# Birotons and \"Dark\" Quantum Hall Hierarchies
## 1 Introduction
Since the quasiparticle charge reduces as the denominator increases proportionally to \(q^{-1}\), the given Haldane-Halperin models predict a decrease in the value of the energy gap between the ground and the excited states with the growing denominator of the fractional state. An alternative hierarchical structure of the FQHE states was proposed by [PERSON]; he developed the conception of composite fermions--quasiparticles consisting of an electron and an even number of the magnetic flux quanta [10]. This model is based on mapping FQHE states onto composite fermions' integer QHE states (the Laughlin-Jain hierarchy mentioned above). The given structure can be generalized as the FQHE states with the filling factor \(\
u\) are mapped onto the new fractional states with the electron filling factor \(\
u_{2}>1\). Hence, successive mappings lead to one of the Laughlin states--the foundation of every hierarchy of the FQHE states. Such a construction corresponds to the system with several \"sorts\" of composite fermions that differ in the number of attached flux quanta. Generally speaking, given an arbitrary fractional filling factor \(\
u\), both described theoretical approaches permit a few ways of bringing a particular FQHE state to the top of its hierarchy. [PERSON] himself suggested selecting real FQHE states on the basis of energy consideration, whereas other studies argued that electron density configurations obtained in different ways are equivalent [11]. To resolve such an uncertainty, [PERSON] and [PERSON] [12] combined the approaches proposed by [PERSON], [PERSON], and [PERSON] and devised a method of matching every fraction to a unique path leading to the top of its hierarchy. Not only does it remove the construction ambiguity, but it also enables qualitative estimation of the energy of the first excited state for every filling factor. In essence, the [PERSON] arrangement is similar to [PERSON]'s, except that at every level of the hierarchy, a new composite particle \"captures\" two extra magnetic flux quanta. Thus, for the child state, the filling factor \(\
u\) can be described by the mediant of the parent state, \(\frac{p}{q}\), and the critical fraction, \(\frac{p^{\prime}}{q^{\prime}},\ q^{\prime}=2m\), as follows: \(\
u=\frac{p}{q}\oplus\frac{p^{\prime}}{q^{\prime}}=\frac{p+p^{\prime}}{q+q^{ \prime}}\). In this case, the energy gap that separates the ground state from the excited states drops for every descendant fraction within its own hierarchy. This rule does not impose principal restrictions on energy gaps in different hierarchies. Hence, there is no direct link between the denominator value of an arbitrary FQHE state and the energy gap. Finding the pattern in energy gaps for a number of filling factors can give us evidence on the credibility of a given hierarchy scheme, even without conjecturing a trial wavefunction for a state at said filling factors, as, e.g., Jain in [10]; therefore, it is important to find the gaps. The choice between the given hierarchical models of the FQHE depends on the development of modern computational resources. However, even for the existing ultra-fast computers, calculating the dispersion dependencies of neutral excitations by solving the many-electron Schrodinger equation with exact diagonalization method proves difficult. It allows covering the range where the momentum corresponding to the energy gap liesonly for the main Laughlin states 1/3 and 2/3. Indeed, explicitly calculating the gap even for the next fractional state 2/5 in the Laughlin-Jain hierarchy presents a considerable computational challenge [13]. To overcome this technical difficulty, we used general assumptions about the dispersion dependencies of neutral excitations for the FQHE states at the zero Landau level regarding the link between the magnitude of the energy gap and the lowest energy at zero momentum. We tested these assumptions numerically for several fractions and built the hierarchy of the FQHE states for \(\
u\) from 1/4 to 3/4--most suitable for the experimental study.