the effect of a magnetic field, and more than one choice of vector potential is possible. Each of these choices constitutes a particular gauge. For a uniform magnetic field, say in the $z$ direction, common choices in Cartesian coordinates are the Landau Gauge, $\vec{\bf A}_1= xB \hat{\bf y}$ (or $\vec{\bf A}_2 = -yB \hat{\bf x}$), or the so-called Symmetric gauge, $\vec{\bf A}_S = -yB/2 \hat{\bf x} + xB/2 \hat{\bf y}$; all of these choices yield the same field $\vec{\bf B} \equiv \vec{\bf \nabla} \times \vec{\bf A} = B {\hat{\bf z}}$.

Naturally all physical quantities obtained in a calculation should be independent of the gauge choice. Typically a gauge choice is made to take advantage of some symmetry in the problem, so that one can proceed analytically. Just as often the physical system is altered to take advantage of the symmetry afforded by the gauge choice. A good example is the seminal paper by Laughlin, [@laughlin1981quantized] where he wanted to address the quantization of the Hall conductivity in a two-dimensional metal. To this end he chose a ribbon bent into a loop (i.e. open boundary conditions in one direction and periodic boundary conditions in the other). The magnetic field is perpendicular 