# Birotons and \"Dark\" Quantum Hall Hierarchies
## 1 Introduction  
Today, the FQHE provides the only experimentally accessible system for observing anyons--quasiparticles with non-Fermi and non-Bose statistics. The pioneering works on the experimental detection of quasiparticles with abelian anyonic statistics \(\pi/3\) in the FQHE state of \(1/3\)[1, 2] opened fundamentally new prospects for incorporating anyons into applied physics. At present, the FQHE states with non-abelian anyons in the fractional states of \(5/2\) and \(12/5\) have been the focus of intense research [3, 4]. In addition to single anyons, multiparticle anyon complexes have been observed, and their collective properties have been investigated [5]. In the very near future, further advancements in the experimental methods of studying anyon matter may well enable the observation of quasiparticles with more sophisticated abelian and non-abelian statistics than that of \(\pi/3\). However, at the current stage of development, FQHE physics is facing challenges related to the thermodynamics and spectral properties of the observed fractional states. These issues raise fundamental questions about the FQHE hierarchical structure and the interrelation between different fractional states. For example, in the spectrum of the neutral excitations of the \(1/3\), \(2/5\), and \(3/7\) fractional states, there have been experimentally observed charge density excitations with abnormally low energies compared to the calculations [6]. Other significant problems arise concerning the interpretation of conductivity activation dependencies for the known FQHE states, close to the filling factor of \(1/2\)[7]. Creating a general hierarchical structure of fractional states requires building its foundation on the lowest spin sublevel of the zero Landau level \(\
u<1\). However, such a task can be considered within several theoretical approaches. The choice of a particular approach shall be based on either experimental observations or the numerical solution of the Schrodinger equation obtained by exact diagonalization for the system of a sufficiently large number of particles. In the present work, we argue that even in the best existing samples, the random potential precludes the observation of the vast majority of possible fractional states. For this reason, the exact diagonalization technique remains the only feasible method of organizing FQHE states into a hierarchical system. All modern conceptions of FQHE states at the lowest Landau level can be summarized by the following well-established facts. First, there exist fractional states belonging to the main Laughlin hierarchy \(\
u=\frac{1}{n}\) (where \(m\) is an odd integer number), as well as their symmetrical counterparts, \(\
u=1-\frac{1}{m}\), emerging due to the electron-hole symmetry. Second, there exists the Jain series. It generalizes the main Laughlin series with the electron filling factors expressed as \(\
u=\frac{m}{2 nm\pm 1}\) and their symmetric pairs of the form \(1-\frac{m}{2 nm\pm 1}\) (where \(n\) is an integer number). Finally, several experimentally detectable \"weak\" fractional states lie outside the main Laughlin-Jain hierarchy, for example, \(4/11,\,4/13,\,5/13\), etc. Hence, it is natural to ask the question of whether the Laughlin-Jain hierarchy is a complete representation of fractional states, with few possible exceptions, or are there any fractional hierarchies for some reason unobservable experimentally (\"dark hierarchies\"). Indeed, since the very discovery of the FQHE, several theoretical approaches have been developed to answer this question. The works of [PERSON] and [PERSON] suggest the following model to account for possible hierarchies of the FQHE. When a large quantity of the charged excitations of the \(\frac{1}{m}\) Laughlin state carrying the charge \(\pm\frac{e}{m}\) (quasi-electrons and quasi-holes) are present in an electronic system, they themselves form a Laughlin-like child state at \(\
u=\frac{p}{q}\)[8; 9].