a typical term contributing to cumulant $C_M$, which can be written in the form $$\label{eqn:C_m}
C_{M,\alpha} = \frac{L}{2 \pi}
\int_{-\frac{\pi}{L}}^{\frac{\pi}{L}} dK \prod_{i=1}^d \langle u_{nK} | \partial^{m_i}_K| u_{nK} \rangle,$$ where $\sum_{i=1}^d m_i = M$ and where we have used the periodic Bloch functions $u_{nK}(x)$ as a basis. The periodic Bloch functions can be written in terms of Wannier functions, $$u_{nK}(x) =  \sum_{p=-\infty}^\infty \exp(iK(p L - x)) a_n(x - p L),$$ where $a_n(x)$ denote the Wannier functions. With this definition it holds that $$\frac{L}{2\pi} \int_{-\pi/L}^{\pi/L} d K \int_0^L d x |u_{nK}(x)|^2 =
\int_{-\infty}^{\infty} d x |a_n(x)| = 1.$$

We can rewrite a scalar product appearing in Eq. (\[eqn:C\_m\]) as $$\begin{aligned}
\nonumber 
\langle u_{nK} | \partial^{m}_K| u_{nK} \rangle =  \sum_{ \Delta p= - \infty}^\infty \exp(-iK \Delta p L )
\int_{-\infty}^\infty dx  \\ a_n^*(x - \Delta p L) (-i x)^m a_n(x). \hspace{2cm}
\label{eqn:scalarprod}\end{aligned}$$ Substituting Eq. (\[eqn:scalarprod\]) $C_{M,\alpha}$ and integrating in $K$ results in $$\begin{aligned}
C_{M.\alpha} = 
\sum_{\Delta p_1 = -\infty}^\infty 
...
\sum_{\Delta p_d = -\infty}^\infty \delta[\Delta P , 0] \hspace{1cm}\\ \hspace{.5cm}
\prod_{j=1}^d 
\left \{ \int_{-\infty}^\infty dx_j (-i x_j)^{m_j} 
a_n^*(x_j - \Delta p_j L)  a_n(x_j) \right \}
, \nonumber\end{aligned}$$ where $\Delta P = \sum_{j=1}^d \Delta p_j$ and $\delta[\Delta P , 0]$ is a Kronecker delta.

