Orbital Landau level dependence of the fractional quantum Hall effect in quasi-two-dimensional electron layers: Finite-thickness effects

The fractional quantum Hall effect (FQHE) in the second orbital Landau level at even-denominator filling factor of 5/2 remains enigmatic and motivates our work. We theoretically consider the effect of the quasi-two-dimensional (2D) nature of the experimental fractional quantum Hall system on a number of FQH states (filling factors: 1/3, 1/5, and 1/2) in the lowest, second, and third orbital Landau levels (LLL, SLL, and TLL, respectively) by calculating the wave-function overlap, as a function of quasi-2D layer thickness, between the exact ground state of a model Hamiltonian and the consensus variational ansatz wave functions, i.e., the Laughlin wave function for 1/3 and 1/5 and the Moore-Read Pfaffian wave function for 1/2. Using large numerical overlap as a stability (or FQHE robustness) criterion, we find that the FQHE does not occur in the TLL (for any quasi-2D layer thickness), is the most robust for zero thickness (strict 2D limit) in the LLL for 1/3 and 1/5 and for 11/5 in the SLL, and is the most robust at finite thickness (4-5 magnetic lengths) in the SLL for the mysterious even-denominator 5/2 state and the presumably more conventional 7/3 state. We do not find any FQHE at 1/2 in the LLL for any thickness for the quasi-2D models considered in our work. Furthermore, we examine the orbital effects of a nonzero in-plane (parallel) magnetic field, finding that its application effectively reduces the quasi-2D layer thickness and, therefore, could destroy the FQHE at 5/2 and 7/3, while it enhances the FQHE at 11/5, in the SLL. The in-plane field also enhances the LLL FQHE states by making the quasi-2D system more purely 2D. The in-plane field effects could thus be qualitatively different in the LLL and the SLL by virtue of magneto-orbital coupling through the finite-thickness effect. Using exact diagonalization on the torus geometry, we show the appearance of the characteristic threefold topological degeneracy expected for the Pfaffian state. This signature is enhanced by nonzero thickness, corroborating our findings from overlap calculations. Our results have ramifications for wave-function engineering, opening the possibility of creating an optimal experimental system where the 5/2 FQHE state is more likely described by the Moore-Read Pfaffian state with obvious applications in the burgeoning field of fault-tolerant topological quantum computing.