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Journal of Experimental and Theoretical Physics Letters

Effective mass of quasiparticles in a Wigner liquid

É. G. Batyev

Published Dec 1, 2002• 1 citation

Abstract

The exchange interaction and effective mass of fermionic excitation in a low-density (r S ≫ 1) system of two-dimensional electrons are estimated from simple considerations. For the ratio of effective (renormalized due to interaction) to band mass, the dependence % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai% aad2gadaahaaWcbeqaaiaacQcaaaaakeaacaWGTbaaaiabg2da9iaa% cIcadaWcgaqaaiaadgeaaeaadaGcaaqaaiaadkhadaWgaaWcbaGaam% 4uaaqabaaabeaaaaGccaGGPaGaciyzaiaacIhacaGGWbGaaiikaGGa% aiab-f7aHnaakaaabaGaamOCamaaBaaaleaacaWGtbaabeaaaeqaaO% Gaaiykaaaa!49B6! $${{m^* } \mathord{\left/ {\vphantom {{m^* } m}} \right. \kern-\nulldelimiterspace} m} = ({A \mathord{\left/ {\vphantom {A {\sqrt {r_S } }}} \right. \kern-\nulldelimiterspace} {\sqrt {r_S } }})\exp (\alpha \sqrt {r_S } )$$ is obtained, where A and α are constants on the order of unity. The effective g factor is independent of r S and is larger than its bare value in the two-valley case (silicon). Comparison with experimental data shows a qualitative agreement with silicon.

DOI: 10.1134/1.1556211

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The exchange interaction and effective mass of fermionic excitation in a low-density (r S ≫ 1) system of two-dimensional electrons are estimated from simple considerations. For the ratio of effective (renormalized due to interaction) to band mass, the dependence % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0xc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcgaqaai% aad2gadaahaaWcbeqaaiaacQcaaaaakeaacaWGTbaaaiabg2da9iaa% cIcadaWcgaqaaiaadgeaaeaadaGcaaqaaiaadkhadaWgaaWcbaGaam% 4uaaqabaaabeaaaaGccaGGPaGaciyzaiaacIhacaGGWbGaaiikaGGa% aiab-f7aHnaakaaabaGaamOCamaaBaaaleaacaWGtbaabeaaaeqaaO% Gaaiykaaaa!49B6!

{{m^* } \mathord{\left/ {\vphantom {{m^* } m}} \right. \kern-\nulldelimiterspace} m} = ({A \mathord{\left/ {\vphantom {A {\sqrt {r_S } }}} \right. \kern-\nulldelimiterspace} {\sqrt {r_S } }})\exp (\alpha \sqrt {r_S } )

is obtained, where A and α are constants on the order of unity. The effective g factor is independent of r S and is larger than its bare value in the two-valley case (silicon). Comparison with experimental data shows a qualitative agreement with silicon.

Within the infinite series of ring (or bubble) diagram approximation for the electronic self-energy as appropriate for the long-range Coulomb interaction, we calculate the density-dependent

T=0

Fermi-liquid quasiparticle effective-mass renormalization in two-dimensional (2D) electron systems in the random-phase approximation. We find that the quasiparticle effective mass increases very strongly with increasing

{r}_{s}

, exhibiting an unexpected divergence at a critical

{r}_{s}

value ($\ensuremath{\sim}16$ in the ideal 2D system). We discuss the possibility for and the consequences of such an interaction-induced effective-mass divergence as the electron density decreases and critically comment on the likely theoretical scenarios which could lead to such a low-density effective-mass divergence in electron liquids.

We discuss the potential advantages of calculating the effective mass of quasiparticles in the interacting electron liquid from the low-temperature free energy vis-a-vis the conventional approach, in which the effective mass is obtained from approximate calculations of the self-energy, or from a quantum Monte Carlo evaluation of the energy of a variational ``quasiparticle wave function''. While raw quantum Monte Carlo data are presently too sparse to allow for an accurate determination of the effective mass, the values estimated by this method are numerically close to the ones obtained in previous calculations using diagrammatic many-body theory. In contrast to this, a recently published parametrization of quantum Monte Carlo data for the free energy of the homogeneous electron liquid yields effective masses that considerably deviate from previous calculations and even change sign for low densities, reflecting an unphysical negative entropy. We suggest that this anomaly is related to the treatment of the exchange energy at finite temperature.

For a charged particle in a homogeneous magnetic field, we construct stationary squeezed states which are eigenfunctions of the Hamiltonian and the non-Hermitian operator % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iqadIfagaqcaiGacoga% caGGVbGaai4CaiabfA6agjabgUcaRiqadMfagaqcaiGacohacaGGPb% GaaiOBaiabfA6agbaa!4894!

\hat X_\Phi = \hat X\cos \Phi + \hat Y\sin \Phi

, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVy0df9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lqpe0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-xir-f0-yqaqVeLsFr0-vr% 0-vr0dc8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGybGbaK% aaaaa!3A82!

\hat X

and Ŷ being the coordinates of the Larmor circle center and Φ is a complex parameter. In the family of the squeezed states, the quantum uncertainty in the Larmor circle position is minimal. The wave functions of the squeezed states in the coordinate representation are found and their properties are discussed. Besides, for arbitrary gauge of the vector potential we derive the symmetry operators of translations and rotations.

Determining the (bare) electron mass

{m}_{0}

in crystals is often hindered by many-body effects since Fermi-liquid physics renormalizes the band mass, making the observed effective mass

{m}^{*}

depend on density. Here, we use a one-dimensional (1D) geometry to amplify the effect of interactions, forcing the electrons to form a nonlinear Luttinger liquid with separate holon and spinon bands, therefore separating the interaction effects from

{m}_{0}

. Measuring the spectral function of gated quantum wires formed in GaAs by means of magnetotunnelling spectroscopy and interpreting them using the 1D Fermi-Hubbard model, we obtain ${m}_{0}=(0.0525\ifmmode\pm\else\textpm\fi{}0.0015){m}_{\text{e}}$ in this material, where

{m}_{\text{e}}

is the free-electron mass. By varying the density in the wires, we change the interaction parameter

{r}_{\text{s}}

in the range from $\ensuremath{\sim}1$--4 and show that

{m}_{0}

remains constant. The determined value of

{m}_{0}

is $\ensuremath{\sim}22%$ lighter than observed in GaAs in geometries of higher dimensionality

D

(

D>1

), consistent with the quasiparticle picture of a Fermi liquid that makes electrons heavier in the presence of interactions.