Chirality of a Weyl point (Quantum ESPRESSO)

In this example, the chirality of two Weyl points of MoP₂ is calculated from first-principles using Quantum ESPRESSO. The Chern number is calculated for three different spheres: The first two surrounding one Weyl point each, and the third containing both Weyl points. Since the two Weyl points have the same chirality (-1), the Chern number adds up to be -2 for the third sphere. The results are shown in the figure below.

This example uses the new interface to Wannier90 2.1 and QE 6. The input files can be found on the github repository.

#!/usr/bin/env python

import os
import shutil
import subprocess
import xml.etree.ElementTree as ET

import matplotlib.pyplot as plt
import numpy as np
import z2pack

# Edit the paths to your Quantum Espresso 6+ and Wannier90 2.1+ here
QEDIR = "/home/greschd/software/qe-6.0/bin/"
WANDIR = "/home/greschd/software/wannier90-2.1.0"

# Commands to run pw, pw2wannier90, wannier90
MPIRUN = "mpirun -np 4 "
PWCMD = MPIRUN + os.path.join(QEDIR, "pw.x ")
PW2WANCMD = MPIRUN + os.path.join(QEDIR, "pw2wannier90.x ")
WANCMD = os.path.join(WANDIR, "wannier90.x")

Z2CMD = (
    WANCMD
    + " MoP2 -pp;"
    + PWCMD
    + "< MoP2.nscf.in >& pw.log;"
    + PW2WANCMD
    + "< MoP2.pw2wan.in >& pw2wan.log;"
)

# creating the results folder, running the SCF calculation if needed
os.makedirs("./plots", exist_ok=True)
os.makedirs("./results", exist_ok=True)
if not os.path.exists("./scf"):
    os.makedirs("./scf")
    print("Running the scf calculation")
    shutil.copyfile("input/MoP2.scf.in", "scf/MoP2.scf.in")
    out = subprocess.call(PWCMD + " < MoP2.scf.in > scf.out", shell=True, cwd="./scf")
    if out != 0:
        raise RuntimeError(
            "Error in SCF call. Inspect scf folder for details, and delete it to re-run the SCF calculation."
        )

# Copying the lattice parameters from MoP2.save/data-file.xml into MoP2.win
cell = ET.parse("scf/MoP2.save/data-file.xml").find("CELL").find("DIRECT_LATTICE_VECTORS")
unit = cell[0].attrib["UNITS"]
lattice = "\n ".join([line.text.strip("\n ") for line in cell[1:]])

with open("input/tpl_MoP2.win") as f:
    tpl_MoP2_win = f.read()
with open("input/MoP2.win", "w") as f:
    f.write(tpl_MoP2_win.format(unit=unit, lattice=lattice))

# Creating the System. Note that the SCF charge file does not need to be
# copied, but instead can be referenced in the .files file.
# The k-points input is appended to the .in file
input_files = ["input/" + name for name in ["MoP2.nscf.in", "MoP2.pw2wan.in", "MoP2.win"]]
system = z2pack.fp.System(
    input_files=input_files,
    kpt_fct=[z2pack.fp.kpoint.qe_explicit, z2pack.fp.kpoint.wannier90_full],
    kpt_path=["MoP2.nscf.in", "MoP2.win"],
    command=Z2CMD,
    executable="/bin/bash",
    mmn_path="MoP2.mmn",
)

# Run the WCC calculations
# a small sphere around one Weyl point
result_1 = z2pack.surface.run(
    system=system,
    surface=z2pack.shape.Sphere(center=(0.6239797, 0.70845137, 0), radius=0.007),
    save_file="./results/res1.json",
    load=True,
)
# a small sphere around the other Weyl point
result_2 = z2pack.surface.run(
    system=system,
    surface=z2pack.shape.Sphere(center=(0.61197818, 0.70080602, 0), radius=0.007),
    save_file="./results/res2.json",
    load=True,
)
# a bigger sphere around both Weyl points
result_3 = z2pack.surface.run(
    system=system,
    surface=z2pack.shape.Sphere(center=(0.61797894, 0.7046287, 0.0), radius=0.014),
    save_file="./results/res3.json",
    load=True,
)

# Combining the two plots
fig, ax = plt.subplots(1, 3, sharey=True, figsize=(14, 4))
for res, axis, title in zip(
    [result_1, result_2, result_3],
    ax,
    ["Sphere around WP1", "Sphere around WP2", "Sphere around both WPs"],
):
    z2pack.plot.chern(res, axis=axis)
    axis.set_title(title)
    axis.set_xlabel(r"$\theta$")
    axis.xaxis.set_ticks([0, 1])
    axis.xaxis.set_ticklabels([r"$-\pi$", r"$0$"])
ax[0].set_ylabel(r"$\bar{\varphi}$", rotation="horizontal")
ax[0].yaxis.set_ticks([0, 1])
ax[0].yaxis.set_ticklabels([r"$0$", r"$2\pi$"])
plt.savefig("plots/plot.pdf", bbox_inches="tight")

print(f"Chern number / Weyl chirality around WP1: {z2pack.invariant.chern(result_1):.2f}")
print(f"Chern number / Weyl chirality around WP2: {z2pack.invariant.chern(result_2):.2f}")
print(
    "Chern number / Weyl chirality around both Weyl points: {:.2f}".format(
        z2pack.invariant.chern(result_3)
    )
)