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(adsbygoogle = window.adsbygoogle || []).push({});

[itex] \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)[/itex]

which is equation (3.9a). In equation 3.10, they use this substitution (changing variables with r=i-j):

[itex]

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)

[/itex]

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

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# A Pierels substitution integral approximation

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