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Synopsis

Quantum Hall states for non-interacting electrons on graphene’s honeycomb lattice are predicted to occur for filling fractions ν = ±(4n + 2), for integer n. Integer and fractional quantum Hall states in graphene are observed in the zeroth Landau level with −2 < ν < 2 and arise from electron-electron interactions. It is generally agreed that there is symmetry breaking in these states, and for the integer quantum Hall state at ν = 0 there has been considerable debate as to whether this arises from quantum Hall ferromagnetism or chiral symmetry breaking (CSB) via magnetic catalysis. There has been relatively little exploration of the orders associated with fractional quantum Hall states in graphene with 0 < |ν| < 1. In this thesis we study the fractional quantum Hall effect in graphene in the presence of CSB orders; in particular charge density wave order (C), easy-axis Neél anti-ferromagnetic order (N) and ferromagnetic order (F).

A feature of incompressible fractional quantum Hall states in e.g. GaAs is that they usually have odd denominators while even denominator states such as ν = 1/2 are compressible. Even denominator fractional quantum Hall states (EDFQH) were recently observed in graphene in a small range of magnetic fields at ν = ±1/2 and ±1/4. The existence of these states is a consequence of degeneracies in the electronic spectrum of graphene that lead to a multicomponent fractional quantum Hall effect. We use a Chern-Simons description of multicomponent fractional quantum Hall states in graphene to investigate the properties of these states and suggest variational wavefunctions that may describe them. We find that the experimentally observed even denominator fractions and standard odd fractions (such as ν = 1/3, 2/5, etc.) can be accommodated within the same flux attachment scheme and argue that they may arise from sublattice or chiral symmetry breaking orders (such as charge-density-wave and antiferromagnetism) of composite Dirac fermions, a phenomenon unifying integer and fractional quantum Hall physics for relativistic fermions in the zeroth Landau level.

With the aim of finding ways to discriminate between different classes of symmetry breaking we study the collective excitations of fractional quantum Hall states in graphene. We focus on states which allow for chiral symmetry breaking orders. We investigate numerically how the collective excitation spectra depend on filling and the flux attachment scheme for two classes of variational states, the Töke-Jain sequence and the Modak-Mandal-Sengupta sequence. We find qualitative similarities between our results and previous work. We propose a hierarchy of stability of states with different flux attachment schemes. We focus on several different ν = 1/3, iii 1/2 and 2/5 states and compare with their observed order of stability in experiments. We find that the stability is largely dominated by the flux attachment and that order parameters play a more minor role. We comment on limitations of our approach.