Fractional quantum Hall effect in higher Landau levels and in graphene

Open Access
Toke, Csaba
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
March 01, 2007
Committee Members:
  • Jainendra Jain, Committee Chair
  • Gerald Dennis Mahan, Committee Member
  • Vincent Henry Crespi, Committee Member
  • Lyle Norman Long, Committee Member
  • fractional quantum Hall effect
  • graphene
  • composite fermions
  • nonabelian states
In this dissertation I try to extend the composite fermion model, the key to our understanding of the fractional quantum Hall effect (FQHE), beyond its primary domain (media with quadratic dispersion relation, lowest Landau level). The poor performance of the standard composite fermion wave functions in the 1/5 to 4/5 filling factor range of the second Landau level of media with quadratic dispersion is caused by the significant residual interaction between composite fermions. A perturbative improvement to the composite fermion wave function describes the exact ground state very accurately for small systems in the disk geometry in this range. Using the same approach in the spherical geometry the excitation gap is estimated at filling factor 7/3. The FQHE at half-filled second Landau level (5/2 and 7/2) has been a long standing enigma. The popular Pfaffian model of Moore and Read is reviewed and scrutinized. In an exact diagonalization study on finite systems it is shown that the Pfaffian model provides an inadequate description of the excitations, whose non-Abelian braiding statistics has been exploited in several proposals for quantum computing. An alternative understanding of the 5/2 FQHE within the composite fermion theory is presented. The residual interaction between composite fermions is found to play a crucial role in establishing incompressibility at 5/2. The low-energy spectrum and the activation gap are estimated with the help of a perturbative procedure that incorporates inter-composite-fermion interactions. This approach is amenable to systematic improvement, and produces ground as well as excited states. It, however, does not relate to non-Abelian statistics in any obvious manner. Graphene, a single-layer hexagonal form of carbon, provides a two-dimensional electron system with two unusual properties: the low-energy electronic states have a linear dispersion, and the two sublattices of the hexagonal lattice introduce a valley degree of freedom. Both have consequences for the FQHE, which remains to be observed. For large Zeeman energy, the low-energy electronic states still have an SU(2) valley symmetry. Phenomena formerly discussed for the vanishing Zeeman energy limit in GaAs are predicted to occur here: purely interaction-induced integer plateaus, large pseudoskyrmions, fractional sequences, even/odd numerator effects, composite fermion pseudoskyrmions, and a pseudospin-singlet composite-fermion Fermi sea. As a consequence of the linear dispersion of the low-energy carriers the |n|=1 Landau level is predicted to show more robust FQHE than the n=1 Landau level of GaAs. If the Zeeman energy is much smaller than the interaction energy scale, the system is approximately SU(4) symmetric. New FQHE states are predicted at fractions n/(2pn+/-1) with n>=3, which involve an essential interplay between the spin and valley degeneracies. Conditions for the observation of these states are outlined, and the structure of these states and their excitations is described. Zero-temperature phase transitions are predicted to occur between these states when the SU(4) symmetry is weakly broken by external fields.