from above are invariant under shifts by integer multiples of $$a_x = \frac{2 \pi}{e B L_y} = \frac{L_x}{n_\Phi}, \
a_y = \frac{2 \pi}{e B L_x} = \frac{L_y}{n_\Phi},$$ in the $x$- and $y$-directions, respectively. Hence, at the quantum level the continuous translation group of the torus is reduced to a discrete subgroup which plays the role of the accidental symmetry group.

In this paper, we treat the gauge field as a classical background field, while only the charged particle is treated quantum mechanically. It is interesting to note that, once the gauge field is also quantized, the transition functions $\varphi_x(y)$ and $\varphi_y(x)$ become fluctuating physical degrees of freedom of the gauge field. Still, as a consequence of $$\begin{aligned}
A_i(x + L_x,y + L_y)&=&A_i(x,y + L_y) - {\partial}_i \varphi_x(y + L_y) \nonumber \\
&=&A_i(x,y) - {\partial}_i \varphi_x(y + L_y) - {\partial}_i \varphi_y(x), \nonumber \\
A_i(x + L_x,y + L_y)&=&A_i(x + L_x,y) - {\partial}_i \varphi_y(x + L_x) \nonumber \\
&=&A_i(x,y) - {\partial}_i \varphi_y(x + L_x) - {\partial}_i \varphi_x(y).\end{aligned}$$ and of $$\begin{aligned}
\Psi(x + L_x,y + L_y)&=&\exp(i e \varphi_x(y + L_y)) \Psi(x,y + L_y) 
\nonumber \\
&=&\exp(i e \varphi_x(y + L_y) + i e \varphi_y(x)) \Psi(x,y), \nonumber \\
\Psi(x + L_x,y + L_y)&=&\exp(i e 