# Birotons and \"Dark\" Quantum Hall Hierarchies
## 2 Materials and Methods  
We considered a two-dimensional system of electrons with Coulomb interaction confined to a parallelogram cell \(\Lambda\
i z=\alpha\tau_{1}+\beta\tau_{2}\), \(0\leq\alpha,\beta\leq 1\), \(z\) being the complex coordinate and \(\tau_{1,2}\) corresponding to coincident sides of the cell, and quantizing magnetic field \(B\) perpendicular to its plane. We applied periodic boundary conditions (PBCs), that is the state of the system is conserved by magnetic translation by \(\tau_{1,2}\). The PBCs are compatible if the cell is pierced by an integer number \(N_{\mathrm{s}}\) of magnetic flux quanta. In such a system, a Hamiltonian of a single electron has a well-known spectrum of \(\hbar\omega_{C}\left(n+\frac{1}{2}\right),\ \omega_{C}=\frac{eB}{m}\) (Landau levels), each energy level having a finite degeneracy equal to \(N_{\mathrm{s}}\). Therefore, for a number of electrons \(N_{e}\), we can define the following base of states diagonalizing the kinetic part of Hamiltonian \(\prod_{j=1}^{N_{\mathrm{s}}}a_{i_{j},n_{j}}^{\dagger}|0\rangle\), the pairwise interaction part given by  
\[\hat{H}_{C}=\sum_{\alpha,\beta}\sum_{i_{1},i_{2},i_{3},i_{4}}\sum_{n_{1},n_{2},n_{3},n_{4}}V_{i_{1}i_{2}i_{3}i_{4}}^{n_{1}n_{2}n_{3}n_{4}}a_{a,i_{1},n_{1}}^ {\dagger}a_{\beta,i_{2},n_{2}}^{\dagger}a_{\beta,i_{3},n_{3}}a_{a}\dot{\alpha }_{i_{4},n_{4}} \tag{1}\]  
where \(a_{a,i_{2},n_{3}}^{\dagger}\) and \(a_{a,i_{3},n_{4}}\) stand for, respectively, the creation and annihilation operators of a spin \(\alpha\) electron \(k\) in the state \(\psi_{i}^{n}\), \(n\) being the number of the Landau level and ranging from 0 to \(\infty\) and \(i\) specifying the number of the states within the Landau level and ranging from 1 to \(N_{\mathrm{s}}\).  
To find the energy spectrum of the given system, it is necessary to calculate matrix elements of Coulomb potential and to diagonalize (1). We assumed that the cyclotron energy \(\hbar\omega_{C}\) and Zeeman splitting are much larger than the Coulomb energy \(\frac{e^{2}}{el_{B}}\); hence, we can neglect the contribution of higher Landau levels and assume that all electrons are spin polarized; therefore, we only need a finite number of matrix elements for exact diagonalization (in what follows, we however present expressions for matrix elements for arbitrary pairs of Landau levels for the sake of completeness). For a multi-electron system with \(N_{e}\) electrons occupying \(N_{\mathrm{s}}\) possible states, the allowed Fock basis comprises \(C_{N_{\mathrm{s}}}^{N_{e}}\) vectors, which makes the exact diagonalization for even simple fractions quite a tedious task. However, an observation due to [PERSON] [15] helps to build a simpler scheme, as there exists a certain momentum-like operator with \(\gcd(N_{e},N_{\mathrm{s}})\) quantum numbers.  
The periodic eigenfunctions of Landau Hamiltonian were built by [PERSON] [16], who also applied the exact diagonalization scheme in the case of a rectangular cell, and later by [PERSON] and [PERSON] [17], the matrix element for Coulomb interaction was computed in the case of a rectangular cell in [18]. However, the formula for a general parallelogram was never published, to our knowledge. Some results concerning the reciprocal vector operator were rederived in a fashion that seems more fitting for the problem.