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arXiv:cond-mat/9911054 · Physical review. B, Condensed matter

From composite fermions to the Calogero-Sutherland model: Edge of the fractional quantum Hall liquid and the dimension reduction

Yue Yu

Published Feb 15, 2000• 6 citations
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Abstract

We derive a microscopic model describing low-lying edge excitations in a fractional quantum Hall liquid with $\ensuremath{\nu}={\ensuremath{\nu}}^{*}/(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varphi}}{\ensuremath{\nu}}^{*}+1).$ For ${\ensuremath{\nu}}^{*}>0,$ it is found that the composite-fermion model reduces to an $\mathrm{SU}({\ensuremath{\nu}}^{*})$ Calogero-Sutherland model in a dimension reduction, whereas it is not exactly soluble for ${\ensuremath{\nu}}^{*}<0.$ However, the ground states in both cases can be found, and the low-lying excitations can show chiral Luttinger-liquid behaviors. On the other hand, we show that the finite-temperature behavior of the $G\ensuremath{-}T$ curve will deviate from the prediction of the chiral Luttinger liquid. We also point out that the suppression of the ``spin'' degrees of freedom agrees with very recent experiments by Chang et al. [Phys. Rev. Lett. 77, 2538 (1996)]. The two-boson model of Lee and Wen (preprint, cond-mat/9801318) is described microscopically.

DOI
10.1103/physrevb.61.4465

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We derive a microscopic model describing low-lying edge excitations in a fractional quantum Hall liquid with \ensuremathν=\ensuremathν∗/(\ensuremathφ\ifmmode~\else˜\fi\ensuremathν∗+1).\ensuremath{\nu}={\ensuremath{\nu}}^{*}/(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varphi}}{\ensuremath{\nu}}^{*}+1).\ensuremathν=\ensuremathν∗/(\ensuremathφ\ifmmode~\else˜\fi​\ensuremathν∗+1). For \ensuremathν∗>0,{\ensuremath{\nu}}^{*}>0,\ensuremathν∗>0, it is found that the composite-fermion model reduces to an SU(\ensuremathν∗)\mathrm{SU}({\ensuremath{\nu}}^{*})SU(\ensuremathν∗) Calogero-Sutherland model in a dimension reduction, whereas it is not exactly soluble for \ensuremathν∗<0.{\ensuremath{\nu}}^{*}<0.\ensuremathν∗<0. However, the ground states in both cases can be found, and the low-lying excitations can show chiral Luttinger-liquid behaviors. On the other hand, we show that the finite-temperature behavior of the G\ensuremath−TG\ensuremath{-}TG\ensuremath−T curve will deviate from the prediction of the chiral Luttinger liquid. We also point out that the suppression of the ``spin'' degrees of freedom agrees with very recent experiments by Chang et al. [Phys. Rev. Lett. 77, 2538 (1996)]. The two-boson model of Lee and Wen (preprint, cond-mat/9801318) is described microscopically.
Based on the composite fermion approach, we derive a microscopic theory describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=ν∗ϕ~ν∗+1ν=\frac{ν^*}{\tildeϕν^*+1}ν=ϕ~​ν∗+1ν∗​. For ν∗>0ν^*>0ν∗>0, it is found that the composite fermion model reduces to an SU(ν∗)(ν^*)(ν∗) Calogero-Sutherland model in the one-dimensional limit, whereas it is not exact soluble for ν∗<0ν^*<0ν∗<0. However, the ground states in both cases can be found and the low-lying excitations can be shown the chiral Luttinger liquid behaviors since a gap exists between the right- and left-moving sectors in each branch of the azimuthal excitations.
We derive a microscopic theory of the composite fermion-type quasiparticles describing the low-lying edge excitations in the fractional quantum Hall liquid with ν = 1/ m . Using the composite fermion transformation, one finds that the edge states of the system in a disc sample are described by the Calogero–Sutherland-like model (CSLM) in the one-dimensional limit. This result presents the consistency between one-dimensional and two-dimensional statistics. It is shown that the low-lying excitations, indeed, have the chiral Luttinger liquid behaviors because there is a gap between the right-and left-moving excitations of the CSLM.
We derive a microscopic theory of the composite particles describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=1/m. Using the composite-particle transformation, one finds that the edge states of the system in a disc sample are described by the Calogero–Sutherland model with other interactions between the composite particles as perturbations. The effects of both the interactions and the confinement potential are considered, and proved unable to change the exponent g=ν=1/m. We also obtain the important chiral condition at the mean field level, and as a result justify microscopically the Chiral Luttinger Liquid model of the FQHE edge states.
This study builds upon the work of Palacios and MacDonald [Phys. Rev. Lett. 76, 118 (1996)], wherein they identify the bosonic excitations of Wen's approach for the edge of the 1∕31∕31∕3 fractional quantum Hall state with certain operators introduced by Stone. Using a quantum Monte Carlo method, we extend this approach to larger systems containing up to 40 electrons and obtain more accurate thermodynamic limits for various matrix elements for a short-range interaction. The results are in agreement with those of Palacios and MacDonald for small systems, but offer further insight into a detailed approach to the thermodynamic limit. For the short range interaction, the results are consistent with chiral Luttinger liquid predictions. We also study excitations using the Coulomb ground state for up to nine electrons to ascertain the effect of interactions on the results; in this case, our tests of the chiral Luttinger liquid approach are inconclusive.
We use bosonic field theories and the infinite system density matrix renormalization group method to study infinite strips of fractional quantum Hall states starting from microscopic Hamiltonians. Finite-entanglement scaling allows us to accurately measure chiral central charge, edge-mode exponents, and momenta without finite-size errors. We analyze states in the first and second levels of the standard hierarchy and compare our results to predictions of the chiral Luttinger liquid theory. The results confirm the universality of scaling exponents in chiral edges and demonstrate that renormalization is subject to universal relations in the nonchiral case. We prove a generalized Luttinger theorem involving all singularities in the momentum-resolved density, which naturally arises when mapping Landau levels on a cylinder to a fermion chain and deepens our understanding of non-Fermi liquids in one dimension.