# Why are maximally localized Wannier functions supposed to be real?

I am computing maximally localized Wannier functions (MLWFs) for several high-pressure phases of a system I am studying using wannier90. For some of these calculations, I see that the imaginary part of the MLWFs obtained is large. This can be seen in the following snippet from the output (Notice the larger than 1 Imaginary/Real ratio):

 *---------------------------------------------------------------------------*
|                               PLOTTING                                    |
*---------------------------------------------------------------------------*

Wannier Function Num:    1       Maximum Im/Re Ratio =    1.336516
Wannier Function Num:    2       Maximum Im/Re Ratio =    1.095717
Wannier Function Num:    3       Maximum Im/Re Ratio =    1.081907
Wannier Function Num:    4       Maximum Im/Re Ratio =    1.179449
Wannier Function Num:    5       Maximum Im/Re Ratio =    1.344709
Wannier Function Num:    6       Maximum Im/Re Ratio =    1.360211
Wannier Function Num:    7       Maximum Im/Re Ratio =    1.161190
Wannier Function Num:    8       Maximum Im/Re Ratio =    1.302460
Wannier Function Num:    9       Maximum Im/Re Ratio =    1.150868
Wannier Function Num:   10       Maximum Im/Re Ratio =    1.230385


From several mailing list threads, tutorials and literature, I see that this means that the MLWFs have converged to a local minimum as opposed to the global minimum we are interested in. For example, in this review it is mentioned that:

It should be noted that the localization functional can display, in addition to the desired global minimum, multiple local minima that do not lead to the construction of meaningful localized orbitals. Heuristically, it is also found that the WFs corresponding to these local minima are intrinsically complex, while they are found to be real, apart from a single complex phase, at the desired global minimum (provided of course that the calculations do not include spin-orbit coupling)

Thus, assuming that the MLWFs I obtained are at a local minimum, I have tried some suggestions from these threads to solve this issue:

• Better choice of initial guess:
• trying different initial guesses and
• using the SCDM approach (a method which doesn't require specifying an initial guess)
• If the system is metallic, include some unoccupied states in the procedure
• The system does not seem to be metallic.

But these have not solved the issue and the large imaginary components are still present in the MLWFs obtained.

The above quoted literature mentions that MLWFs are supposed to be real at the global minimum, but I do not understand the reason for this. Is there a reason for the MLWFs to be real at the global minimum? Any literature regarding this would be helpful.

Note: Alternatively, if there are any other methods that avoid these issues which I can use to obtain the MLWFs corresponding to the global minimum, I would also consider that an answer to my immediate problem.

• WFs are always found to be real whenever we arrived at a global minimum. So this is claimed to be a conjecture based on empirical findings. In this, they claimed to prove the conjecture. Commented Mar 27, 2023 at 9:54
• @AbdulMuhaymin Thanks for the reference. It was helpful. If possible, please add it as an answer and it will be considered for the bounty. Otherwise, the bounty might go unused. Commented Mar 31, 2023 at 5:14
• It's best if each post contains only one question rather than two (this is why I flagged for Tyberius to make the edit that he did). I personally would have even removed the second question, and the only answer also did address only one out of the two questions here. I emailed Nicola Marzari to try to get an answer, but his response was to check the Wannier mailing list (which you have already done). Also related is the answer by ChatGPT: chat.stackexchange.com/transcript/message/63266671#63266671. I also emailed Arash Mostofi but didn't get an answer. You can try others. Commented Apr 1, 2023 at 0:32