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arXiv:cond-mat/9912129 · Physical Review Letters

Excitation Spectrum and Effective Mass of the Even-Fraction Quantum Hall Liquid

Masaru Onoda, T. Mizusaki, Takaharu Otsuka, Hideo Aoki

Published Apr 24, 2000• 11 citations
arXivHTMLPDF

Abstract

To probe the nature of the even-fraction quantum Hall system, we have investigated the low-lying excitation spectrum by exact diagonalization for finite systems. We have found (i) a striking one-to-one correspondence (i.e., a shell structure) between the spectrum and those for free (composite) fermions, (ii) a surprisingly straight scaling plot for the excitation energy that gives a zero gap (metal) in the thermodynamic limit, (iii) the effective mass evaluated from the scaling becoming heavier for nu = 1/2,1/4,1/6, but (iv) some deviations from the single-mode or the Hartree-Fock composite fermion approximation.

DOI
10.1103/physrevlett.84.3942

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To probe the nature of the even-fraction quantum Hall system, we have investigated the low-lying excitation spectrum by exact diagonalization for finite systems. We have found (i) a striking one-to-one correspondence (i.e., a shell structure) between the spectrum and those for free (composite) fermions, (ii) a surprisingly straight scaling plot for the excitation energy that gives a zero gap (metal) in the thermodynamic limit, (iii) the effective mass evaluated from the scaling becoming heavier for nu = 1/2,1/4,1/6, but (iv) some deviations from the single-mode or the Hartree-Fock composite fermion approximation.
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 compute energy gaps for spin-polarized fractional quantum Hall states in the lowest Landau level (LL) at filling fractions $\ensuremath{\nu}=\frac{1}{3},$ 25,\frac{2}{5},52​, 37,\frac{3}{7},73​, and 49\frac{4}{9}94​ using exact diagonalization of systems with up to 16 particles and extrapolation to the infinite system-size limit. The gaps calculated for a pure Coulomb interaction and ignoring finite width effects, disorder and LL mixing agree well with the predictions of composite fermion theory provided the logarithmic corrections to the effective mass are included. This is in contrast with previous estimates, which, as we show, overestimated the gaps at $\ensuremath{\nu}=2/5$ and 3/7 by around 15%. We also study the reduction of the gaps as a result of the nonzero width of the two-dimensional (2D) layer. We show that these effects are accurately accounted for using either Gaussian or ``$z\ifmmode\times\else\texttimes\fi{}$ Gaussian'' (zG)(z\mathrm{G})(zG) trial wave functions, which we show are significantly better variational wave functions than the Fang-Howard wave function. The Gaussian and zGz\mathrm{G}zG wave functions give Haldane pseudopotential parameters which agree with those of self-consistent local density approximation calculations to better than $\ifmmode\pm\else\textpm\fi{}0.2%.$ For quantum well parameters typical of heterostructure samples, we find gap reductions of around 20%. The experimental gaps, after accounting heuristically for disorder, are still around 40% smaller than the computed gaps. However, for the case of tetracene layers in metal-insulator-semiconductor (MIS) devices we find that the measured activation gaps are close to those we compute. We discuss possible reasons why the difference between computed and measured activation gaps is larger in GaAs heterostructures than MIS devices. Finally, we present calculations using systems with up to 18 electrons of the gap at $\ensuremath{\nu}=\frac{5}{2}$ including width corrections.
Exact diagonalization studies have revealed that the energy spectrum of interacting electrons in the lowest Landau level splits, nonperturbatively, into bands, which is responsible for the fascinating phenomenology of this system. The theory of nearly free composite fermions has been shown to be valid for the lowest band, and thus to capture the low-temperature physics, but it overpredicts the number of states for the excited bands. We explain the state counting of higher bands in terms of composite fermions with an infinitely strong short-range interaction between an excited composite-fermion particle and the hole it leaves behind. This interaction, the form of which we derive from the microscopic composite-fermion theory, eliminates configurations containing certain tightly bound composite-fermion excitons. With this modification, the composite-fermion theory reproduces, for all well defined excited bands seen in exact diagonalization studies, an exact counting for $\ensuremath{\nu}>1/3$, and an almost exact counting for $\ensuremath{\nu}\ensuremath{\le}1/3$. The resulting insight clarifies that the corrections to the nearly free composite-fermion theory are not thermodynamically significant at sufficiently low temperatures, thus providing a microscopic explanation for why it has proved successful for the analysis of the various properties of the composite-fermion Fermi sea.
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.
We use the dynamical structure factors of the quantum Hall states at $\ensuremath{\nu}=1/3$ and 1/21/21/2 in the lowest Landau level to study their excitation spectrum. Using the density matrix renormalization group in combination with the time-dependent variational principle on an infinite cylinder geometry, we extract the low-energy properties. At $\ensuremath{\nu}=1/3$, a sharp magnetoroton mode and the two-roton continuum are present and the finite-size effects can be understood using the fractional charge of the quasiparticle. At $\ensuremath{\nu}=1/2$, we find low-energy modes with linear dispersion and the static structure factor $\overline{s}(q)\ensuremath{\sim}{(q\ensuremath{\ell})}^{3}$ in the limit $q\ensuremath{\ell}\ensuremath{\rightarrow}0$. The properties of these modes agree quantitatively with the predictions of the composite-fermion theory placed on the infinite cylinder.
Using the Hamiltonian formulation for composite fermions developed recently, the temperature dependence of the spin polarization is computed for the translationally invariant fractional quantum Hall states at ν = 1/3 and ν = 2/5 in two steps. In the first step, the effect of particle-hole excitations on the spin polarization is computed in a composite-fermion Hartree-Fock approximation. The computed magnetization for ν = 1/3 lies above the experimental results for intermediate temperatures indicating the importance of long-wavelength spin fluctuations which are not correctly treated in the Hartree-Fock approximation. In the second step, spin fluctuations beyond the Hartree-Fock approximation are included for ν = 1/3 by mapping the problem onto the coarse-grained continuum quantum ferromagnet. The parameters of the description in terms of the effective continuum quantum ferromagnet are extracted from the preceding Hartree-Fock analysis. After the inclusion of spin fluctuations in a large- N approach, the results for the finite-temperature spin polarization are in quite good agreement with the experiments.