x}^{\square}/mLx□​/m, and independent of xxx. At m=3,5m=3, 5m=3,5, the ground-state has essentially lower energy per electron than the Laughlin, uniform liquid, ground-state; the same holds at m=1m=1m=1. At m≥3m \geq 3m≥3, the compound form of the many-body ground-state wave function leads to the compound structure of each electron within the main strip (MS; Lx□×LyL_{x}^{\square} \times L_{y}Lx□​×Ly​). Obtained compound exciton and compound spin-exciton states show finite excitation gaps. The excited compound electron (hole) is composed, within MS, from mmm strongly correlated quasielectrons (quasiholes) of the total charge e/me/me/m (−e/m-e/m−e/m) each. The activation gap is obtained: it is given by the excitation gap of relevant compound exciton, at m≥3m \geq 3m≥3, and by the gap of pertinent compound spin-exciton, at m=1m=1m=1. Quantized Hall conductance σH=e2/(2mπℏ)σ_{H}=e^{2}/(2 m π\hbar)σH​=e2/(2mπℏ) is obtained. The theory is in good agreement with experiments.
Liquid-Crystal State of ν=1/mν=1/mν=1/m Quantum Hall Effects
At filling factor ν=1/mν=1/mν=1/m, mmm odd integer, I present variational ground-state and excited-state wave functions, of two-dimensional electron system with homogeneous ion background, that show the condensation into a liquid-crystal state. For m=1,3,5m=1, 3, 5m=1,3,5, the ground-state energy per electron is substantially lower than the Laughlin one, for uniform liquid state.
The fractional quantum Hall effect was experimentally discovered in 1982. It was observed that the Hall conductivity σyx of a two-dimensional electron system is quantized, σyx=e2/3h, in the vicinity of the Landau level filling factor ν=1/3. In 1983, Laughlin proposed a trial many-body wave function, which he claimed described a “new state of matter”—a homogeneous incompressible liquid with fractionally charged quasiparticles. Here, I develop an exact diagonalization theory that allows one to calculate the energy and other physical properties of the ground and excited states of a system of N two-dimensional Coulomb interacting electrons in a strong magnetic field. I analyze the energies, electron densities, and other physical properties of the systems with N≤7 electrons continuously as a function of magnetic field in the range 1/4≲ν<1. The results show that both the ground and excited states of the system resemble a sliding Wigner crystal whose parameters are influenced by the magnetic field. Energy gaps in the many-particle spectra appear and disappear as the magnetic field changes. I also calculate the physical properties of the ν=1/3 Laughlin state for N≤8 and compare the results with the exact ones. This comparison, as well as an analysis of some other statements published in the literature, show that the Laughlin state and its fractionally charged excitations do not describe the physical reality, neither at small N nor in the thermodynamic limit. The results obtained shed new light on the nature of the ground and excited states in the fractional quantum Hall effect.
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