Hi I'm trying to understand the reasoning behind Peierls substitution. The final result seems to be simply replacing the hopping elements

$$t_{ij} \to t_{ij} e^{i \frac{q}{\hbar} \int_i^j \vec{A} \cdot d \vec{r}}$$

I'm trying to understand why the line integral in the exponent is taken along the straight line connecting the two sites? Aside from having the shortest euclidean distance, I do not see what is so special about this path? From my understanding, the difference of the line integrals along two different paths is proportional to the magnetic flux enclosed by the two paths.

$$\oint \vec{A} \cdot d\vec{r} = \iint \vec{B} \cdot d\vec{S}$$

So it seems to me the choice of path is actually quite important.

I'm trying to understand the reasoning behind Peierls substitution. The final result seems to be simply replacing the hopping elements

$$t_{ij} \to t_{ij} e^{i \frac{q}{\hbar} \int_i^j \vec{A} \cdot d \vec{r}}$$

The line integral in the exponent is taken along the straight line connecting the two sites. Aside from having the shortest euclidean distance, I do not see what is so special about this path? From my understanding, the difference of the line integrals along two different paths is proportional to the magnetic flux enclosed by the two paths.

$$\oint \vec{A} \cdot d\vec{r} = \iint \vec{B} \cdot d\vec{S}$$

So it seems to me the choice of path is actually quite important.