Creating a Z2Pack System¶

The first step in creating a calculation with Z2Pack is to define the system which you are going to study. Currently, there are three different types of systems or models available:

Models with an explicit Hamilton matrix - z2pack.hm¶

The simplest way of creating a system is when it can be explicitly described by a matrix Hamiltonian $$\mathcal{H}(\mathbf{k})$$. For example, an effective model for an isotropic Weyl point is given by

$\mathcal{H}(\mathbf{k}) = \sum_{i\in \{x, y, z\}} k_i \cdot \sigma_i$

The class z2pack.hm.System is constructed by passing the function which describes this Hamiltonian as the first argument:

import z2pack
import numpy as np

def hamiltonian(k):
kx, ky, kz = k
return np.array([
[kz, kx - 1j * ky],
[kx + 1j * ky, -kz]
])

system = z2pack.hm.System(hamiltonian)


By default, the lower-lying half of the states will be used for constructing the Wilson loop and Wannier charge centers. In this case, since the matrix has size 2, only the lowest band contributes. This behaviour can be changed by setting the bands keyword in the constructor. Either the number of “occupied” states can be given, or the relevant bands can be selected by index.

This means that

system = z2pack.hm.System(hamiltonian, bands=2)


includes WCC from both bands, and

system = z2pack.hm.System(hamiltonian, bands=[1])


includes only the upper band (because the index starts at 0).

Note

Keyword arguments which were not discussed here are described in the Reference. Throughout this tutorial, only the basic keywords will be covered. Clicking on a method or class name like z2pack.hm.System will take you to the relevant section of the reference.

Tight-binding models - z2pack.tb¶

For tight-binding models, the TBmodels package (which started its life as a part of Z2Pack) is used. TBmodels uses its tbmodels.Model class to describe a tight-binding model. There are several ways to create those, described in the TBmodels tutorial . Instances of tbmodels.Model can be used to construct Z2Pack systems, using the z2pack.tb.System class.

The following code shows how to create a Z2Pack system from a tight-binding model given in Wannier90’s *_hr.dat format.

import z2pack
import tbmodels

model = tbmodels.Model.from_wannier_files(hr_file='path_to_directory/wannier90_hr.dat')
system = z2pack.tb.System(model)


First-principles calculations - z2pack.fp¶

In order to calculate topological invariants reliably using first-principles, Z2Pack needs to dynamically make calls to the first-principles code. This means that one must provide a way of calling the first-principles code automatically from within Z2Pack. The z2pack.fp.System class aims to make this as simple as possible.

There are four steps involved in each call to a first-principles code:

1. Input files created by the user are copied into the working folder

2. A string specifying the k - points is either appended to one of those files or put in a separate file

3. The first - principles code is called and Wannier90 creates the .mmn file

4. Z2Pack reads the overlap matrices from the .mmn file

1. Preparing input files¶

For the first step, the user needs to create input files for an NSCF run calling Wannier90. These input files should also contain a reference to the density file acquired in a previous SCF run. However, the k-points used in the NSCF run should not be in these files. The reason for this is that the k-points will change many times during a Z2Pack calculation. When creating the z2pack.fp.System instance, the input files should be listed in the input_files keyword argument (as a list of strings).

The Wannier90 input file should contain the exclude_bands tag, such that only the bands for which the topological invariant should be calculated are included. Usually, this means that the unoccupied bands are excluded.

Starting from version 2.1, Wannier90 has a dedicated interface to specify which overlap matrices should be computed. To use this interface, use the k-point function wannier90_full() as described in the next section.

Wannier90 2.0 and before¶

For older versions of Wannier90, the interface to explicitly specify which overlaps are computed does not exist. This must be done manually, by setting the right input flags. The goal is that overlap matrices between neighbouring k-points along the line are computed exactly once, i.e. no overlaps are computed from one k-point to the neighbour’s equivalent point in another unit cell. In most cases this can be done by setting shell_list 1.If the unit cell is very long in a certain direction, however, it can happend that this setting will just compute overlaps between equivalent points in different unit cells. In that case, you have two options:

• Add more k-points to the line using the iterator parameter. For example, iterator=range(20, 51, 2) would mean that the calculation starts with 20 k-points instead of 8. Of course, this increases the cost of the computation

• Set the parameter search_shells instead of shell_list. It should be large enough s.t. the direct neighbours are included, but not so large that the neighbour’s equivalent points are included.

2. Preparing k-points input¶

If you are using VASP, ABINIT or Quantum Espresso, you can use the functions provided in z2pack.fp.kpoint to create k-points input. Else, you will need to specify a function producing the input for specifying the k-points.

In both cases, the function itself should be given as the kpts_fct input variable, while the file the k-points string should be printed to is given as kpts_path. If you need the k-points input to be written to more than one file, you can let kpts_fct be a list of functions, and kpts_path a list of file names.

The function given in kpt_fct must have the following syntax:

def function_name(kpt):
...
return string


where kpt is a list containing the desired k-points including the periodic image of the first point. Hence to compute a string with N k-points, N + 1 points are given, and the last point is a periodic image of the first. Note thus that the function should be constructed in such a way that the first-principles code will not use the last point in its calculation.

3. Call to the first-principles code¶

The call to the first-principles code is simple: just provide Z2Pack with the command line input (as a string) of how to call the first-principles code you are using. This is the command keyword argument to fp.System.

Finally, Z2Pack needs the path to where the overlap file wannier90.mmn will be (Keyword argument mmn_path). By default, it is assumed to be in the top level of the build directory.

Combining these four steps, we get the following example (for VASP):

system = z2pack.fp.System(
input_files=[
"input/CHGCAR",
"input/INCAR",
"input/POSCAR",
"input/POTCAR",
"input/wannier90.win"
],                              # Step 1
kpt_fct=z2pack.fp.kpoint.vasp,  # Step 2
kpt_path="KPOINTS",             # Step 2
command="mpirun \$VASP >& log",  # Step 3
mmn_path='wannier90.mmn'        # Step 4 (this is the default setting)
)


First-principles codes¶

Depending on which first-principles code you use, there are a few things that you should look out for, and input parameters that must be set. In general, the easiest way to get started is by using one of the examples provided. Here is a short description of the special input flags needed for different codes.

Quantum Espresso¶

Of the first-principles codes which have been tested, Quantum Espresso currently has the best integration with Z2Pack. Starting with version 6.0 (with Wannier90 version 2.1 or higher) it supports calculating topological invariants on arbitrary surfaces. To enable this, use the new Wannier90 interface described above (wannier90_full()), and add the option regular_mesh = .false. to the pw2wannier90 input.

VASP¶

Required input arguments:

LWANNIER90 = .TRUE.
LWRITE_MMN_AMN = .TRUE.
ISYM = -1


Now that you know how to construct the various systems, it’s time to get to work: Let’s run some calculations!