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<p>Dear Patric,</p>
<p><br>
</p>
<p>For the Z2 calculation, the surface should go only across half
the Brillouin zone. So, the surface should probably be <tt>lambda
s, t: [s / 2, t, 0]</tt>.</p>
<p><br>
</p>
<p>The reason for this is that the second half is a mirror image of
the first. When you go across the full BZ, the result for the Z2
invariant will always be zero.</p>
<p><br>
</p>
<p>Best regards,</p>
<p>Dominik<br>
</p>
<br>
<div class="moz-cite-prefix">On 28.09.2018 12:04, Yifan GAO wrote:<br>
</div>
<blockquote type="cite"
cite="mid:TY2PR0101MB23027C53CD1F67E03AAF39A3AFEC0@TY2PR0101MB2302.apcprd01.prod.exchangelabs.com">
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<p style="margin-top:0;margin-bottom:0">Dear Dominik,</p>
<p style="margin-top:0;margin-bottom:0"><br>
</p>
<p style="margin-top:0;margin-bottom:0">Thanks for your reply!
I'm really a rookie in this area.</p>
<p style="margin-top:0;margin-bottom:0"><br>
</p>
<p style="margin-top:0;margin-bottom:0">Now I'm testing
silicene(<span>10.1103/PhysRevLett.107.076802</span>) with
Z2Pack. I successfully get the edge states by wannier90
calculation. The Z2 invariant should be 1 according to the
reference. I set
<i>num_band=num_wann=8</i> to integrate the bands under the
fermi level and chose <span style="color: rgb(33, 33, 33);
font-family: wf_segoe-ui_normal, "Segoe UI",
"Segoe WP", Tahoma, Arial, sans-serif, serif,
EmojiFont; font-size: 15px;"><i>lambda s, t: (s, t, 0)</i><span
style="font-family: Calibri, Helvetica, sans-serif,
EmojiFont, "Apple Color Emoji", "Segoe UI
Emoji", NotoColorEmoji, "Segoe UI Symbol",
"Android Emoji", EmojiSymbols; font-size: 16px;"> and
to calculate lines along ky.</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span style="color:
rgb(33, 33, 33); font-family: wf_segoe-ui_normal,
"Segoe UI", "Segoe WP", Tahoma, Arial,
sans-serif, serif, EmojiFont; font-size: 15px;"><span
style="font-family: Calibri, Helvetica, sans-serif,
EmojiFont, "Apple Color Emoji", "Segoe UI
Emoji", NotoColorEmoji, "Segoe UI Symbol",
"Android Emoji", EmojiSymbols; font-size: 16px;"><br>
</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span style="color:
rgb(33, 33, 33); font-family: wf_segoe-ui_normal,
"Segoe UI", "Segoe WP", Tahoma, Arial,
sans-serif, serif, EmojiFont; font-size: 15px;"><span
style="font-family: Calibri, Helvetica, sans-serif,
EmojiFont, "Apple Color Emoji", "Segoe UI
Emoji", NotoColorEmoji, "Segoe UI Symbol",
"Android Emoji", EmojiSymbols; font-size: 16px;">The
result given by Z2Pack shows Z2 = 0. I don't know if
there's anything wrong with my calculation?</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span style="color:
rgb(33, 33, 33); font-family: wf_segoe-ui_normal,
"Segoe UI", "Segoe WP", Tahoma, Arial,
sans-serif, serif, EmojiFont; font-size: 15px;"><span
style="font-family: Calibri, Helvetica, sans-serif,
EmojiFont, "Apple Color Emoji", "Segoe UI
Emoji", NotoColorEmoji, "Segoe UI Symbol",
"Android Emoji", EmojiSymbols; font-size: 16px;"><br>
</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span style="color:
rgb(33, 33, 33); font-family: wf_segoe-ui_normal,
"Segoe UI", "Segoe WP", Tahoma, Arial,
sans-serif, serif, EmojiFont; font-size: 15px;"><span
style="font-family: Calibri, Helvetica, sans-serif,
EmojiFont, "Apple Color Emoji", "Segoe UI
Emoji", NotoColorEmoji, "Segoe UI Symbol",
"Android Emoji", EmojiSymbols; font-size: 16px;">All
the parameters are similar to the Bi example. I changed <i>search_shell</i>
in range of 24~96 in order to finish the calculation.</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span style="color:
rgb(33, 33, 33); font-family: wf_segoe-ui_normal,
"Segoe UI", "Segoe WP", Tahoma, Arial,
sans-serif, serif, EmojiFont; font-size: 15px;"><span
style="font-family: Calibri, Helvetica, sans-serif,
EmojiFont, "Apple Color Emoji", "Segoe UI
Emoji", NotoColorEmoji, "Segoe UI Symbol",
"Android Emoji", EmojiSymbols; font-size: 16px;"><br>
</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span style="color:
rgb(33, 33, 33); font-family: wf_segoe-ui_normal,
"Segoe UI", "Segoe WP", Tahoma, Arial,
sans-serif, serif, EmojiFont; font-size: 15px;"><span
style="font-family: Calibri, Helvetica, sans-serif,
EmojiFont, "Apple Color Emoji", "Segoe UI
Emoji", NotoColorEmoji, "Segoe UI Symbol",
"Android Emoji", EmojiSymbols; font-size: 16px;">Best
regards,</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span style="color:
rgb(33, 33, 33); font-family: wf_segoe-ui_normal,
"Segoe UI", "Segoe WP", Tahoma, Arial,
sans-serif, serif, EmojiFont; font-size: 15px;"><span
style="font-family: Calibri, Helvetica, sans-serif,
EmojiFont, "Apple Color Emoji", "Segoe UI
Emoji", NotoColorEmoji, "Segoe UI Symbol",
"Android Emoji", EmojiSymbols; font-size: 16px;">Patric</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span style="color:
rgb(33, 33, 33); font-family: wf_segoe-ui_normal,
"Segoe UI", "Segoe WP", Tahoma, Arial,
sans-serif, serif, EmojiFont; font-size: 15px;"><br>
</span></p>
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<hr style="display:inline-block;width:98%" tabindex="-1">
<div id="divRplyFwdMsg" dir="ltr"><font style="font-size:11pt"
color="#000000" face="Calibri, sans-serif"><b>From:</b>
Dominik Gresch <a class="moz-txt-link-rfc2396E" href="mailto:greschd@phys.ethz.ch"><greschd@phys.ethz.ch></a><br>
<b>Sent:</b> Thursday, September 27, 2018 11:22:31 AM<br>
<b>To:</b> Yifan GAO; <a class="moz-txt-link-abbreviated" href="mailto:z2pack@lists.phys.ethz.ch">z2pack@lists.phys.ethz.ch</a><br>
<b>Subject:</b> Re: topological invariants calculation on 2D
system</font>
<div> </div>
</div>
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<p>Dear Patric,</p>
<p><br>
</p>
<p>When calculating 2D invariants, the only Chern and Z2
invariants that can be calculated are those on the 2D
Brillouin zone itself. Since you are using VASP, the 2D system
is represented as a 3D system, where one dimension is very
long in real space, and correspondingly very narrow in
reciprocal space. As a result, the bands will be almost flat
in that direction. So, the surface you need to choose for
calculating the 2D invariant should be extended in the
remaining two dimensions.</p>
<p><br>
</p>
<p>As a simple examle, if your unit cell is</p>
<p><br>
</p>
<p>a_1 = (1, 0, 0)</p>
<p>a_2 = (0, 1, 0)</p>
<p>a_3 = (0, 0, 200)</p>
<p><br>
</p>
<p>then you want the surface to span k_x and k_y, e.g. lambda s,
t: (s, t, 0) for a Chern number. The k_z value can be chosen
to be zero, but it could really be anything since the system
should not depend on k_z.</p>
<p><br>
</p>
<p>Best regards,</p>
<p>Dominik<br>
</p>
<br>
<div class="x_moz-cite-prefix">On 27.09.2018 14:38, Yifan GAO
wrote:<br>
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<p style="margin-top:0; margin-bottom:0">Hello,</p>
<p style="margin-top:0; margin-bottom:0"><br>
</p>
<p style="margin-top:0; margin-bottom:0">I don't quite
understand how can I get the topological <span>invariants,
like Z2 and Chern, within a 2D system? I'm feeling
puzzled when building the surface. </span></p>
<p style="margin-top:0; margin-bottom:0"><span>BTW I'm using
vasp. </span></p>
<p style="margin-top:0; margin-bottom:0"><span><br>
</span></p>
<p style="margin-top:0; margin-bottom:0"><span>Thank you,</span></p>
<p style="margin-top:0; margin-bottom:0"><span>Patric</span></p>
<p style="margin-top:0; margin-bottom:0"><span><br>
</span></p>
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