# How to understand Peierls substitution? [duplicate]

The Peierls substitution is given by the following formula: $$t_{12}\rightarrow t'_{12}=\exp\left[\int_{{\bf R}_{1}}^{{\bf R}_{2}}{\bf A}({\bf r})\cdot d{\bf r}\right]t_{12}.$$

What are the meanings of the $${\bf R}_{1}$$ and and $${\bf R}_{2}$$? For example, let us assume that I have a NaCl crystal with body center cubic structure. The fractional coordinates are $$(0, 0, 0)$$ for Na and $$(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$$ for Cl, and the lattice vectors are $$({\bf a},{\bf b},{\bf c})$$. Now, the $$t_{12}$$ is defined as the hopping from the $$p$$ orbital of Cl [in the cell $${\bf R}(lmn)=l{\bf a}+m{\bf b}+n{\bf c}$$] to the $$s$$ orbital of Na [in the cell of $${\bf R}(uvw)=u{\bf a}+v{\bf b}+w{\bf c}$$]. Now my question is: If we want to do Peierls substitution, should I use $${\bf R}_{1}=l{\bf a}+m{\bf b}+n{\bf c}+\frac{1}{2}{\bf a}+\frac{1}{2}{\bf b}+\frac{1}{2}{\bf c}$$ and $${\bf R}_{2}=u{\bf a}+v{\bf b}+w{\bf c}$$, or should I use $${\bf R}_{1}=l{\bf a}+m{\bf b}+n{\bf c}$$ and $${\bf R}_{2}=u{\bf a}+v{\bf b}+w{\bf c}$$?

Don't know if this is still relevant to you after all the years. But I think $$R1=la+mb+nc+1/2a+1/2b+1/2c$$ and $$R2=ua+vb+wc$$ is the way to go. Because we should make sure that the accumulated phase due to the presence of $$\vec{A}$$ can reproduce flux $$\phi$$ through any loop. Thus only by using $$R_1$$ and $$R_2$$ as their actual positions can you ensure that the total phase $$\Pi_{i \in \text{loop}}exp(\int_{R_i}^{R_{i+1}} \vec{A} \cdot d\vec{r})$$ along the loop is accumulated to $$\phi = \oint \vec{A} \cdot d\vec{r}$$