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<div>Thank you very much for explanation.</div><div><br></div><div>I found another problem. According to the paper "Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface" DOI: 10.1038/NPHYS1270 </div><div>There is a band inversion at gamma point for Bi2Se3. So the parity product at gamma point is different for no SOC case and SOC case.</div><div><br></div><div>However, I count the valence parity as below</div><div><br></div><div>no SOC</div><div><br></div><div>uugggugguuguguguguugguu product=+</div><div><br></div><div>SOC</div><div><br></div><div>++--++----++++++----++--++--++--++--++--++---- product=+</div><div><br></div><div>They are the same. But I calculated the surface band, dirac cone do exist in SOC case.</div><div><br></div><div>What is wrong with my counting?</div><div><br></div><div>below is the output of bands.x for no SOC and SOC at gamma point</div><div>--------------------------</div><div><br></div><div>no SOC gamma symmetry, highest occupied level 7.8539</div><div><br></div><div><div>Band symmetry, D_3d (-3m) point group:</div><div><br></div><div> e( 1 - 2) = -15.33380 eV 2 --> E_u L_3' </div><div> e( 3 - 4) = -15.33248 eV 2 --> E_g L_3 </div><div> e( 5 - 5) = -15.32190 eV 1 --> A_1g L_1 </div><div> e( 6 - 6) = -15.31898 eV 1 --> A_2u L_2' </div><div> e( 7 - 8) = -15.30949 eV 2 --> E_g L_3 </div><div> e( 9 - 10) = -15.30663 eV 2 --> E_u L_3' </div><div> e( 11 - 11) = -6.57512 eV 1 --> A_1g L_1 </div><div> e( 12 - 12) = -5.32387 eV 1 --> A_2u L_2' </div><div> e( 13 - 13) = -5.22664 eV 1 --> A_1g L_1 </div><div> e( 14 - 14) = -2.00339 eV 1 --> A_2u L_2' </div><div> e( 15 - 15) = -1.26798 eV 1 --> A_1g L_1 </div><div> e( 16 - 16) = 5.51014 eV 1 --> A_2u L_2' </div><div> e( 17 - 17) = 5.51634 eV 1 --> A_1g L_1 </div><div> e( 18 - 19) = 5.81972 eV 2 --> E_u L_3' </div><div> e( 20 - 21) = 6.26517 eV 2 --> E_g L_3 </div><div> e( 22 - 23) = 6.96103 eV 2 --> E_u L_3' </div><div> e( 24 - 24) = 7.91582 eV 1 --> A_2u L_2' </div><div> e( 25 - 25) = 8.11337 eV 1 --> A_1g L_1 </div><div> e( 26 - 27) = 9.02988 eV 2 --> E_g L_3 </div><div> e( 28 - 28) = 10.34421 eV 1 --> A_2u L_2' </div><div> e( 29 - 30) = 10.59126 eV 2 --> E_u L_3' </div><div> e( 31 - 31) = 13.58144 eV 1 --> A_1g L_1 </div><div> e( 32 - 32) = 14.17587 eV 1 --> A_2u L_2' </div></div><div><br></div><div>------------------------</div><div><br></div><div>SOC gamma symmetry, highest occupied level 7.7444</div><div><br></div><div><div>Band symmetry, D_3d (-3m) double point group:</div><div><br></div><div> e( 1 - 4) = -16.63750 eV 4 --> G_5+ L_4+ </div><div> e( 1 - 4) = -16.63750 eV 4 --> G_6+ L_5+ </div><div> e( 1 - 4) = -16.63750 eV 4 --> G_5- L_4- </div><div> e( 1 - 4) = -16.63750 eV 4 --> G_6- L_5- </div><div> e( 5 - 6) = -16.62527 eV 2 --> G_4+ L_6+ </div><div> e( 7 - 8) = -16.62265 eV 2 --> G_4- L_6- </div><div> e( 9 - 10) = -13.66107 eV 2 --> G_4- L_6- </div><div> e( 11 - 12) = -13.65905 eV 2 --> G_4+ L_6+ </div><div> e( 13 - 14) = -13.64175 eV 2 --> G_4+ L_6+ </div><div> e( 15 - 16) = -13.63940 eV 2 --> G_5+ L_4+ </div><div> e( 15 - 16) = -13.63940 eV 2 --> G_6+ L_5+ </div><div> e( 17 - 18) = -13.63741 eV 2 --> G_4- L_6- </div><div> e( 19 - 20) = -13.63672 eV 2 --> G_5- L_4- </div><div> e( 19 - 20) = -13.63672 eV 2 --> G_6- L_5- </div><div> e( 21 - 22) = -6.40199 eV 2 --> G_4+ L_6+ </div><div> e( 23 - 24) = -5.17683 eV 2 --> G_4- L_6- </div><div> e( 25 - 26) = -5.15601 eV 2 --> G_4+ L_6+ </div><div> e( 27 - 28) = -1.79777 eV 2 --> G_4- L_6- </div><div> e( 29 - 30) = -1.02045 eV 2 --> G_4+ L_6+ </div><div> e( 31 - 32) = 5.18513 eV 2 --> G_4- L_6- </div><div> e( 33 - 34) = 5.40747 eV 2 --> G_4+ L_6+ </div><div> e( 35 - 36) = 6.03590 eV 2 --> G_4- L_6- </div><div> e( 37 - 38) = 6.14972 eV 2 --> G_4+ L_6+ </div><div> e( 39 - 40) = 6.19631 eV 2 --> G_5- L_4- </div><div> e( 39 - 40) = 6.19631 eV 2 --> G_6- L_5- </div><div> e( 41 - 42) = 6.51310 eV 2 --> G_5+ L_4+ </div><div> e( 41 - 42) = 6.51310 eV 2 --> G_6+ L_5+ </div><div> e( 43 - 44) = 6.80580 eV 2 --> G_4- L_6- </div><div> e( 45 - 46) = 7.20470 eV 2 --> G_5- L_4- </div><div> e( 45 - 46) = 7.20470 eV 2 --> G_6- L_5- </div><div> e( 47 - 48) = 7.50637 eV 2 --> G_4+ L_6+ </div><div> e( 49 - 50) = 8.05825 eV 2 --> G_4- L_6- </div><div> e( 51 - 52) = 9.13333 eV 2 --> G_4+ L_6+ </div><div> e( 53 - 54) = 9.27043 eV 2 --> G_4- L_6- </div><div> e( 55 - 56) = 9.86223 eV 2 --> G_5+ L_4+ </div><div> e( 55 - 56) = 9.86223 eV 2 --> G_6+ L_5+ </div><div> e( 57 - 58) = 11.09081 eV 2 --> G_4- L_6- </div><div> e( 59 - 60) = 11.46791 eV 2 --> G_5- L_4- </div><div> e( 59 - 60) = 11.46791 eV 2 --> G_6- L_5- </div><div> e( 61 - 62) = 13.77176 eV 2 --> G_4+ L_6+ </div><div> e( 63 - 64) = 14.35855 eV 2 --> G_4- L_6- </div></div>
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