# Composite fermion basis for two-component Bose gases

## TL;DR

This paper develops a mathematically rigorous basis for composite fermion states in two-component Bose gases, enabling better analysis of their structure and overlaps with exact solutions in quantum Hall systems.

## Contribution

It introduces multiple methods to construct a linearly independent basis for simple CF states, addressing a key mathematical challenge in the field.

## Key findings

- Constructed linearly independent basis for CF states
- Analyzed the structure of lowest-lying states
- Studied overlap scaling with exact wave functions

## Abstract

Despite its success, the composite fermion (CF) construction possesses some mathematical features that have, until recently, not been fully understood. In particular, it is known to produce wave functions that are not necessarily orthogonal, or even linearly independent, after projection to the lowest Landau level. While this is usually not a problem in practice in the quantum Hall regime, we have previously shown that it presents a technical challenge for rotating Bose gases with low angular momentum. These are systems where the CF approach yields surprisingly good approximations to the exact eigenstates of weak short-range interactions, and so solving the problem of linearly dependent wave functions is of interest. It can also be useful for studying higher bands of fermionic quantum Hall states. Here we present several ways of constructing a basis for the space of so-called "simple" CF states for two-component rotating Bose gases in the lowest Landau level, and prove that they all give sets of linearly independent wave functions that span the space. Using this basis, we study the structure of the lowest-lying state using so-called restricted wave functions. We also examine the scaling of the overlap between the exact and CF wave functions at the maximal possible angular momentum for simple states.