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