# Birotons and \"Dark\" Quantum Hall Hierarchies
## 2 Materials and Methods
### Magnetic Translations and One-Electron Wavefunctions  
For a single electron in a constant magnetic field, the Hamiltonian \(\hat{H}=\frac{1}{2m}\big{(}\hat{p}-\frac{c}{c}A\big{)}^{2}\). We work in the Landau gauge \(\hat{A}=(^{0}_{\mathbb{Z}})\), so one-electron states at the lowest Landau level can be represented as [16]  
\[\psi(x,y)=f(z)\exp\left(-\frac{y^{2}}{2l_{B}^{2}}\right),\ z=x+iy \tag{2}\]  
Because of the PBC, we have \(\hat{t}_{m}(\tau_{1,2})\psi=\psi\), where magnetic translations:  
\[\hat{t}_{m}(\tau)=\exp\left(\frac{i}{\hbar}\big{(}\hat{p}-\frac{e}{c}A\big{)} \tau\right). \tag{3}\]  
By the Campbell-Hausdorff formula (here, \(\hat{t}(\tau)\) stands for an ordinary translation by \(\tau\) and \(\tau=\tau_{x}+i\tau_{y}\)):  
\[\hat{t}_{m}(\tau)=\exp\left(\frac{i\tau_{x}\tau_{y}}{2l_{B}^{2}}\right)\exp \left(\frac{i\tau_{y}x}{l_{B}^{2}}\right)\hat{t}(\tau) \tag{4}\]  
\[\hat{t}_{m}(\tau_{1})\hat{t}_{m}(\tau_{2})=\exp\left(\frac{i(\tau_{1x}\tau_{2 y}-\tau_{1y}\tau_{2x})}{l_{B}^{2}}\right)\hat{t}_{m}(\tau_{2})\hat{t}_{m}( \tau_{1}) \tag{5}\]  
For a state to be conserved by two magnetic translations by \(\tau_{1}\) and \(\tau_{2}\), they must commute; therefore:  
\[\tau_{1x}\tau_{2y}-\tau_{1y}\tau_{2x}=2\pi N_{s}l_{B}^{2},\ N_{s}\in\mathbb{Z} \tag{6}\]  
Applying (4) to (2) and taking \(\hat{t}_{m}(\tau_{1,2})\psi=\psi\) into account, we have the following condition on \(f\):  
\[\frac{f(z+\tau_{i})}{f(z)}=\exp\left(-\frac{i(\tau_{i}+2z)\tau_{iy}}{2l_{B}^{2 }}\right) \tag{7}\]  
Specifying \(\tau_{1}\in\mathbb{R},\ \tau_{2}=|\tau_{2}|e^{i\theta}\) (note that (6) yields \(\tau_{1}|\tau_{2}|\sin\theta=2\pi N_{s}l_{B}^{2}\)), we obtain \(\frac{f(z+\tau_{1})}{f(z)}=1\) and, hence, represent \(f\) as a Fourier series \(f(z)=\sum_{k\in\mathbb{Z}}c_{k}\exp\left(i\frac{2\pi kz}{\tau_{1}}\right)\). To find its coefficients, substitute it into (7), which gives  
\[c_{k+N_{s}}=\exp\left(\frac{i\tau_{2}\pi N_{s}}{\tau_{1}}\right)\exp\left(i \frac{2\pi k\tau_{2}}{\tau_{1}}\right)c_{k} \tag{8}\]  
Consequently \(f(z)\) corresponding to LLL states with PBCs constitutes a linear space of dimension \(N_{s}\), with one example of a base delivered by  
\[f_{r}(z)=\sum_{m\in\mathbb{Z}}\exp\left(i\frac{m(m|\tau_{2}|\sin\theta+2X_{r} )|\tau_{2}|e^{i\theta}}{2l_{B}^{2}}\right)\exp\left(i\frac{(X_{r}+m|\tau_{2}| \sin\theta)z}{l_{B}^{2}}\right), \tag{9}\]  
\(X_{r}=\frac{2\pi l_{B}^{2}r}{\tau_{1}}\).  
Finally, the LLL wavefunctions are  
\[\psi_{r}(x,y)=\left(\frac{1}{\tau_{1}\sqrt{\pi}l_{B}}\right)^{ \frac{1}{2}}\sum_{m\in\mathbb{Z}}\exp\left(i\frac{(m^{2}|\tau_{2}|^{2}\sin \theta+2X_{r}m|\tau_{2}|)\cos\theta}{2l_{B}^{2}}\right)\times\\ \times\exp\left(i\frac{(X_{r}+m|\tau_{2}|\sin\theta)x}{l_{B}^{2}} \right)\exp\left(-\frac{(y+X_{r}+m|\tau_{2}|\sin\theta)^{2}}{2l_{B}^{2}} \right). \tag{10}\]  
Similarly, at the \(n\)-th LL (\(H_{n}\) is the \(n\)-th Hermite polynomial):\[\psi_{r}^{n}=\left(\frac{1}{L_{1}\sqrt{\pi 2^{n}n!l!_{B}}}\right)^{ \frac{1}{2}}\sum_{m\in\mathbb{Z}}e^{im\left(2\pi j\frac{L_{2}\cos\theta}{L_{1}} \right)}e^{i\frac{n^{2}l_{2}^{2}\sin 2\theta}{4l_{B}^{2}}}\exp\left(i\frac{(X_{r}+mL_{2} \sin\theta)x}{l_{B}^{2}}\right)\\ \exp\left(-\frac{(y+X_{r}+mL_{2}\sin\theta)^{2}}{2l_{B}^{2}} \right)H_{n}\left(\frac{y+X_{r}+mL_{2}\sin\theta}{l_{B}}\right) \tag{11}\]