# Questionable Taylor expansion for Peierls substitution

In this paper, on page 3, the authors go from the tight binding model w the Peierls substitution $$H = \sum_{i,j} \sum_{a,b} t_{a,b} \exp\left(i \int_{\textbf{R}_{j,b}}^{\textbf{R}_{i,a}} dr'_{\mu} A_{\mu} (\textbf{r}',t ) \right) c_{i,a}^\dagger c_{j,b}$$ to $$H= H_0 + \sum_{i,j} \sum_{a,b} t_{a,b} (L_\mu^A A_\mu + 1/2 L_{\mu v}^{AA} A_\mu A_v +..) c_{i,a}^\dagger c_{j,b}$$ where $$H_0$$ is $$H$$ without the exponential, and then define $$L_{\mu}^A = ( \partial_{A_\mu} H)|_{A=0}$$ and so on.

However, shouldn't this second line just be $$H= H_0 + L_\mu^A A_\mu + 1/2 L_{\mu v}^{AA} A_\mu A_v +...$$ since the partial derivative is w res to $$H$$? or am I missing something since we're taking a derivative over $$A$$ instead of $$r$$?

• Seems like the paper is expanding the exponential, instead of H. I am not too sure of their notation either.
– Bio
May 18, 2022 at 18:37
• Yes, it's a typo. And the notation is unclear. This is pre-print, so you could let the authors know, or if it got published somewhere you could find the actual journal article and see if it is fixed already.
– hft
May 19, 2022 at 3:03

In this paper, on page 3, the authors go from the tight binding model w the Peierls substitution $$H = \sum_{i,j} \sum_{a,b} t_{a,b} \exp\left(i \int_{\textbf{R}_{j,b}}^{\textbf{R}_{i,a}} dr'_{\mu} A_{\mu} (\textbf{r}',t ) \right) c_{i,a}^\dagger c_{j,b}$$

I note here, for completeness, that in their expression $$t$$ also depends on $$i$$ and $$j$$, and they use the notation $$t_{ab}(i,j)$$.

to $$H= H_0 + \sum_{i,j} \sum_{a,b} t_{a,b} (L_\mu^A A_\mu + 1/2 L_{\mu v}^{AA} A_\mu A_v +..) c_{i,a}^\dagger c_{j,b}$$ where $$H_0$$ is $$H$$ without the exponential, and then define $$L_{\mu}^A = ( \partial_{A_\mu} H)|_{A=0}$$ and so on.

However, shouldn't this second line just be $$H= H_0 + L_\mu^A A_\mu + 1/2 L_{\mu v}^{AA} A_\mu A_v +...$$

Yes, their expression is wrong, for the reason you already know. This is probably just a typo.

The expansion is obtained (as they say in the paper) by using the Taylor series expansion of the exponential: $$H = \sum_{i,j} \sum_{a,b} t_{a,b}(i,j) \left(1 + i \int_{\textbf{R}_{j,b}}^{\textbf{R}_{i,a}} dr'_{\mu} A_{\mu} (\textbf{r}',t ) - \frac{1}{2}\int_{\textbf{R}_{j,b}}^{\textbf{R}_{i,a}} dr'_{\mu} A_{\mu} (\textbf{r}',t ) \int_{\textbf{R}_{j,b}}^{\textbf{R}_{i,a}} dr''_{\mu} A_{\mu} (\textbf{r}'',t ) +\ldots \right) c_{i,a}^\dagger c_{j,b}$$

Your expression is more correct, but also looks a little questionable to me, probably because the notation is not super clear. You are dotting $$L_\mu$$ with $$A_\mu$$, but there is no free $$A_\mu(\vec r,t)$$ so I suppose your expression probably should have (or implicitly has) a convolution as well as a dot product. I suppose this is the case, since I also suppose the $$\partial_{A_\mu}$$ means a functional derivative. And so, it must be convolved with the $$A$$ field to make sense.

• yes as for the last paragraph, ty May 20, 2022 at 21:40