Example

As an example, let us calculate a $Z_{2}$ invariant of a 3D topological insulator Bi$_2$Se$_3$ [83].

SCF calculation

You can perform the SCF calculation using 'Bi2Se3-Z2.dat' stored in the directory 'work' with the following keyword:

    HS.fileout      on      #on|off, default=off
After finishing the SCF calculation normally, the out file 'Bi2Se3-Z2.scfout' is generated.


Calculation of Z$_2$ invariant
Before computing the $Z_{2}$ invariant by using the code 'Z2FH.c', please compile the code in directory 'source' as

    % make Z2FH
After the compilation, you may obtain the excutable file 'Z2FH' in the directory 'work'. Then, let us move on the calculation of the Z$_2$ invariant as
   % ./Z2FH Bi2Se3-Z2.scfout
 or
   % ./Z2FH Bi2Se3-Z2.scfout < Z2FH.in > Z2FH.out
 or
   % mpirun -np 4 ./Z2FH Bi2Se3-Z2.scfout < Z2FH.in > Z2FH.out

The file 'Z2FH.in' contains parameters which are requested by 'Z2FH' such as
   5
   1
   1
In the first case above, you will be interactively asked from the program as follows:
    ******************************************************************
    ******************************************************************
     Z2FH:
     code for calculating the Z2 invariant of bulk systems
     by Fukui-Hatsugai method.
     Copyright (C), 2019, Hikaru Sawahata, Naoya Yamaguchi,
     Fumiyuki Ishii and Taisuke Ozaki 
     This is free software, and you are welcome to         
     redistribute it under the constitution of the GNU-GPL.
     
     Please cite the following article:
     H. Sawahata, N. Yamaguchi, H. Kotaka and F. Ishii,
     Jpn. J. Appl. Phys. 57, 030309 (2018).
    ******************************************************************
    ******************************************************************
    Mesh1 Number(Half Direction):5
    Calculate All plane?(0:No,1:Yes)1
    Restart?[1:x0,2:xpi,3:y0....]1
    
    Read the scfout file (Bi2Se3-Z2.scfout)
    ***
    The file format of the SCFOUT file:  3
    And it supports the following functions:
    - jx
    - polB
    - kSpin
    - Z2FH
    - calB
    ***


Parameters for Z2FH

Let us explain the parameters in the input file 'Z2FH.in'.


Output files

After the calculation by 'Z2FH', the following files are generated.

As an example, we show the integer-valued field $n({\bf k})$ on $k_1=0$ in Fig. 77, which can be obtained by the procedure explained above as 'gnuplot LCNum1.pl'. Since $n({\bf k})$ is the gauge dependent value, you may find a different result in your calculation, while the Z$_2$ invariant should be reproduced.