# A Pierels substitution integral approximation

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1. Aug 4, 2017

### DeathbyGreen

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):

$H_{ext} = \sum_{k_1k_2\alpha\beta}c^{\dagger}_{k_1\alpha}c_{k_2\beta}\frac{1}{N}\sum_{rj}e^{i(k_2-k_1)j - ik_1r}h^{\alpha\beta}_r(t)i\int_j^{j+r}A_p(t)dl\\ \approx \sum_{k_1k_2\alpha\beta}c^{\dagger}_{k_1\alpha}c_{k_2\beta}\frac{1}{N}\sum_{rj}e^{i(k_2-k_1)j - ik_1r}h^{\alpha\beta}_r(t)irA_p(j +\frac{r}{2},t)$

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

Last edited: Aug 4, 2017
2. Aug 4, 2017

The factor 1/2 you have in the last expression on the right should not be there. ($A(s,t)$ is simply getting evaluated at the midpoint of the interval.)

3. Aug 4, 2017

### DeathbyGreen

Thank you! Do you have a source for that? The equation I wrote

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

Is straight from the textbook, no typos.

4. Aug 4, 2017

Then they goofed. This is basic calculus. The 1/2 in the 3rd expression (in front of the $A$), is in error.

5. Aug 4, 2017

### king vitamin

That textbook is full of errors. They did a really bad job.

6. Aug 8, 2017

### MathematicalPhysicist

From my lecturers and students in QFT,Electromagnetism, Statistical Mechanics and Solid State Physics every expression should be regarded as "upto multiple numerical factor" correct.

In your case it's only half, that's superb! :-D