While the fractional quantum Hall effect (FQHE) is ubiquitous in two-dimensional electron systems, its properties depend significantly on the host material. Its occurence in bilayer graphene, in particular, is exciting because of the effective tunability of the electron-electron interaction in this host. First, bilayer graphene possesses an eightfold degenerate--spin, valley, and orbital--central Landau level (LL) octet, where novel incompressible states may arise that exploit the orbital degree of freedom. Second, bilayer graphene has a band gap that is tunable by an external electric field. This extra knob may drive quantum phase transitions among different gapped and gapless correlated phases in the system. Third, its LL spectrum exhibits a number of level crossings as a function of the external magnetic field and the potential energy difference between the layers. Fourth, bilayer graphene has an additional energy scale, the leading interlayer hopping, which is about 0.4 eV. As a consequence, the orbital structure of each LL is tuned via B, and so are the matrix elements of the interaction. Fifth, the exchange field of the sea of filled negative energy LLs influences the orbital structure of partially filled levels similarly to the Lamb-shift in atomic physics. Finally, as we can obtain the same filling factor at different charge carrier/magnetic field values (by gating), and, unlike in the monolayer, the kinetic energy and the Coulomb interaction energy scale with different powers of B, we can also control the signicance of LL mixing and screening by inter-layer transitions.
From the first experimental observations of FQHE in bilayer graphene no consistent picture emerges, which is not surprising because of the large parameter space. Both odd and even-denominator fractions have been seen, both in the central LL octet and in higher LLs, and also some phase transitions in dual-gated setting. Particle-hole symmetry was sometimes obeyed, sometimes not.
Our major goals are: (i) We intend to systematically elaborate the phase diagram at odd-denominator fractions within the central Landau level octet, with the orbital degree of freedom, the effect of the sea of filled levels, LL mixing and external elds fully taken into account. (ii) We will investigate the occurence of multi-component states at LL crossings. (iii) We will explore the possibility stabilizing nonabelian FQHE states at even-denominator fractions. In this respect, we have go beyond the often inconclusive overlap integral calculations in small systems, which already proliferate in the literature. (iv) Finally, we will try to resolve paradoxes at integer filling factors. So far, electron correlation has proven irrelevant at integers in all other two-dimensional systems, but particular recent experimental findings might change this.
Reliable background in quantum mechanics and solid-state physics, programming skills, experience with computer algebra softwares.