Integral Approximation in Peierls Substitution

Question

In the textbook "Topological Insulators and Topological Superconductors" by B. Andrei Bernevig and Taylor L. Hughes, there is a chapter titled "Hall conductance and Chern Numbers". In section 3.1.2 (page 17) they are discussing including an external field in a tight binding model, the Peierls substitution. They make the statement that "if the vector potential" A(s,t) "does not vary widely over the integration path" (when moving from lattice site R to R') we can use the approximation

$$\int_R^{R'}A(s,t)\cdot ds \approx (R-R')\cdot \frac{1}{2}(A(R',t)+A(R,t)) \approx. (R'-R)\cdot \frac{1}{2}A\left(\frac{R'+R}{2},t\right)$$

which is equation (3.9a). In equation 3.10, they use this substitution (changing variables with r=i-j): $$ H_{ext} = \sum_{k_1k_2\alpha\beta}c^{\dagger}_{k_1\alpha}c_{k_2\beta}\frac{1}{N}\sum_{rj}e^{i(k_2-k_1)j - ik_1r}h^{\alpha\beta}_r(t)i\int_j^{j+r}A_p(l,t)dl\\ \approx \sum_{k_1k_2\alpha\beta}c^{\dagger}_{k_1\alpha}c_{k_2\beta}\frac{1}{N}\sum_{rj}e^{i(k_2-k_1)j - ik_1r}h^{\alpha\beta}_r(t)irA_p(j +\frac{r}{2},t) $$

I don't understand where the $ \frac{1}{2}$ goes. It seems to disappear going from equation 3.9a to 3.10.

Answer 1

I think that what is wrong is the approximation above. Let's take the function: $f(x)=x$ to see it clearly.

You say that: $$A(R′,t)+A(R,t) \approx A\left(\dfrac{R′+R}{2},t\right)$$ (it can be easily deduced from the equation shown)

That would mean that: $$R'+R\approx\dfrac{R'+R}{2}$$

Don't you think that here the $\frac{1}{2}$ should be removed in the approximation and that the rest is OK as it is?

Answer 2

I'm not sure if this is just a more longwinded restating of ccorbella's answer, but I there might be a little room for confusion.

I think the problem is in the last part of your first expression.

Assuming that $A$ is slowly varying we can Taylor expand about $\frac{R+ R'}{2}$, so we have \begin{equation} \frac{1}{2}\left(A(R, t) + A(R', t)\right) \approx \frac{1}{2}\left(A(\frac{R + R'}{2}, t) + \frac{d}{dx}A(x, t)|_{x=\frac{R + R'}{2}}\left[R - \frac{R + R'}{2}\right] + A(\frac{R + R'}{2}, t) + \frac{d}{dx}A(x, t)|_{x=\frac{R + R'}{2}}\left[R' - \frac{R + R'}{2}\right] + \cdots\right) \end{equation} To first order, the derivative terms cancel and we have \begin{equation} \frac{1}{2}\left(A(R, t) + A(R', t)\right) = \frac{1}{2}\left(2 A(\frac{R + R'}{2}, t) + O([R-R']^2)\right) \approx A(\frac{R + R'}{2}, t) \end{equation}

This is all to say that if $A$ is slowly varying, you can replace the average of $A$ at $R$ and at $R'$ with $A$ evaluated at the average of $R$ and $R'$. So the first equation is off by an overall factor of 2, but the second equation is correct.