Why are maximally localized Wannier functions supposed to be real?

Question

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.

Answer 1

This is not a direct solution to your problem but a partial answer to your question Is there a reason for the MLWFs to be real at the global minimum?.

Marzari and Vanderbilt first reported that

We also find that at the true global minimum the Wannier functions always turn out to be real, apart from a trivial overall phase; while at the false local minima, they are typically complex, only being real if the initial conditions described in Sec. V B have been used. In summary, while false local minima can occur in our minimization scheme, they do not seem to pose any foreseeable problem in actual calculations.

This empirical conjecture has been proved by Ri and Ri in this arxiv paper: Proof that the maximally localized Wannier functions are real.