# Birotons and \"Dark\" Quantum Hall Hierarchies
## 4 Discussion  
Calculation results for biproton energies were compared to the activation dependencies of conductivity in fractional states of the Laughlin-Jain hierarchy. In this case, the doubled activation energy was considered approximately equal to the energy of a biproton with zero momentum [24]. Hence, we found that the activation energies closely agree with the calculation for the main fractions, \(1/3\) and \(2/3\). However, approaching the filling factor of \(1/2\) leads to increasing deviation between the calculated energies and the experimental values of activation energies. Thus, for example, for the fractional states of \(5/11\) and \(6/11\), the discrepancy between the calculated and experimental data already exceeds an order of magnitude (Figure 2). The observed disagreement cannot be explained by the non-locality of electronic wavefunctions due to the finite width of the quantum wells used in the experiment. Indeed, the change in biproton energies due to the quantum well width is virtually independent of the FQHE state (Figure 2). It should also be noted that in the experiment, a linear approximation of the activation energy of fractional states in the Laughlin-Jain hierarchy to the filling factor \(1/2\) leads to non-physical, negative values [7],  
Figure 1: (**a**) Dispersion dependencies of the five lowest-energy excitations in Laughlin fractional QHE states \(1/3\) and \(2/3\), calculated for nine electrons in a \(\delta\)-function GaAs/AlGaAs quantum well at a magnetic field of \(10\) T. (**b**) Calculated energies of birotons with zero momentum and rotons in Laughlin state \(1/3\), at a magnetic field of \(10\) T, plotted versus the number of electrons. The dashed line indicates the smallest number of electrons needed to observe the minimum in the dispersion of the roton branch. The solid lines are included as a matter of convenience. which does not occur in the numerical modeling (Figure 3). A plausible explanation of this effect is that, for electron filling factors in the vicinity of \(\
u=1/2\), we are observing a temperature-induced reorganization of electron density rather than activation dependency. In that case, the ground state is brought to the FQHE states of a close filling factor that belong to a \"dark\" (experimentally undetectable) FQHE state. The transition between different fractional states should be accompanied by the appearance of bulk conductivity in a two-dimensional electronic system. This effect is insignificant for the activation energies of the main fractional states of the Laughlin-Jain hierarchy, 1/3 and 2/3. Indeed, energy gaps between the ground states of these fractions and the \"dark\" fractional states of close filling factors is approximately equal to the activation energies themselves. However, as the electron filling factor approaches \(\
u=1/2\), the given energy gaps separating fractions of the Laughlin-Jain hierarchy and close to them \"dark\" fractional states are substantially reduced, leading to the decrease in \"activation\" energy observed in the experiment (Figure 2). Figure 2: (**a**) Dispersion dependencies of the five lowest-energy excitations in the Laughlin fractional QHE state 2/5, calculated for twelve electrons in a \(\delta\)-function GaAs/AlGaAs quantum well at a magnetic field of 10 T. The dashed line marks the boundary of an elementary cell in inverse space. (**b**) Calculated energies of birotons with zero momentum for FQHE state 2/5 at a magnetic field of 10 T plotted as a function of the number of electrons. (**c**) The doubled experimental activation energy of fractional states from the Laughlin–Jain hierarchy measured in [7] (red dots) normalized by the calculated energies of birotons with zero momentum. (**d**) Energy dependencies of birotons with zero momentum for the FQHE states of 1/3 (black circles), 2/5 (black triangles), and 13/27 (black squares). For comparison, red circles indicate the energy dependency of the roton minimum for FQHE 1/3. The solid lines are included for convenience.