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In GW-BSE calculations, people could analyze the exciton wave function $|S\rangle$ by plotting the BSE eigenvector $A_{cv\mathbf{k}}$ that satisfies $|S\rangle = \sum_{cv\mathbf{k}}A_{cv\mathbf{k}}|cv\rangle$. In genreal, $A_{cv\mathbf{k}}$ is complex number, usually the modulus or squared modulus is plotted. I'm wondering how to add the phase information on the plot like Fig. 10 in PRB 93, 235435 (2016).

exciton wave function

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I think these figures are plotting the square of $A_{vc\vec{k}}$, therefore no phase information is contained in these figures. Note that the figures that you are showing are marking each excitonic state with $s$, $p$, $d$, and $f$, which represent exactly the square of the wave function in the problem of the hydrogen atom.

The exciton character and composition in reciprocal space is given by the so-called exciton weights

\begin{equation} \omega_{v\vec{k}}^\lambda=\sum_c |A_{vc\vec{k}}^\lambda|^2 \tag{1} \end{equation}

and

\begin{equation} \omega_{c\vec{k}}^\lambda=\sum_v |A_{vc\vec{k}}^\lambda|^2 \tag{1} \end{equation} which contain information about the BSE eigenvectors and represent the contributions to a given electronic transition to the $\lambda$th solution of the BSE.

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    $\begingroup$ The overall shape can be obtained by the A squared. But where do the negative values come from? $\endgroup$ – Xiaoming Wang May 4 at 5:32
  • $\begingroup$ @XiaomingWang I can't figure out that :) $\endgroup$ – Jack 2 days ago
  • $\begingroup$ Let's see if anyone has experience with that. $\endgroup$ – Xiaoming Wang 2 days ago
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    $\begingroup$ In the figure, the blue and red denotes the phases, so it is not squared. They look like real-valued versions of the wave-function, though, not the complex valued. $\endgroup$ – Greg 2 days ago
  • $\begingroup$ @Greg The wave-function are complex valued fore sure. $\endgroup$ – Xiaoming Wang yesterday

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