\theta_y - i e B L_x y \right) \Psi(x,y).\end{aligned}$$ Hence, consistency of the boundary condition requires $$\exp\left(- i e B L_x L_y \right) = 1 \ \Rightarrow \
B = \frac{2 \pi  n_\Phi}{e L_x L_y}, \ n_\Phi \in {{\mathbb{Z}}}.$$ The total magnetic flux through the torus $$\Phi = B L_x L_y = \frac{2 \pi n_\Phi}{e},$$ is hence quantized in integer units of the elementary magnetic flux quantum $2 \pi/e$. Interestingly, the spectrum of a charged particle in a constant magnetic field is discrete (but infinitely degenerate) already in the infinite volume. As we will see, in the finite periodic volume it has only a finite $|n_\Phi|$-fold degeneracy determined by the number of flux quanta.

A quantum mechanical charged particle is sensitive to the complex phases defined by the Polyakov loops $$\exp(i e \Phi_x(y)) = \exp(i e B L_x y - i \theta_x), \
\exp(i e \Phi_y(x)) = \exp(i e B L_y x - i \theta_y),$$ which are measurable in Aharonov-Bohm-type experiments. Remarkably, the Polyakov loops explicitly break the translation invariance of the torus at the quantum level. This is reminiscent of the quantum mechanical breaking of CP invariance due to the $\theta$-vacuum angle in non-Abelian gauge theories. The complex phases 