<div dir="ltr"><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">Dear developers and fellow Wannier90 users,</font><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br></font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">I have recently used Wannier90 to extract tight-binding parameters and develop a model for nonadiabatic small polaron transport in my system, and so far I've seen great success in developing this model. Now, I am moving forward and considering possible superexchange mechanisms, and I believe I can use the final unitary matrices (from both the disentanglement and wannierisation phases) calculated with Wannier90 to connect my MLWFs to the original Bloch states. However, I am a bit confused in a very specific way as to how the code is using the unitary matrices during the procedure, so I wanted to run my thoughts by anyone who can help.</font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br></font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">I'll focus on just the disentanglement since it seems the wannierisation phase will follow suit, and the dimensionality of the disentanglement phase is nontrivial. Based on your published papers, I understand that disentanglement constructs new |u_opt_nk> states through <span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:small;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial;float:none;display:inline">U^{dis(k)}</span>, specifically taking U_{mn}^{dis(k)} |u_mk> and summing over m. I also understand the matrix U^{dis(k)} is rectangular, with indices m > n. Based on this notation from the papers, it seems the transformation from the "old" to "new" basis is something like </font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br></font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">(New) = U (Old), </font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br>where U is shorthand for <span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:small;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;font-weight:400;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;background-color:rgb(255,255,255);text-decoration-style:initial;text-decoration-color:initial;float:none;display:inline">U^{dis(k)}</span>. However, the dimensions here don't make sense if U is an m x n matrix and (Old) represents a set of m-length column vectors. Furthermore, my understanding of unitary transformations from my courses in quantum mechanics is that, instead of the above, I should actually see something like:</font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br></font></div><div><div style="font-size:small;font-style:normal;font-variant-ligatures:normal;font-variant-caps:normal;letter-spacing:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px;text-decoration-style:initial;text-decoration-color:initial"><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">(New) = U^dagger (Old), </font></div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br class="gmail-Apple-interchange-newline">which is shown to be true in J.J. Sakurai's 2nd edition QM book (page 37). Lastly, the code for Wannier90 contains a comment that says:</font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br></font></div><div>
<font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><span></span>
</font><p class="gmail-p1" style="margin:0px;font-style:normal;font-variant:normal;font-stretch:normal;font-size:13px;line-height:normal"><span class="gmail-s1" style="font-variant-ligatures:no-common-ligatures;background-color:rgb(255,255,255)"><font face="arial, helvetica, sans-serif" color="#000000">Note: |psi> U_opt = |psitilde> and obviously <span class="gmail-Apple-converted-space"> </span></font></span></p><p class="gmail-p1" style="margin:0px;font-style:normal;font-variant:normal;font-stretch:normal;font-size:13px;line-height:normal"><span style="font-variant-ligatures:no-common-ligatures;color:rgb(0,0,0);font-family:arial,helvetica,sans-serif"> <psitilde| = (U_opt)^dagger <psi|</span><span class="gmail-Apple-converted-space" style="font-variant-ligatures:no-common-ligatures;color:rgb(0,0,0);font-family:arial,helvetica,sans-serif"> </span></p>
<font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br></font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">which seems to imply that |psi> is a row vector instead of the normal column vector, and U_opt is multiplied on the right instead of the left to yield the new basis. </font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br></font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">Because of these ambiguities, I'm confused as to how I should use these unitary matrices going forward (i.e. which one of the expressions above is actually correct?), and I would appreciate any insight you have. </font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)"><br></font></div><div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">Many thanks,</font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">Peyton Cline</font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">PhD Student</font></div><div><font face="arial, helvetica, sans-serif" color="#000000" style="background-color:rgb(255,255,255)">University of Colorado Boulder</font></div></div></div>