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arXiv:cond-mat/9306020 · Physical Review Letters

Tunneling into a two-dimensional electron system in a strong magnetic field

Song He, P. M. Platzman, Bertrand I. Halperin

Published Aug 2, 1993• 123 citations
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Abstract

We investigate the properties of the one-electron Green's function in an interacting two-dimensional electron system in a strong magnetic field, which describes an electron tunneling into such a system. From finite-size diagonalization, we find that its spectral weight is suppressed near zero energy, reaches a maximum at an energy of about 0.2${\mathit{e}}^{2}$/\ensuremath{\epsilon}${\mathit{l}}_{\mathit{c}}$, and decays exponentially at higher energies. We propose a theoretical model to account for the low-energy behavior. For the case of Coulomb interactions between the electrons, at even-denominator filling factors such as \ensuremath{\nu}=1/2, we predict that the spectral weight varies as ${\mathit{e}}_{0}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\omega}}}$/\ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert}, for \ensuremath{\omega}\ensuremath{\rightarrow}0.

DOI
10.1103/physrevlett.71.777

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Physical review. B, Condensed matter•Feb 15, 1993•145 citations
We investigate the properties of the one-electron Green's function in an interacting two-dimensional electron system in a strong magnetic field, which describes an electron tunneling into such a system. From finite-size diagonalization, we find that its spectral weight is suppressed near zero energy, reaches a maximum at an energy of about 0.2e2{\mathit{e}}^{2}e2/\ensuremath{\epsilon}lc{\mathit{l}}_{\mathit{c}}lc​, and decays exponentially at higher energies. We propose a theoretical model to account for the low-energy behavior. For the case of Coulomb interactions between the electrons, at even-denominator filling factors such as \ensuremath{\nu}=1/2, we predict that the spectral weight varies as e0\ensuremath−\ensuremathω{\mathit{e}}_{0}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\omega}}}e0\ensuremath−\ensuremathω​/\ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert}, for \ensuremath{\omega}\ensuremath{\rightarrow}0.
Using a path integral approach and bosonization, we calculate the low-energy asymptotics of the one particle Green's function for a "magnetically incoherent" one dimensional strongly interacting electron gas at temperatures much greater than the typical exchange energy but much lower than the Fermi energy. The Green's function exhibits features reminiscent of spin-charge separation, with exponential spatial decay and scaling behavior with interaction dependent anomalous exponents inconsistent with any unitary conformal field theory. We compute the tunneling density of states at low energies and find that it is a power law in energy with exponent 1/(4g)-1, where g is the Luttinger interaction parameter in the charge sector. The underlying physics is made transparent by the simplicity of the approach. Our results generalize those of Cheianov and Zvonarev [Phys. Rev. Lett. 92, 176401 (2004)]].
We present an effective theory describing the low-energy properties of an interacting two-dimensional electron gas at large noninteger filling factors \ensuremath{\nu}\ensuremath{\gg}1. Assuming that the interaction is sufficiently weak, ${\mathit{r}}_{\mathit{s}}$1, we integrate out all the fast degrees of freedom, and derive the effective Hamiltonian acting in the Fock space of the partially filled Landau level only. This theory enables us to find two energy scales controling the electron dynamics at energies less than \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. The first energy scale, (\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/\ensuremath{\nu})ln(\ensuremath{\nu}${\mathit{r}}_{\mathit{s}}$), appears in the one-electron spectral density as the width of a pseudogap. The second scale, rs{\mathit{r}}_{\mathit{s}}rs​\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is parametrically larger, it characterizes the exchange-enhanced spin splitting and the thermodynamic density of states.
The problem of interacting electrons moving under the influence of a strong magnetic field in two dimensions on a finite disk is reconsidered. First, the results of exact diagonalizations for up to N = 9 electrons for Coulomb as well as for a short–range interaction are used in the search for a peculiar ground state corresponding to filling factor 1/3. Not for the Coulomb, but only for the short–range interaction, can the 1/3– state be safely identified amongst the spectra of various filling factors close to 1/3. Second, the propositions of the concept of quasiparticles, as used in the hierarchical theory, are examined in view of the exact results for the disk geometry. Whereas the theory for the quasiholes is in complete accordance with the spectra, for the quasielectrons, finite size corrections make an analysis difficult. For the quasielectron energy, an extrapolation to N → ∞ is given and compared with the corresponding extrapolations of three different proposals for trial wave functions. While the limiting value for the best trial wave function is very close to the limit of the exact results, the behavior of the finite size corrections of the exact energies and of the trial wave functions, respectively, is qualitatively rather different.
A many-body theory is given to show that the magnetic properties of a two-dimensional electron system are dependent on the filling factor ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$ defined as the ratio of the ideal Fermi energy against the field energy. Explicit results are obtained for absolute zero and large values of this factor. The internal energy is evaluated explicitly in consideration of the ideal, first-order exchange and ring contributions. The result converges for all rs{r}_{s}rs​ because for the ring contribution no small-momentum-transfer assumption is made and for low densities a rigorous analytical continuation is adopted. When plotted against ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$, the internal energy shows parabolic variations in a periodic way until it approaches the average kinetic energy for large ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$. The ideal susceptibility shows zigzag changes characteristic of absolute zero when it is plotted against ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$. It is zero at ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}=2m+1$, where it jumps from $\ensuremath{-}2m$ to 2(m+1)2(m+1)2(m+1), mmm being an integer. The Coulomb interaction suppresses the zigzag variations for small rs{r}_{s}rs​ and inverts the oscillating pattern at rs{r}_{s}rs​ \ensuremath{\sim}0.7. The oscillations thereafter may be considered characteristic of a liquid state rather than a gaseous state as they are somewhat different from those for small rs{r}_{s}rs​. The susceptibility is convergent for all rs{r}_{s}rs​.
We report finite cluster exact diagonalization studies under periodic boundary conditions of two layers of electrons in a strong magnetic field at a half-filled Landau level. For the system studied by Suen et al., the ground states have more than 97 per cent of its spectral weight in the symmetric single particle levels. The eigenfunctions and eigenvalues up to a constant factor for different aspect ratios are essentially the same as previous numerical results of a single layer of electrons of zero thickness at half-filling.
We study the nature of the many-body electron states (and quantum phase transitions between them) in a double-quantum-well structure under a strong external magnetic field as a function of the materials parameters that define the two-layer system, namely, the thickness of individual layers, the separation between them, the individual well potentials confining the electrons, and the potential barrier between them. Motivated by two recent experiments, we consider two different situations, one with (almost) no quantum tunneling between the wells, and the other with substantial interwell quantum tunneling. We use the spherical system finite-size exact diagonalization technique for our calculations. By calculating the overlap of our exact small-system numerical wave functions with various (postulated) analytic wave functions, we comment on the nature of the incompressibility for various Landau-level filling factors (\ensuremath{\nu}). We investigate, in particular details, the possibility of \ensuremath{\nu}=1/2 (i.e., 1/4 average occupancy in each well), and, 1 (i.e., 1/2 occupancy in each well) incompressible states where \ensuremath{\nu} is the total filling factor for the system. We also provide results for the \ensuremath{\nu}=2/3 situation. Our conclusion, based on our use of realistic system parameters in our calculations, is that in both the recent experimental realizations of the \ensuremath{\nu}=1/2 fractional quantum Hall effect, the relevant ground state is the so-called ``331'' state which is stabilized by the competition between intrawell and interwell electron-electron correlations.We calculate for various double-quantum-well structural parameters and magnetic fields the \ensuremath{\nu}=1/2 excitation spectra and the energy gaps, and compare it with the recent experimental data, obtaining excellent quantitative agreement. We provide a detailed quantum phase diagram for the \ensuremath{\nu}=1 state as a function of the interwell Coulomb interaction and the symmetric-antisymmetric single-particle gap, and find that the \ensuremath{\nu}=1 quantum Hall effect observed in the two recent experiments, in contrast to the \ensuremath{\nu}=1/2 effect, belongs in some sense to two different universality classes. We also predict the existence of a reentrant \ensuremath{\nu}=1 quantum Hall effect in double-quantum-well systems.