Theoretical Search for Nested Quantum Hall Effect of Composite Fermions

TL;DR

This paper explores the theoretical possibility of nested quantum Hall states of composite fermions, which could lead to new quantum Hall effects beyond the well-understood integral states, using an advanced computational method.

Contribution

The paper develops an exact method to study low energy spectra of interacting composite fermions without assuming their internal structure, enabling analysis of large systems at various filling factors.

Findings

Indications that incompressibility at certain fractions may have a different origin than previously thought.

The method allows analysis of large systems with minimal assumptions, advancing theoretical understanding.

Results suggest new types of quantum Hall states beyond traditional models.

Abstract

Almost all quantum Hall effect to date can be understood as {\em 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 $\nu^*=n+\bar \nu$ 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 62 $hc/e$, 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.