Bug: magnetic unit cell commensuration in the Hofstadter model

[pelegemanuel]

Bug description
The HofstadterFermions class supports 3 different gauge choices: landau_x, landau_y, symmetric, which should all be physically equivalent. I have used ED simulations on small (3 by 3) systems to confirm that this is indeed the case. While the results for landau_x and landau_y always agree, the results of the symmetric gauge choice agree only if boundary conditions are open in both x and y.

I believe the problem has to do with the incommensuration of the magnetic unit cell and the lattice. As one passes through mx sites of a magnetic unit cell [mx, 1] in the landau_x gauge, a phase of (2pi / q) * mx, which is a multiple of 2pi, is acquired. As one passes through mx sites in the symmetric gauge, only (0.5 * 2pi / q) * mx is acquired, which is an integer multiple of pi, rather than 2pi. Hence, if Lx is an odd integer multiple of mx, a phase of -1 is acquired encircling the lattice, and the boundary conditions are antiperiodic. A different model is implemented implicitly, though physical equivalence is expected, since one only made a different gauge choice.

PR Description

1. Assert commensuration of the phases. Namely, for the symmetric gauge, assert L_i % (2 m_i) == 0, with i=x,y and otherwise raise an error. Add a similar assertion to the other gauges.
2. Add a test for the equivalence of different gauges. Done using ED for fermions only, the bosonic Hofstadter model gets too large.