Theoretical search for the nested quantum Hall effect of composite fermions

Almost all quantum Hall effect to date can be understood as integral quantum Hall effect of appropriate particles, namely, electrons or composite fermions. This paper investigates theoretically the feasibility of nested states of composite fermions which would lead to a quantum Hall effect that cannot be understood as integral quantum Hall effect of composite fermions. The weak residual interaction between composite fermions will play a crucial role in the establishment of such quantum Hall states by opening a gap in a partially filled composite-fermion level. To treat the problem of interacting composite fermions, we develop a powerful method that allows us to obtain the low-energy spectra at composite-fermion fillings of ${\nu }^{*}=n+\overline{\nu }$ without making any assumption regarding the structure of composite fermions in the topmost partially filled level. The method is exact aside from neglecting the composite-fermion Landau level mixing, and enables us to study rather large systems, for example, 24 particles at a total flux of $62hc/e,$ for which the dimension of the lowest Landau level Hilbert space is $\sim {10}^{17}.$ We have investigated, for fully spin-polarized composite fermions, several filling factors between 1/3 and 2/5 using this approach. The results indicate that any possible incompressibility at these fractions is likely to have a fundamentally different origin than that considered earlier.