File format

Geometry

The file name in the Tutorial is geometry.dat. When we use Standard mode of mVMC/{\mathcal H}\Phi, this file is generated automatically. Therefore we do not have to care it.

1.000000     0.000000     0.000000  (1)
0.000000     1.000000     0.000000  (1)
0.000000     0.000000     1.000000  (1)
0.000000     0.000000     0.000000  (2)
4 0 0                               (3)
0 4 0                               (3)
0 0 1                               (3)
0.000000     0.000000     0.000000  (4)
1.000000     0.000000     0.000000  (4)
2.000000     0.000000     0.000000  (4)
3.000000     0.000000     0.000000  (4)
0.000000     1.000000     0.000000  (4)
1.000000     1.000000     0.000000  (4)
2.000000     1.000000     0.000000  (4)
3.000000     1.000000     0.000000  (4)
0.000000     2.000000     0.000000  (4)
1.000000     2.000000     0.000000  (4)
2.000000     2.000000     0.000000  (4)
3.000000     2.000000     0.000000  (4)
0.000000     3.000000     0.000000  (4)
1.000000     3.000000     0.000000  (4)
2.000000     3.000000     0.000000  (4)
3.000000     3.000000     0.000000  (4)
  1. The unit lattice vectors. Arbitrary unit.
  2. The phase for the one-body term across boundaries of the simulation cell (degree unit).
  3. Three integer vector specifying the shape of the simulation cell. They are the same as the input parameters a0W, a0L, a0H, a1W... in Standard mode.
  4. The position of each site. The fractional coordinate is used.

One- and Two-body correlation function in the site representation

Specify the index of correlation function to be computed

Specify the index of correlation functions computed with mVMC/{\mathcal H}\Phi. When we use the standard mode, this file is generated automatically. The general description is written in the manuals for mVMC/{\mathcal H}\Phi. The file names in the Tutorial are greenone.def (one body) and greentwo.def (two body).

For calculating correlation functions in Supported quantities, indices must be specified as follows:

  • \langle {\hat c}_{{\bf k} \uparrow}^{\dagger} {\hat c}_{{\bf k} \uparrow}\rangle

    \langle {\hat c}_{i \uparrow}^{\dagger} {\hat c}_{j \uparrow}\rangle with (i, j) ranging on the whole site.

  • \langle {\hat c}_{{\bf k} \downarrow}^{\dagger} {\hat c}_{{\bf k} \downarrow}\rangle

    \langle {\hat c}_{i \downarrow}^{\dagger} {\hat c}_{j \downarrow}\rangle with (i, j) ranging on the whole site.

  • \langle {\hat \rho}_{\bf k} {\hat \rho}_{\bf k}\rangle and \langle {\hat S}_{\bf k}^{z} {\hat S}_{\bf k}^{z} \rangle

    \langle {\hat c}_{i \sigma}^{\dagger} {\hat c}_{i \sigma} {\hat c}_{j \sigma'}^{\dagger} {\hat c}_{j \sigma'}\rangle with (i, j) ranging on the whole site and (\sigma, \sigma') ranging from \uparrow to \downarrow.

  • \langle {\hat S}_{\bf k}^{+} {\hat S}_{\bf k}^{-} \rangle and \langle {\hat {\bf S}}_{\bf k} \cdot {\hat {\bf S}}_{\bf k} \rangle

    For {\mathcal H}\Phi, \langle {\hat c}_{i \sigma}^{\dagger} {\hat c}_{i -\sigma} {\hat c}_{j -\sigma}^{\dagger} {\hat c}_{j \sigma}\rangle with (i, j) ranging on the whole site and \sigma ranging from \uparrow to \downarrow. For mVMC, \langle {\hat c}_{i \sigma}^{\dagger} {\hat c}_{j \sigma} {\hat c}_{j -\sigma}^{\dagger} {\hat c}_{i -\sigma}\rangle with (i, j) ranging on the whole site and \sigma ranging from \uparrow to \downarrow. In the both cases, please care the order of operators.

In the default settings of Standard mode (outputmode="corr"), the above indices are specified automatically. Therefore we do not have to care it.

Results of correlation function in the site representation

The correlation functions having the indices specified in Specify the index of correlation function to be computed are computed by mVMC/{\mathcal H}\Phi, and written to files. The general description of this file is written in the manuals of mVMC/{\mathcal H}\Phi. File names in the Tutorial are output/zvo_cisajs_001.dat and output/zvo_cisajscktalt_001.dat (mVMC), or output/zvo_cisajs.dat and output/zvo_cisajscktalt.dat ({\mathcal H}\Phi).

The utility fourier reads these files before the calculation. If some of the correlation functions with indices written in Specify the index of correlation function to be computed are lacking (for example, because Standard mode was not used), this utility assume them as 0.

Correlation functions in the primitive Brillouin zone

This file contains the Fourier-transformed correlation function and generated by the utility fourier. The file name in the Tutorial is output/zvo_corr.dat.

#HPhi          16                                              (1)
# kx[1] ky[2] kz[3](Cart.) UpUp[4,5] (Re. Im.) DownDown[6,7]   (2)
# Density[8,9] SzSz[10,11] S+S-[12,13] S-S+[14,15]             (2)
#k-offset      0.0000000      0.0000000      0.0000000         (3)
0.00000E+00    0.00000E+00    0.00000E+00    0.31250E-01  .... (4)
0.15708E+01    0.00000E+00    0.00000E+00    0.31250E-01  .... (4)
:                                                               :
  1. "#HPhi" for the output of HPhi, "#mVMC" for the output of vmc.out The subsequent integer indicate the number of k points in the primitive Brillouine zone.
  2. The description of the quantities in each column.
  3. The k offset for the one-body correlation function. That is to say, the one-body correlation function in the 4-7 columns are those at the k point shifted from that point in the 1-3 column.
  4. The k point (Cartesian) and correlation functions. The real- and the imaginary-part of each correlation function are written.

k-point file for corplot

This file is generated by fourier and read by corplot when the correlation function is plotted. The file name is kpoint.dat.

81           9                                      (1)
0.62832E+01    0.00000E+00    0.00000E+00           (2)
0.00000E+00    0.62832E+01    0.00000E+00           (2)
0.00000E+00    0.00000E+00    0.62832E+01           (2)
-0.62832E+01   -0.62832E+01    0.00000E+00      1   (3)
-0.47124E+01   -0.62832E+01    0.00000E+00      2   (3)
-0.31416E+01   -0.62832E+01    0.00000E+00      3
-0.15708E+01   -0.62832E+01    0.00000E+00      4
0.00000E+00   -0.62832E+01    0.00000E+00      1
0.15708E+01   -0.62832E+01    0.00000E+00      2
0.31416E+01   -0.62832E+01    0.00000E+00      3
0.47124E+01   -0.62832E+01    0.00000E+00      4
  1. The total number of k points plotted by corplot and the number of columns for displaying by splot of gnuplot.
  2. Reciprocal lattice vectors (Cartesian coordinate).
  3. The k vector (Cartesian) and the index of the equivalent k point in the primitive Brillouin zone. This number is the same as that in Correlation functions in the primitive Brillouin zone

gnuplot script

This file is generated by corplot, and read from gnuplot launched automatically. We also can launch gnuplot independently and load this script. The file name is correlation.gp.

#set terminal pdf color enhanced \    (1)
#dashed dl 1.0 size 20.0cm, 20.0cm    (1)
#set output 'correlation.pdf'         (1)
#set view 60.0, 30.0                  (1)

set view equal xy
set ticslevel 0
set hidden3d
set xlabel 'kx'
set ylabel 'ky'
set zrange [    0.25000E-10:    0.18435E+00]

set pm3d
set pm3d interpolate 5, 5
set view 0.0, 0.0

#####  Set Brillouin-Zone Boundary  #####

set arrow from    -0.31416E+01,   -0.31416E+01,    ...
set arrow from    -0.31416E+01,    0.31416E+01,    ...
:
#####  End Set Brillouin-Zone Boundary  #####

splot \
'correlation.dat' u 1:2:3 w l tit '1' (2)
pause -1
  1. When we want to write the figure to a file, this line is uncommented. For pasting this figure on the paper etc., we write the setting of font, line-color, and so on. For more details, please see the manual of gnuplot.
  2. Plotting the file in Correlation function at wide range of k.

Correlation function at wide range of k

This file is generated by corplot, and read from gnuplot through gnuplot script. The file name is correlation.dat.

-0.62832E+01   -0.62832E+01    0.18435E+00    0.00000E+00
-0.47124E+01   -0.62832E+01    0.36159E-01    0.00000E+00
-0.31416E+01   -0.62832E+01    0.20921E-01    0.00000E+00
-0.15708E+01   -0.62832E+01    0.36159E-01    0.00000E+00
 0.00000E+00   -0.62832E+01    0.18435E+00    0.00000E+00
 0.15708E+01   -0.62832E+01    0.36159E-01    0.00000E+00
 0.31416E+01   -0.62832E+01    0.20921E-01    0.00000E+00
 0.47124E+01   -0.62832E+01    0.36159E-01    0.00000E+00
 0.62832E+01   -0.62832E+01    0.18435E+00    0.00000E+00

-0.62832E+01   -0.47124E+01    0.36159E-01    0.00000E+00
-0.47124E+01   -0.47124E+01    0.20921E-01    0.00000E+00
-0.31416E+01   -0.47124E+01    0.11372E-01    0.00000E+00
:

The 1st and the 2nd column contains the k vector (Cartesian). 3rd and the 4th column contains the correlation function and its standard error, respectively.