A Numerical Study of the Composite Fermi Liquid

Abstract

The Fractional Quantum Hall Effect (FQHE) describes the behavior of a strongly correlated 2D electron gas subject to a high perpendicular magnetic field. This results in the effective 'quenching' of kinetic energy by creating a flat-band situation where electrons are forced into discrete, massively degenerate Landau Levels. At near-zero temperatures, the electrons occupy the Lowest Landau Level (LLL) at partial filling represented by a fractional filling factor. A characteristic feature of this phenomenon is the incompressibility of the FQH phase due to the presence of robust gaps in the many-body spectrum, a consequence of the interaction-mediated lifting of degeneracy in the partially filled LLL. This problem can be formulated as an Integer Quantum Hall Effect (IQHE) of bound particles called composite fermions (CFs), where the many-body gap is replaced by the cyclotron gap separating CF Landau Levels for a reduced effective field. This thesis is an attempt to utilize two commonly employed numerical techniques in condensed matter physics, Exact Diagonalization (ED) and infinite-system Density Matrix Renormalization Group (iDMRG) to examine the behavior of an incompressible FQH (Fractional Quantum Hall) phase approaching the special case of the half-filled Landau Level. In particular, we examine the scaling of gaps for fractionally charged excitations (quasi-holes and quasi-particles) above the ground state for filling fractions along the Jain sequence p/2p+1. For large p, the sequence approaches half-filling, where the vanishing effective field leads to gap closure and the formation of a compressible Fermi sea of composite fermions.

Previous theoretical investigations show that the Chern-Simons field theory formulation of the half-filled FQH phase introduces renormalizations in effective mass such that it obtains a p-dependence whose exact form depends on the nature of the underlying electron-electron interaction. By explicitly calculating the excitation gaps and probing their scaling behavior upon varying p under different interactions, the project aims to check whether the findings of numerical studies are consistent with theoretical predictions, and whether smooth variations in the interaction can capture some unique trends in scaling behavior.