# How to plot exciton wave function with phase information?

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).

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

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

and

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

• The overall shape can be obtained by the A squared. But where do the negative values come from? – Xiaoming Wang May 4 at 5:32
• @XiaomingWang I can't figure out that :) – Jack May 5 at 23:35
• Let's see if anyone has experience with that. – Xiaoming Wang May 5 at 23:40
• 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. – Greg May 6 at 4:57
• @Greg The wave-function are complex valued fore sure. – Xiaoming Wang May 7 at 14:06

Found a small script for BGW to do the plot https://github.com/BerkeleyGW/bgwtools/blob/master/bgwtools/BSE/plot_envelope.py.

• Have you figured out the phase problem? – Jack Jun 4 at 5:19