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arXiv:cond-mat/9507063 · Physical Review Letters

Charge Density Wave Behavior in the Integer Quantum Hall Effect Edge States

M. Franco, Luis Brey

Published Aug 12, 1996• 14 citations
arXivHTMLPDF

Abstract

We analyze the effect of Coulomb interactions on edge excitations of an electron gas confined in a bar of thickness $W$, and in the presence of a magnetic field corresponding to unit filling factor in the quantum Hall effect. We find that such long-range interactions between the edges result in a ground state with quasi-long-range order similar to a charge density wave. The spectral density of states vanishes at zero frequency and increases with frequency faster than any power law. We also find that the conductance of an infinitely long system is zero.

DOI
10.1103/physrevlett.77.1358

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We analyze the effect of Coulomb interactions on edge excitations of an electron gas confined in a bar of thickness WWW, and in the presence of a magnetic field corresponding to unit filling factor in the quantum Hall effect. We find that such long-range interactions between the edges result in a ground state with quasi-long-range order similar to a charge density wave. The spectral density of states vanishes at zero frequency and increases with frequency faster than any power law. We also find that the conductance of an infinitely long system is zero.
Fradkin's formulation of statistical field theory is applied to the Coulomb interacting electron gas in a magnetic field. The electrons are confined to a plane in normal three-dimensional space and also interact with the physical three-dimensional electromagnetic field. The magnetic-translation-group Ward identities are derived. By using them, it is shown that the exact electron propagator is diagonalized in the basis of the wave functions of the free electron in a magnetic field whenever the magnetic-translation-group symmetry is unbroken. The general tensor structure of the polarization operator is obtained and used to show that the Chern-Simons action always describes the Hall-effect properties of the system. A general proof of the Str\^eda formula for the Hall conductivity is presented. It follows that the coefficient of the Chern-Simons terms in the long-wavelength approximation is exactly given by this relation. Such a formula, expressing the Hall conductivity as a simple derivative, in combination with a diagonal form of the full propagator, allows us to obtain a simple expression for the filling factor and the Hall conductivity. Indeed, these results, after assuming that the chemical potential lies in a gap of the density of states, lead to the conclusion that the Hall conductivity is given without corrections by ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$=\ensuremath{\nu}e2{\mathit{e}}^{2}e2/h, where \ensuremath{\nu} is the filling factor. In addition, it follows that the filling factor is independent of the magnetic field if the value of the chemical potential remains in the gap.
States of strongly interacting particles are of fundamental interest in physics and can produce exotic emergent phenomena and topological structures. We consider here two-dimensional electrons in a magnetic field, and, departing from the standard practice of restricting to the lowest LL, introduce a model short-range interaction that is infinitely strong compared to the cyclotron energy. We demonstrate that this model lends itself to an exact solution for the ground as well as excited states at arbitrary filling factors ν<1/2p and produces a fractional quantum Hall effect at fractions of the form ν=n/(2pn+1), where n and p are integers. The fractional quantum Hall states of our model share many topological properties with the corresponding Coulomb ground states in the lowest Landau level, such as the edge physics and the fractional charge of the excitations.
Effects of long-range Coulomb interactions on the Fermi-edge singularity in optical spectra are investigated theoretically for one-dimensional spin-1/2 fermion systems with the use of the Tomonaga-Luttinger bosonization technique. Low-energy excitation spectrum near the Fermi level shows that dispersion of the charge-density fluctuation remains gapless but is nonlinear when the electron-electron (e-e) Coulomb interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., an infinite force range). Temporal behavior of the current-current correlation function is calculated analytically for arbitrary force ranges, ${\ensuremath{\lambda}}_{\mathit{e}}$ and ${\ensuremath{\lambda}}_{\mathit{h}}$, of the e-e and the electron-hole (e-h) Coulomb interactions. (i) When both the e-e and the e-h interactions have large but finite force ranges (${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\infty} and ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\infty}), the correlation function yields a power-law decay only for a long-time regime, which is determined by the force ranges as t>max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]/vF{\mathit{v}}_{\mathit{F}}vF​. Corresponding optical spectrum near the Fermi edge (within an energy range of \ensuremath{\Elzxh}vF{\mathit{v}}_{\mathit{F}}vF​/max[${\ensuremath{\lambda}}_{\mathit{e}}$,${\ensuremath{\lambda}}_{\mathit{h}}$]) exhibits the power-law divergence or the power-law convergence, which is an ordinary Fermi-edge singularity. (ii) When either the e-e or the e-h interaction is of the ${\mathit{x}}^{\mathrm{\ensuremath{-}}1}$ type (i.e., ${\ensuremath{\lambda}}_{\mathit{e}}$\ensuremath{\rightarrow}\ensuremath{\infty} and/or ${\ensuremath{\lambda}}_{\mathit{h}}$\ensuremath{\rightarrow}\ensuremath{\infty}), an exponent of the correlation function is dependent on time to lead the faster decay than that of any power laws. Then the optical spectra show no power-law dependence and always converge (become zero) at the Fermi edge, which is in striking contrast to the ordinary power-law singularity. Relations between the spectral shape and a measurement time (frequency resolution) are also clarified. \textcopyright{} 1996 The American Physical Society.
We study equilibration of quantum Hall edge states at integer filling factors, motivated by experiments involving point contacts at finite bias. Idealizing the experimental situation and extending the notion of a quantum quench, we consider time evolution from an initial nonequilibrium state in a translationally invariant system. We show that electron interactions bring the system into a steady state at long times. Strikingly, this state is not a thermal one: Its properties depend on the full functional form of the initial electron distribution and not simply on the initial energy density. Further, we demonstrate that measurements of the tunneling density of states at long times can yield either an overestimate or an underestimate of the energy density, depending on details of the analysis, and discuss this finding in connection with an apparent energy loss observed experimentally. More specifically, we treat several separate cases: for filling factor $\ensuremath{\nu}=1$ we discuss relaxation due to finite-range or Coulomb interactions between electrons in the same channel, and for filling factor $\ensuremath{\nu}=2$ we examine relaxation due to contact interactions between electrons in different channels. In both instances we calculate analytically the long-time asymptotics of the single-particle correlation function. These results are supported by an exact solution at arbitrary time for the problem of relaxation at $\ensuremath{\nu}=2$ from an initial state in which the two channels have electron distributions that are both thermal but with unequal temperatures, for which we also examine the tunneling density of states.
We study the effect of electron-electron interaction on the charge and spin structures at the edge of integer quantum Hall liquids, under three different kinds of confining potentials. Our exact diagonalization calculation for small systems indicates that the low-energy excitations of the $\ensuremath{\nu}=1$ ferromagnetic state are bosonic edge spin waves. Instabilities of the ferromagnetic state with altering confinement strength result from the softening of these edge spin waves and formation of edge spin textures. In the $\ensuremath{\nu}\ensuremath{\lesssim}2$ regime, exact diagonalization on edge electron systems indicates that compact Hartree-Fock states with different total spin always become ground states in some regions of parameter space, and the ground states appearing between two compact states are their edge spin waves. The initial $\ensuremath{\nu}=2$ instability is toward the compact state with total spin 1. Larger systems are studied using a microscopic trial wave function, and some quantitative predictions on the edge instabilities for a certain type of confining potential are reached in the thermodynamic limit.
We study the ground state and low-energy excitations of fractional quantum Hall systems on a disk at a filling fraction nu = 5/2, with Coulomb interaction and a background confining potential. We find the Moore-Read ground state is stable within a finite but narrow window in parameter space. The corresponding low-energy excitations contain a fermionic branch and a bosonic branch, with widely different velocities. A short-range repulsive potential can stabilize a charge +e/4 quasihole at the center, leading to a different edge excitation spectrum due to the change of boundary conditions for Majorana fermions, clearly indicating the non-Abelian nature of the quasihole.