Models and Methods extending the Composite Fermion theory

Abstract

This thesis attempts to abstract and generalize some key features of the composite fermion (CF) wavefunctions. CF theory successfully explains the experimentally observed fractional quantum Hall effect (FQHE) in the lowest Landau level. We present a short-range strongly interacting model for 2D electron gas in a magnetic field. The model is exactly solvable on the disk, and the eigenstates abstract several key features of the CF wavefunctions. Exact diagonalization of the interaction shows that it produces FQHE at the Jain sequence of filling fractions. Unlike parent Hamiltonians for Laughlin, Moore Read, Read Rezayi, and projected $2/5$ states, the model allows exact solutions for all charged excitations, neutral modes, and incompressible states. The model interaction can be extended to other geometries, and the low-energy spectrum shows the same qualitative features you see in the disk. However, the disk eigenfunctions do not generalize to compact geometries like the sphere and torus. Eigenfunctions in these geometries can nevertheless be written for QPs of $1/m$ state. In the second half, we study the properties of a generalization of CF states, namely parton states. Parton states are seen as a candidate for many non-Abelian filling fractions in FQHE, which require multiple Landau levels. Specifically, we study the real space entanglement spectra (RSES) of these states using an efficient Monte Carlo method. By computing the RSES of simple non-Abelian parton states, namely $\Phi_{2}^2$, $\Phi_{2}^3$ and $\Phi_{3}^2$, we verify that the edge-spectra of partons states of kind $\Phi_{n}^k$ are indeed given by highest-weight representation of $\widehat{su}(n)_k$ Kac-Moody algebra. Finally, we perform extensive benchmarks of the accuracy of the Monte Carlo technique by comparing the RSESs of $2/5$ CF state computed using the efficient Monte Carlo method and brute force methods. We present an approximate projection method that generalizes the Jain-Kamilla projection method, allowing efficient computation of RSES of projected CF states for systems as large as $100$ particles.