# Topological nonsymmorphic insulator versus Dirac semimetal in KZnBi

Rahul Verma  Department of Condensed Matter Physics and Materials Science,
Tata Institute of Fundamental Research, Mumbai 400005, India  Bikash Patra
Department of Condensed Matter Physics and Materials Science, Tata Institute
of Fundamental Research, Mumbai 400005, India  Bahadur Singh
<EMAIL_ADDRESS>Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Mumbai 400005, India

###### Abstract

KZnBi was discovered recently as a new three-dimensional (3D) Dirac semimetal
with a pair of bulk Dirac fermions in contrast to the $\mathbb{Z}_{2}$ trivial
insulator reported earlier. In order to address this discrepancy, we have
performed electronic structure and topological state analysis of KZnBi using
the local, semilocal, and hybrid exchange-correlation (XC) functionals within
the density functional theory framework. We find that various XC functionals,
including the SCAN meta-GGA and hybrid functionals with 25% Hartree-Fock (HF)
exchange, resolve a topological nonsymmorphic insulator state with the glide-
mirror protected hourglass surface Dirac fermions. By carefully tuning the
modified Becke-Jhonson (mBJ) potential parameters, we recover the correct
orbital ordering and Dirac semimetal state of KZnBi. We further show that
increasing the default HF exchange in hybrid functionals ($>40\%$) can also
capture the desired Dirac semimetal state with the correct orbital ordering of
KZnBi. The calculated energy dispersion and carrier velocities of Dirac states
are found to be in excellent agreement with the available experimental
results. Our results demonstrate that KZnBi is a unique topological material
where large electron correlations are crucial to realize the Dirac semimetal
state.

##  I Introduction

Interest in symmetry-protected nontrivial states of topological materials has
driven search and discovery efforts for finding materials with nontrivial
electronic properties useful for fundamental studies and device applications
Singh  _et al._ (2023); Hasan and Kane (2010); Armitage  _et al._ (2018). Many
topological materials are now predicted in high-throughput materials searches
ranging from insulators to metals and semimetals, where topological states are
protected not only by the free-fermion symmetries such as time-reversal or
particle-hole symmetries but also by crystalline symmetries Singh  _et al._
(2023); Hasan and Kane (2010); Armitage  _et al._ (2018); Tang  _et al._
(2019); Vergniory  _et al._ (2019); Zhang  _et al._ (2019); Fu (2011); Hsieh
_et al._ (2012); Shiozaki and Sato (2014); Wang  _et al._ (2019). Among
various topological materials, three-dimensional (3D) Dirac semimetals support
four-fold band crossings described by pseudo-relativistic Dirac fermions with
linear energy dispersion in their electronic structure. These band crossings
are protected by crystalline symmetries such as rotational or nonsymmorphic
symmetries other than time-reversal and inversion symmetries Young  _et al._
(2012); Wang  _et al._ (2012); Yang and Nagaosa (2014); Xu  _et al._ (2015);
Liu  _et al._ (2014a); Singh  _et al._ (2018a). Systematically reducing these
symmetries can transition a topological Dirac semimetal into a nontrivial
insulator or Weyl semimetal Murakami (2007); Wang  _et al._ (2012); Huang  _et
al._ (2015); Singh  _et al._ (2012); Mardanya  _et al._ (2019). The
nonsymmorphic crystalline symmetries that accompany a symmetry operation with
a fractional translation can protect topological states in nontrivial
insulators. Such topological nonsymmorphic insulators can realize special
hourglass-like Dirac fermions over the material’s surface that respect
nonsymmorphic symmetries Liu  _et al._ (2014b); Alexandradinata  _et al._
(2016); Wang  _et al._ (2016); Ma  _et al._ (2017); Liang  _et al._ (2017);
Singh  _et al._ (2018b); Ezawa (2016).

Among various families of topological materials, the AB$X$-type hexagonal
honeycomb materials attract significant attention due to their greater
materials and topological state tunability Zhang  _et al._ (2011); Yan  _et
al._ (2012); Zhang  _et al._ (2018); Gao  _et al._ (2018); Bennett  _et al._
(2012). Specifically, these materials can realize $\mathbb{Z}_{2}$ topological
insulators, topological nonsymmorphic crystalline insulators, topological
Dirac semimetals, and three-dimensional quantum spin-Hall insulators as well
as provide an opportunity to realize topological phase transitions from a
nontrivial insulator state to a Dirac or Weyl semimetal state by pressure or
systematically lowering the crystalline symmetries. For example, KHg$X$ ($X$ =
As, Sb, or Bi) materials with Hg$X$ honeycomb layer host topological non-
symmorphic crystalline insulator with hourglass-like surface Dirac fermions
and undergo a phase transition to topological Dirac semimetal state under
external pressure Wang  _et al._ (2016); Ma  _et al._ (2017); Liang  _et al._
(2017); Kuo  _et al._ (2019). Due to diverse chemical bondings, the
topological state in these materials is highly-sensitive to the materials’
parameters and electron interactions Zhang  _et al._ (2018). The distinction
between the topological Dirac semimetal or topological insulating state is
thus not very obvious in these materials.

KZnBi, which is of particular interest to this study, was predicted
theoretically as a $\mathbb{Z}_{2}$ trivial insulator Zhang  _et al._ (2018)
or a metal Bennett  _et al._ (2012). A recent experimental work, however,
reported it as a 3D topological Dirac semimetal with a pair of bulk Dirac
fermions in the bulk Brillouin zone (BZ) Song  _et al._ (2021a). The angle-
dependent magneto-transport measurements further reveal a phase transition to
realize chiral fermion states with anomalous Hall effect under an applied
magnetic field Song  _et al._ (2021b). The nontrivial features in KZnBi can be
tuned by varying the direction and flux of the Berry curvature field Song  _et
al._ (2021b); Park  _et al._ (2022). These results indicate that KZnBi
realizes a topological Dirac semimetal state in disagreement with
$\mathbb{Z}_{2}$ trivial insulator or a metallic state reported in the first-
principles studies. In this context, the density functional theory (DFT) based
first-principles calculations, in principle, can reproduce the ground state of
materials accurately provided the exact exchange-correlation (XC) functionals.
Proper inclusion of exchange and correlation effects may thus be crucial to
describe the topological Dirac semimetal state of KZnBi.

In this work, we provide a comprehensive investigation of the electronic
structure and topological properties of KZnBi based on first-principles
calculations. By carefully considering the different XC functionals including
the strongly constrained and appropriately normed (SCAN) and hybrid (with 25%
Hartree-Fock (HF) exchange) functionals, we delineate the electronic and
topological state of KZnBi. Our systematic analysis shows that SCAN density
functional reproduces the structural parameters in agreement with the
experimental results. However, it produces a topological nonsymmorphic
insulator state in disagreement with the experiments. By tuning the modified
Becke-Jhonson (mBJ) potential parameters, we recover the correct orbital
ordering and Dirac semimetal state of KZnBi. We further show that the Dirac
semimetal state can also be captured using the hybrid functionals with 42%
Hartree-Fock (HF) exchange. Our results demonstrate the importance of XC
functionals in the first-principles calculations to reproduce the topological
Dirac semimetal state of KZnBi in agreement with the experiments.

##  II Methodology

Electronic structure calculations are performed within the Kohn-Sham DFT
(KSDFT) framework Kohn and Sham (1965) using the projector augmented-wave
(PAW) method to model the ionic potentials as implemented in Vienna ab-initio
simulation package (VASP) Blöchl (1994); Kresse and Furthmüller (1996). The
accuracy of the KSDFT calculations depends solely on the types of the XC
density functionals, which can be arranged along different rungs of Jacob’s
ladder Perdew and Schmidt (2001); Marzari  _et al._ (2021). The XC functionals
are broadly categorized into semilocal, hybrid, and post-hybrid functionals.
The semilocal XC functionals can be written as

$E_{\rm xc}[\rho_{\uparrow},\rho_{\downarrow}]=\int d^{3}r\rho({\bf
r})\epsilon_{\rm
xc}(\rho_{\uparrow},\rho_{\downarrow},\nabla\rho_{\uparrow},\nabla\rho_{\downarrow},\tau_{\uparrow},\tau_{\downarrow})$
(1)

where $\rho_{\uparrow}({\bf r})$ and $\rho_{\downarrow}({\bf r})$ are the
electron spin densities and constitute the local density approximation (LDA)
Perdew and Zunger (1981); Ceperley and Alder (1980). Apart from the electron
spin densities, spin density gradients ($\nabla\rho_{\uparrow}({\bf r})$ and
$\nabla\rho_{\downarrow}({\bf r})$) are included in generalized gradient
approximations (GGA) Perdew  _et al._ (1996a). In the meta-GGAs additional
kinetic energy densities
($\tau_{\sigma=\uparrow,\downarrow}=\sum_{i=1}^{occ}\frac{1}{2}|\nabla\psi_{i,\sigma}|^{2}$)
are added Sun  _et al._ (2015). Notably, semilocal XC functionals can explain
the equilibrium geometry and symmetries of materials although they can fail to
reproduce the location of the highest occupied and lowest unoccupied state due
to the inherent self-interaction error Perdew and Zunger (1981). The hybrid
functionals are designed to overcome this issue by suitably mixing the
semilocal part with the exact HF exchange energy Perdew  _et al._ (1996b);
Heyd  _et al._ (2003). The range-separated hybrid functional can be written as

$E_{\rm xc}^{\rm HSE}=\alpha E_{\rm x}^{\rm SR,HF}(\omega)+(1-\alpha)E_{\rm
x}^{\rm SR,PBE}(\omega)+E_{\rm x}^{\rm LR,PBE}+E_{\rm c}^{\rm PBE}$ (2)

where $\alpha$ defines the fraction of HF exchange mixed at the short range
and $\omega$ determines the range-separation parameter. Another way to correct
the band gap at the semilocal level is to directly approximate the exchange
potential Talman and Shadwick (1976). In this spirit, modified Becke-Jhonson
(mBJ) exchange potential is proposed with the following form

$v_{{\rm x},\sigma}^{\rm TB-mBJ}({\bf r})=cv_{{\rm x},\sigma}^{\rm BR}({\bf
r})+(3c-2)\frac{1}{\pi}\sqrt{\frac{5}{12}}\sqrt{\frac{2\tau_{\sigma}({\bf
r})}{\rho_{\sigma}({\bf r})}}$ (3)

where, $v_{x,\sigma}^{BR}({\bf r})$ is the Becke-Roussel potential Becke and
Johnson (2006). The value of parameter $c$ ($c_{mBJ}$, henceforth) can be
determined from the average of $\nabla\rho/\rho$ over the unit cell Tran and
Blaha (2009). The first term in Eq. 3 denotes the average HF potential,
whereas the second term gives the screening potential. Equation 3 can be
regarded as a hybrid potential where the parameter $c_{mBJ}$ controls how much
the HF and the screening component are mixed.

We use the aforementioned XC functionals lying on different rungs of the
Jacob’s ladder Perdew and Schmidt (2001) to calculate the electronic and
topological state of KZnBi (see Table 1). The kinetic energy cut-off of 420 eV
for the plane-wave basis set and Gaussian smearing with a smearing width of 50
meV are used. We consider the van der Waals corrections within the DFT-D3
method of Grimme with zero damping function Grimme  _et al._ (2010). The spin-
orbit coupling (SOC) is added self-consistently to include the relativistic
effects Perdew  _et al._ (1996a). The optimization of lattice geometry is
performed until the forces on each atom are less than $10^{-2}$ eV$\AA^{-1}$.
A tolerance for electronic energy minimization is $10^{-6}$ eV and a
$\Gamma-$centred $11\times 11\times 9$ $k$-mesh for the BZ sampling is
considered. The topological properties are calculated using the materials-
specific tight-binding model generated using the vasp2wannier interface. The
atom-centred Wannier functions are generated using the Zn-$s$ and Bi-$p$
orbitals Mostofi  _et al._ (2008). We calculate the surface states using the
iterative Green’s functions method with WannierTools package Sancho  _et al._
(1985); Wu  _et al._ (2018).

##  III Results

###  III.1 Crystal structure and symmetries

KZnBi belongs to the non-symmorphic space group P$6_{3}/mmc$ (No. 194;
$D^{4}_{6h}$) and consists of ZnBi planer honeycomb layers with K atoms lying
between them Song  _et al._ (2021a). The {$-A-B-$}n stacking of the ZnBi
honeycomb layers with weak interlayer coupling mediated by K atoms in the
$\hat{z}$ direction forms the KZnBi crystal structure (see Fig. 1(a)). The K,
Zn, and Bi atoms occupy Wyckoff positions $2a~{}(0,0,0)$,
$2d~{}(\frac{1}{3},\frac{2}{3},\frac{3}{4})$, and
$2c~{}(\frac{1}{3},\frac{2}{3},\frac{1}{4})$, respectively, in the lattice.
Upon considering the valence electron configuration of the K, Zn, and Bi
atoms, we find that the K atom forms a monovalent cationic state (K+) with one
electron transferred to the honeycomb lattice ([ZnBi]-). Such a charge
transfer satisfies the octet rule of electron filling and suggests that KZnBi
may realize a small gap insulator or band overlap semimetal Papoian and
Hoffmann (2000).

The crystal symmetries of KZnBi include an inversion $\mathcal{\hat{I}}$
centred at K atom, six-fold screw rotation $\tilde{C}_{6z}$:
$(x,y,z)\rightarrow(x\ cos(\theta)-y\ sin(\theta),x\
sin(\theta)+ycos(\theta),z+c/2)$ with $\theta=2\pi/6$, a glide reflection
$\tilde{M}_{x}$: $(x,y,z)\rightarrow(-x,y,z+c/2)$, $M_{y}$:
$(x,y,z)\rightarrow(x,-y,z)$, and $\tilde{M_{z}}$:
$(x,y,z)\rightarrow(x,y,-z+1/2)$. Note that here $\tilde{M}_{x}$ and
$\tilde{C}_{6z}$ are non-symmorphic symmetries that involve unremovable
lattice translation even by shifting the origin of the unit cell. The top view
of the crystal structure is shown in Fig. 1(b) with a transformed orthorhombic
unit cell. The (100) and (010) planes (surface normal is given in the
cartesian coordinate system) represent the truncated ZnBi honeycomb lattice in
zigzag and armchair directions. The bulk BZ and its associated (010) and (100)
surface BZs of KZnBi are shown in Fig. 1(c). The grey plane centred at
$k_{z}=0$ shows the $\tilde{M_{z}}$ mirror plane. A detailed discussion of
surface symmetries and their associated electronic features is given below.

Figure 1: Crystal lattice, symmetries, and band structure of KZnBi. Bulk
crystal structure of KZnBi with the space group P$6_{3}/mmc$ (left). The ZnBi
honeycomb layers mediated by K layers are stacked along the hexagonal
$\hat{z}$ axis. The minimal unit cell arrangement and symmetries are shown in
the right panel. Maroon dotted plane and solid grey line denote the
$\tilde{M}_{x}$ and $\tilde{M}_{z}$ mirror planes. The dark yellow line shows
the three-fold rotational axis. (b) Top view of the crystal structure with
truncated zigzag and armchair directions. (c) Bulk Brillouin zone (BZ) and
associated (100) (blue color), and (001) (green color) plane projected surface
BZs of KZnBi. The high-symmetry points are marked. The grey plane inside the
bulk BZ denotes the $\tilde{M_{z}}$ mirror plane. (d) Calculated band
structure of KZnBi with SCAN+vdW density functional using fully optimized
structural parameters. Red and sky-blue colors indicate the contribution from
the Bi-$p$ and Zn-$s$ states.

We present the optimized lattice parameters of KZnBi obtained with different
XC density functionals in Table 1. The parameters obtained with the LDA are
smaller than the corresponding experimental values. The calculated bond
lengths of planer Zn-Bi bond ($d_{Zn-Bi}$) and out-of-plane K-{ZnBi} bond
($d_{K-ZnBi}$) are smaller than their associated experimental values as
expected since the LDA is known to overestimates bond strengths in many cases.
The lattice parameters obtained with GGA are overestimated almost by the same
amount as the LDA underestimated them. Since KZnBi has a layered structure,
the van der Waals (vdW) interaction plays a significant role in its structure
parameters. The mismatch in lattice parameters with experimental values
decreases when vdW Grimme  _et al._ (2010) interactions are included during
geometry optimization with GGA. While considering the meta-GGA, SCAN and
R2SCAN density functionals provide lattice parameters close to their
experimental values. Importantly, the SCAN+vdW provides an excellent match
with the corresponding experimental parameters. We thus describe the
electronic properties of KZnBi using the SCAN+vdW optimized parameters in our
later discussion.

Table 1: Calculated lattice parameters for KZnBi in P$6_{3}/mmc$ (No. 194; $D^{4}_{6h}$) space group using different XC density functionals. $a$ and $c$ are the hexagonal in-plane and out-of-plane lattice constants. $d_{Zn-Bi}$ and $d_{K-ZnBi}$ are bond lengths of in-plane Zn-Bi and out-of-plane K-{ZnBi} bonds.  |  $a$ (Å) |  $c$ (Å) |  $d_{Zn-Bi}$ (Å) |  $d_{K-ZnBi}$ (Å)
---|---|---|---|---
LDA | 4.591 |  10.262 | 2.650 | 2.565
GGA | 4.743 |  10.913 | 2.738 | 2.728
GGA+vdW | 4.711 |  10.692 | 2.720 | 2.673
SCAN | 4.652 |  10.706 | 2.685 | 2.676
R2SCAN | 4.678 |  10.725 | 2.700 | 2.681
SCAN+vdW | 4.636 |  10.601 | 2.676 | 2.650
Exp. Song  _et al._ (2021a) | 4.676 |  10.597 | 2.699 | 2.649

###  III.2 Topological non-symmorphic crystalline insulator state

The bulk band structure of KZnBi using fully relaxed structural parameters is
shown in Fig. 1(d). It realizes an insulating state with an indirect bulk gap
of 445 meV. The bands near the Fermi level consist of strongly hybridized
Bi-$p$ and Zn-$s$ states with a band inversion at the $\Gamma$ and $A$ points.
The presence of both time-reversal and inversion symmetries in KZnBi enforces
two-fold Kramers’s degeneracy in all the bands. The non-symmorphic symmetries
enforce additional degeneracy in the valence and conduction bands at the BZ
boundaries. Such band degeneracies are evident along the $L-A$ symmetry line
in Fig. 1(d). To discuss the topological state, we present the irreducible
representations of the space group characterizing band symmetries along the
$\Gamma-A$ line in Fig. 2(a). The bands at $\Gamma$ and $A$ points are two and
four-fold degenerate, respectively. At $A$ point, there are two types of
representations, $A_{6}$ and $A_{4}+A_{5}$ ($A_{4/5}$, in short). As one moves
away from A to $\Gamma$ at any intermediate point $\Delta$ (0, 0, $\delta
k_{z}$), the four-fold bands split into two two-fold bands
$A_{6}\rightarrow\Delta_{8}+\Delta_{9}$ and
$A_{4}+A_{5}\rightarrow\Delta_{7}+\Delta_{7}$. At the $\Gamma$ point, the
three $\Delta$ point representations evolve to $\Gamma_{i}^{\pm}$ ($i=7-12$),
where $\pm$ represents parity eigenvalue of the states. Importantly, $A_{6}$
and $\Gamma_{11,12}^{-}$ bands are inverted and lie in the valence region
without having any band crossings along the $\Gamma-A$ direction. The parity
eigenvalue analysis gives a $\mathbb{Z}_{2}=0$, revealing band inverted
$\mathbb{Z}_{2}$ trivial insulator state of KZnBi.

To characterize the exact topological state of KZnBi, we present the Wannier
charge centers (WCC) evolution of the occupied states along $X-U-Z-\Gamma-X$
direction in Fig. 2(b). The nontrivial connection in the WCC spectrum is
clearly seen. Note that the glide reflection $\tilde{M}_{x}$ combined with the
time-reversal $\mathcal{\hat{T}}$ symmetry defines a $\mathbb{Z}_{4}$
invariant ($\chi$) while the mirror operator $\tilde{M}_{z}$ ensures a well-
defined mirror Chern number (MCN) $C_{m}$ on the $k_{z}=0$ and $\pi/c$ planes.
The relation between the WCC spectrum and topological invariants is given as
Wieder  _et al._ (2018); Kuo  _et al._ (2019),

$\chi=2(n_{XU}^{+}+n_{Z\Gamma}^{+})+n_{\Gamma X}~{}{mod}~{}4$ (4)

and

$C_{m}=(n_{-X\Gamma X}^{+i}-n_{-X\Gamma X}^{-i})/2$ (5)

The numbers $n$’s in Eq. 4 and 5 can be evaluated from the WCC spectrum as
follows. If we choose the $\tilde{M}_{x}$ glide subspace with eigenvalue
$+ie^{(-ik_{z}c/2)}$ along $X\rightarrow U$ and $Z\rightarrow\Gamma$
direction, $n_{XU}^{+}$ and $n_{Z\Gamma}^{+}$ is defined as the difference
between the number of times an arbitrary horizontal reference line crosses the
bands with positive and negative slopes. $n_{\Gamma X}$ is the difference
between the number of times the reference line crosses positive and negative
slope bands in the $\Gamma\rightarrow X$ direction. Similarly, in the
$\tilde{M_{z}}$ mirror subspace along $-X\rightarrow\Gamma\rightarrow X$
direction, $n_{-X\Gamma X}^{+i}$ and $n_{-X\Gamma X}^{-i}$ gives the
difference between the number of times this line crosses the positive and
negative slope bands. Since the system is time-reversal symmetric,
$n_{-X\Gamma X}^{+i}=-n_{-X\Gamma X}^{-i}$. Note that while taking this
difference, one should compute {positive slope} - {negative slope} bands.
Following this convention, the values deduced from the WCC spectrum are
$n_{XU}^{+}=0$, $n_{Z\Gamma}^{+}=0$, $n_{\Gamma X}=2$ and $n_{-X\Gamma
X}^{+i}=-2$. The obtained topological invariants are $\chi=2$ and $C_{m}=-2$.
This shows that KZnBi is a topological non-symmorphic crystalline insulator
similar to KHgX Wang  _et al._ (2016); Ma  _et al._ (2017); Liang  _et al._
(2017); Kuo  _et al._ (2019). The robustness of this topological state is
further verified by calculating the electronic structure and WCCs spectrum
using different XC density functionals (LDA, GGA, R2SCAN) including the hybrid
density functional with 25% HF exchange (results not shown for brevity).

Figure 2: Topological non-symmorphic crystalline insulator state of KZnBi. (a)
Closeup of the bulk bands along the $\Gamma-A$ line with irreducible
representations. The Bi-$p$ and Zn-$s$ orbital characters are given in red and
sky-blue, respectively. (b) Calculated Wannier charge centers (WCC) spectrum
along various high-symmetry lines. The dashed red line identifies an arbitrary
reference line to calculate the topological invariants (see text for details).
Closeups of the spectrum in the light red, yellow, and green boxes are given
in the insets. (c) Band structure of the (010) surface of KZnBi. The inset
shows a zoomed-in view of the surface Hourglass fermion. (d) Constant surface
band contours at various energies. (e) Band structure of the (100) surface of
KZnBi. (f) Surface band contour at 100 meV (yellow line $e$) with the
associated spin texture. The four counter-propagating spin-polarized states
represent the double quantum spin-Hall states. (g) Evaluation of the WCCs on
the $k_{z}=0$ plane with nonzero mirror Chern number $C_{m}=-2$.

The nontrivial surface states in topological crystalline insulators (non-
symmorphic and symmorphic) appear only on the crystal surfaces that respect
crystalline symmetries protecting the topological state. To showcase these
nontrivial states, we calculate the surface band structure of (010) and (100)
surfaces of KZnBi (Figs. 2(c) and (e)). On the (010) surface, the glide-mirror
$\tilde{M}_{x}$ and mirror $\tilde{M}_{z}$ are preserved. Since the time-
reversal $\mathcal{\hat{T}}$ and $\tilde{M}_{x}$ symmetries commute $i.~{}e.$
$[\mathcal{\hat{T}},\tilde{M}_{x}]=0$. This results
($\mathcal{\hat{T}}\tilde{M}_{x})^{2}$ =$\mathcal{\hat{T}}^{2}$
$\tilde{M}_{x}^{2}$ = t(c$\hat{z}$)=$e^{-ik_{z}c}$. At $k_{z}=\pi/c$,
$(\mathcal{\hat{T}}\tilde{M}_{x})^{2}=-1$ and thus, a Kramer-like double
degeneracy is enforced at every wave vector (k) on the
$\overline{Z}-\overline{U}$ line as seen in Figs. 2(c). We observe Mobius
twisted bands along the $\tilde{M}_{x}$ invariant $\overline{X}-\overline{U}$
and $\overline{Z}-\overline{\Gamma}$ paths. The eigenvalue of $\tilde{M}_{x}$
operator at any wave vector along these lines is $\pm ie^{-ik_{z}c/2}$. When
$k_{z}=0$, the eigenvalues are $\pm i$ and when $k_{z}=\pi/c$, the eigenvalues
are $\pm 1$. This eigenvalues switching enforces bands to come in pair of
four, giving rise to hourglass surface fermions in KZnBi. The hourglass band
dispersion is clearly revealed in the zoom-in view of bands along
$\overline{Z}-\overline{\Gamma}$ in Fig. 2(c). The evolution of hourglass
surface states in the 2D surface BZ is shown in Fig. 2(d).

To resolve the $\tilde{M_{z}}$ mirror symmetry protected surface states, we
present the calculated band structure of the (100) surface in Fig. 2(e). On
the $k_{z}=0$ plane (gray plane in Fig. 1(c)), the bulk bands split into $+i$
or $-i$ subspace of $\tilde{M_{z}}$. By decomposing the Hamiltonian
$H(k)=H^{+i}(k)+H^{-i}(k)$, where $H^{\pm i}(k)$ is spanned by the mirror
eigenstates $\ket{\psi^{\pm i}_{k,n}}$ satisfying $\tilde{M_{z}}\ket{\psi^{\pm
i}_{k,n}}=\pm{i}\ket{\psi^{\pm i}_{k,n}}$, and calculating the associated
Chern numbers, we obtain the mirror Chern number $C_{m}=-2$ (Fig. 2(g)). Due
to this nontrivial $C_{m}$ on the $k_{z}=0$ plane, topological nontrivial
Dirac cone states are seen. However, these bands are almost non-dispersive
along the $k_{z}$ direction (${\overline{\Gamma}}-\overline{Z}$ line),
resulting in a highly anisotropic band dispersion with a nearly flat band
dispersion along ${\overline{\Gamma}}-\overline{Z}$. We show the constant
energy contours at 100 meV above the Fermi level on the $k_{y}-k_{z}$ plane in
Fig. 2(f). The two pairs of spin-polarized counter-propagating surface states
that are well-spaced in momentum space are clear. These states reflect the
double quantum spin-Hall insulator state in KZnBi Wang  _et al._ (2016).

###  III.3 Electronic state tuning and topological Dirac semimetal state

The preceding analysis demonstrates that KZnBi realizes a topological
nonsymmorphic crystalline insulator with different XC functionals in their
default setting. This is in disagreement with the recent ARPES experiments
where a topological Dirac semimetal state is reported Song  _et al._ (2021a).
To resolve this issue, we now present the bulk bands obtained by changing the
parameter $c_{mBJ}$ of mBJ potential in Fig. 3. This parameter mixes the
average HF and screening potential and can be tuned manually to obtain the
correct band ordering Tran and Blaha (2009); Koller  _et al._ (2011). Figure
3(a) presents the evolution of bulk bands with $c_{mBJ}$ along $A-\Gamma-A$
line. In the atomic limit of KZnBi, the space group representations of the
valence bands are $\Delta_{7}$ and $\Delta_{7}$, whereas the conduction band
are $\Delta_{8}$ and $\Delta_{9}$ at an intermediate point $\Delta$ on this
line. The associated $\hat{C}_{6z}$ rotational eigenvalues are

$\displaystyle\Delta_{7}=\\{e^{i\pi/2},e^{-i\pi/2}\\}$ (6)
$\displaystyle\Delta_{8}=\\{e^{i5\pi/6},e^{-i5\pi/6}\\}$
$\displaystyle\Delta_{9}=\\{e^{i\pi/6},e^{-i\pi/6}\\}$

Here, we omit the phase factor $e^{(-ik_{z}c/2)}$ arising from the lattice
translation along $c$ direction. Due to inversion and time-reversal
symmetries, these eigenvalues are degenerate in pairs. The band crossings
between different $\Delta_{i}$ bands will not hybridize and can result in
symmetry-protected band crossings. At the $c_{mBJ}$ value of 1.15,
$\Delta_{8}$ and $\Delta_{9}$ bands lie in the valence region, developing
double band inversions at both $\Gamma$ and $A$ points. With an increase in
$c_{mBJ}$ to 1.20, $\Delta_{7}$ and $\Delta_{8}$ bands cross at the Fermi
level to form stable band crossings. With a further increase in $c_{mBJ}$ to
1.26, $\Delta_{7}$ band crosses with $\Delta_{9}$ band to generate the stable
band crossings at the Fermi level. When $c_{mBJ}$ is increased to 1.35, both
$\Delta_{7}$ bands move to the valence region and generate a trivial insulator
state.

Figure 3: Evolution of band crossings and topological phase tuning in KZnBi.
(a) Calculated band structure along $A-\Gamma-A$ line for various values of
$c_{mBJ}$. The space group representations are noted on bands. (b) Band gap at
$\Gamma$ ($\delta_{i,~{}{i=1-4}}$) and $A$ points as a function of $c_{mBJ}$
(see text for details). The horizontal grey line at $\delta=0$ marks the
reference line separating the normal and inverted states. The solid markers
represent the calculated values, whereas solid lines are guides to the eye.
TDSM1, TDSM2, and TDSM3 represent topological Dirac semimetal with different
ordering of the crossings bands. TNCI represents a topological non-symmorphic
crystalline insulator. (c) Bands structure obtained with $c_{mBJ}=1.265$ and
hybrid density functional with $\omega=0.2$ Å-1 and $\alpha=42\%$ (see Eq. 2).
Closeup of the bulk bands along $\Gamma-A$ is shown in the right panel.

To clarify the distinct ordering of the crossings bands, we define energy gap
$\delta_{i}~{}(i=1-5)$ at $\Gamma$ and $A$ points as

$\displaystyle\delta_{1}=\Gamma_{9}^{+}-\Gamma_{10}^{-}$ (7)
$\displaystyle\delta_{2}=\Gamma_{11}^{-}-\Gamma_{7}^{+}$
$\displaystyle\delta_{3}=\Gamma_{9}^{+}-\Gamma_{7}^{+}$
$\displaystyle\delta_{4}=\Gamma_{11}^{-}-\Gamma_{10}^{-}$
$\displaystyle\delta_{5}=A_{6}-A_{4/5}$

The evolution of these $\delta_{i}$’s as function of $c_{mBJ}$ is shown in
Fig. 3(b). All $\delta_{i}$’s are negative at lower $c_{mBJ}$ values
($c_{mBJ}<1.17$), and whenever $\delta_{i}$’s changes sign, a band inversion
gets unlocked. The critical values of $c_{mBJ}$ at the phase transition points
are marked by arrows in Fig. 3(b). At lower $c_{mBJ}$ values, a topological
nonsymmorphic crystalline insulator with $\chi=2$ and $C_{m}=-2$ is formed. At
the intermediate values of $c_{mBJ}$, a topological Dirac semimetal state with
$C_{m}=-1$ is realized. However, the ordering of crossings bands that form
Dirac states changes with $c_{mBJ}$. At larger values of $c_{mBJ}>1.32$, a
trivial insulator is realized. On comparing with the experimental results, we
find that the topological Dirac semimetal phase realized with
$1.26<c_{mBJ}<1.32$ better reflects the experimental situation. Such a Dirac
semimetal phase and band orderings are reproduced by increasing the default HF
exchange to 42% with $\omega=0.2$ Å-1 in hybrid density functionals, as shown
in Fig. 3(c). These results clearly reveal that enhanced electron correlations
are essential to restore the correct orbital ordering and topological Dirac
semimetal state of KZnBi.

We discuss topological properties of the Dirac semimetal state in Fig. 4 and
compare them with the experimental results. The two symmetry-protected four-
fold degenerate Dirac nodes lie on the $A-\Gamma-A$ rotational axis at
$(0,0,\pm 0.143)$ Å-1 (Fig. 4(a)). The energy dispersion around the Dirac
nodes is highly anisotropic between in-plane and out-of-plane directions as
resolved in Fig. 4(b). Since energy dispersion on the $k_{z}=0$ plane is
gapped, we can define the $\mathbb{Z}_{2}$ number similar to insulators. The
calculated WCC spectrum reveals a nontrivial $\mathbb{Z}_{2}=1$ on this plane.
This demonstrates that the Dirac semimetal state of KZnBi is topologically
protected, similar to Na3Bi and Cd3As2 Dirac semimetals Xu  _et al._ (2015);
Liu  _et al._ (2014a). Figures 4(d) and 4(e) show the surface band structure
of (001) and (010) surfaces of KZnBi. The two Dirac nodes project at the
$\overline{\Gamma}$ point on the (001) plane. On the (010) surface, the Dirac
nodes project in the $\Gamma-Z$ line. The topological surface states emanating
from a projected Dirac node are seen in Fig. 4(e). The associated constant
energy spectrum at various energies in Fig. 4(f) reveals the double Fermi arcs
states that emanate and terminate at the projected Dirac nodes at the Fermi
level. The Fermi arc states change topology as one moves away from the Fermi
level.

Figure 4: Topological Dirac semimetal state of KZnBi. (a) Location of the
four-fold Dirac nodes in the hexagonal Brillouin zone on the rotational axis
(red sphere). (b) Energy dispersion of the Dirac cones along the out-of-plane
(left) and in-plane (right) directions. (c) Evolution of Wannier charge
centers (WCCs) on the $k_{z}=0$ plane along $\Gamma-X$ line. The WCCs cross
the red reference line one-time, revealing nontrivial $\mathbb{Z}_{2}=1$.
(d)-(e) Calculated band structure of (001) and (010) surface of KZnBi.
Topological surface states emanating from the projected Dirac nodes are
resolved. (f) Evolution of surface band contours as a function of energy.
Double Fermi arc surface states connecting the projected bulk Dirac nodes are
resolved at the Fermi energy.

We present the local energy dispersion of Dirac bands along $k_{x}$, $k_{y}$,
and $k_{z}$ directions in Figs. 5(a)-(c) and associated Fermi velocities
$v_{f}=\partial E/\partial{k}$ in Figs. 5(d)-(f). The Fermi velocity of bands
at the Dirac nodes is $9.09\times 10^{5}\ m/s$, $8.78\times 10^{5}\ m/s$, and
$1.31\times 10^{5}\ m/s$ along $k_{x}$, $k_{y}$ and $k_{z}$ directions,
respectively. The energy dispersion and Fermi velocity reveal the anisotropic
nature of Dirac carriers. The calculated location, nature, and carrier
velocities of the Dirac states are in excellent agreement with their
corresponding experimental results Song  _et al._ (2021a).

Figure 5:  Energy dispersion and Fermi velocity of Dirac states. Calculated
energy dispersion of the Dirac bands along (a) $k_{x}$, (b) $k_{y}$ and (c)
$k_{z}$ directions. The center of the axis is located at a Dirac node.
Calculated carrier velocity along (d) $k_{x}$, (e) $k_{y}$ and (f) $k_{z}$
directions.

##  IV Conclusion

We have investigated the electronic structure of KZnBi within the framework of
density functional theory, with a focus on delineating its exact topological
state. Our structural optimization based on various levels of XC density
functionals shows that the lattice parameters obtained with SCAN+vdW are in
excellent agreement with the experimental parameters. The associated
electronic state realizes an insulator state with inverted band ordering in
bulk BZ. The band symmetries and WCC spectrum show that the system realizes a
topological nonsymmorphic crystalline insulator with topological invariants
$\chi=2$ and $C_{m}=-2$. This calculated electronic and topological state is
found to be distinct from the recent experiments that report KZnBi as a
topological Dirac semimetal. We address this discrepancy by showing that
enhanced XC effects are essential to reproduce the topological Dirac semimetal
state with correct orbital ordering in KZnBi. Specifically, a higher value of
$c_{mBJ}$ parameter that mixes the HF exchange and screening potentials
reproduces the correct orbital ordering and Dirac semimetal state. Our
calculated Dirac nodes, their location, and associated carrier velocities are
found to be in excellent agreement with available experimental results. Our
study thus demonstrates that enhanced XC effects are essential to reproduce
the correct orbital ordering and topological Dirac semimetal state of KZnBi.

## Acknowledgements

This work is supported by the Department of Atomic Energy of the Government of
India under Project No. 12-R$\&$D-TFR-5.10-0100 and benefited from the
computational resources of TIFR Mumbai.

## References

  * Singh  _et al._ (2023) Bahadur Singh, Hsin Lin, and Arun Bansil, “Topology and symmetry in quantum materials,” Adv. Mater. 35, 2201058 (2023).
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
  * Armitage  _et al._ (2018) N. P. Armitage, E. J. Mele, and Ashvin Vishwanath, “Weyl and dirac semimetals in three-dimensional solids,” Rev. Mod. Phys. 90, 015001 (2018).
  * Tang  _et al._ (2019) Feng Tang, Hoi Chun Po, Ashvin Vishwanath, and Xiangang Wan, “Comprehensive search for topological materials using symmetry indicators,” Nature 566, 486–489 (2019).
  * Vergniory  _et al._ (2019) M. G. Vergniory, L. Elcoro, Claudia Felser, Nicolas Regnault, B. Andrei Bernevig, and Zhijun Wang, “A complete catalogue of high-quality topological materials,” Nature 566, 480–485 (2019).
  * Zhang  _et al._ (2019) Tiantian Zhang, Yi Jiang, Zhida Song, He Huang, Yuqing He, Zhong Fang, Hongming Weng, and Chen Fang, “Catalogue of topological electronic materials,” Nature 566, 475–479 (2019).
  * Fu (2011) Liang Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011).
  * Hsieh  _et al._ (2012) Timothy H. Hsieh, Hsin Lin, Junwei Liu, Wenhui Duan, Arun Bansil, and Liang Fu, “Topological crystalline insulators in the SnTe material class,” Nat. Commun. 3, 982 (2012).
  * Shiozaki and Sato (2014) Ken Shiozaki and Masatoshi Sato, “Topology of crystalline insulators and superconductors,” Phys. Rev. B 90, 165114 (2014).
  * Wang  _et al._ (2019) Baokai Wang, Bahadur Singh, Barun Ghosh, Wei-Chi Chiu, M. Mofazzel Hosen, Qitao Zhang, Li Ying, Madhab Neupane, Amit Agarwal, Hsin Lin, and Arun Bansil, “Topological crystalline insulator state with type-II dirac fermions in transition metal dipnictides,” Phys. Rev. B 100, 205118 (2019).
  * Young  _et al._ (2012) S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J. Mele, and A. M. Rappe, “Dirac semimetal in three dimensions,” Phys. Rev. Lett. 108, 140405 (2012).
  * Wang  _et al._ (2012) Zhijun Wang, Yan Sun, Xing-Qiu Chen, Cesare Franchini, Gang Xu, Hongming Weng, Xi Dai, and Zhong Fang, “Dirac semimetal and topological phase transitions in ${A}_{3}$Bi (${A}=\text{Na}$, K, Rb),” Phys. Rev. B 85, 195320 (2012).
  * Yang and Nagaosa (2014) Bohm-Jung Yang and Naoto Nagaosa, “Classification of stable three-dimensional dirac semimetals with nontrivial topology,” Nat. Commun. 5, 4898 (2014).
  * Xu  _et al._ (2015) Su-Yang Xu, Chang Liu, Satya K. Kushwaha, Raman Sankar, Jason W. Krizan, Ilya Belopolski, Madhab Neupane, Guang Bian, Nasser Alidoust, Tay-Rong Chang, Horng-Tay Jeng, Cheng-Yi Huang, Wei-Feng Tsai, Hsin Lin, Pavel P. Shibayev, Fang-Cheng Chou, Robert J. Cava, and M. Zahid Hasan, “Observation of fermi arc surface states in a topological metal,” Science 347, 294–298 (2015).
  * Liu  _et al._ (2014a) Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M. Weng, D. Prabhakaran, S.-K. Mo, H. Peng, P. Dudin, T. Kim, M. Hoesch, Z. Fang, X. Dai, Z. X. Shen, D. L. Feng, Z. Hussain, and Y. L. Chen, “A stable three-dimensional topological dirac semimetal Cd3As2,” Nat. Mater. 13, 677–681 (2014a).
  * Singh  _et al._ (2018a) Bahadur Singh, Sougata Mardanya, Chenliang Su, Hsin Lin, Amit Agarwal, and Arun Bansil, “Spin-orbit coupling driven crossover from a starfruitlike nodal semimetal to dirac and weyl semimetal state in CaAuAs,” Phys. Rev. B 98, 085122 (2018a).
  * Murakami (2007) Shuichi Murakami, “Phase transition between the quantum spin hall and insulator phases in 3D: emergence of a topological gapless phase,” New J. Phys. 9, 356 (2007).
  * Huang  _et al._ (2015) Shin-Ming Huang, Su-Yang Xu, Ilya Belopolski, Chi-Cheng Lee, Guoqing Chang, BaoKai Wang, Nasser Alidoust, Guang Bian, Madhab Neupane, Chenglong Zhang, Shuang Jia, Arun Bansil, Hsin Lin, and M. Zahid Hasan, “A weyl fermion semimetal with surface fermi arcs in the transition metal monopnictide TaAs class,” Nat. Commun. 6, 7373 (2015).
  * Singh  _et al._ (2012) Bahadur Singh, Ashutosh Sharma, H. Lin, M. Z. Hasan, R. Prasad, and A. Bansil, “Topological electronic structure and Weyl semimetal in the TlBiSe2 class of semiconductors,” Phys. Rev. B 86, 115208 (2012).
  * Mardanya  _et al._ (2019) Sougata Mardanya, Bahadur Singh, Shin-Ming Huang, Tay-Rong Chang, Chenliang Su, Hsin Lin, Amit Agarwal, and Arun Bansil, “Prediction of threefold fermions in a nearly ideal dirac semimetal BaAgAs,” Phys. Rev. Mater. 3, 071201 (2019).
  * Liu  _et al._ (2014b) Chao-Xing Liu, Rui-Xing Zhang, and Brian K. VanLeeuwen, “Topological nonsymmorphic crystalline insulators,” Phys. Rev. B 90, 085304 (2014b).
  * Alexandradinata  _et al._ (2016) A. Alexandradinata, Zhijun Wang, and B. Andrei Bernevig, “Topological insulators from group cohomology,” Phys. Rev. X 6, 021008 (2016).
  * Wang  _et al._ (2016) Zhijun Wang, A. Alexandradinata, R. J. Cava, and B. Andrei Bernevig, “Hourglass fermions,” Nature 532, 189–194 (2016).
  * Ma  _et al._ (2017) Junzhang Ma, Changjiang Yi, Baiqing Lv, ZhiJun Wang, Simin Nie, Le Wang, Lingyuan Kong, Yaobo Huang, Pierre Richard, Peng Zhang, Koichiro Yaji, Kenta Kuroda, Shik Shin, Hongming Weng, Bogdan Andrei Bernevig, Youguo Shi, Tian Qian, and Hong Ding, “Experimental evidence of hourglass fermion in the candidate nonsymmorphic topological insulator KHgSb,” Sci. Adv. 3, e1602415 (2017).
  * Liang  _et al._ (2017) A. J. Liang, J. Jiang, M. X. Wang, Y. Sun, N. Kumar, C. Shekhar, C. Chen, H. Peng, C. W. Wang, X. Xu, H. F. Yang, S. T. Cui, G. H. Hong, Y.-Y. Xia, S.-K. Mo, Q. Gao, X. J. Zhou, L. X. Yang, C. Felser, B. H. Yan, Z. K. Liu, and Y. L. Chen, “Observation of the topological surface state in the nonsymmorphic topological insulator KHgSb,” Phys. Rev. B 96, 165143 (2017).
  * Singh  _et al._ (2018b) Bahadur Singh, Barun Ghosh, Chenliang Su, Hsin Lin, Amit Agarwal, and Arun Bansil, “Topological hourglass dirac semimetal in the nonpolar phase of Ag2BiO3,” Phys. Rev. Lett. 121, 226401 (2018b).
  * Ezawa (2016) Motohiko Ezawa, “Hourglass fermion surface states in stacked topological insulators with nonsymmorphic symmetry,” Phys. Rev. B 94, 155148 (2016).
  * Zhang  _et al._ (2011) Hai-Jun Zhang, Stanislav Chadov, Lukas Müchler, Binghai Yan, Xiao-Liang Qi, Jürgen Kübler, Shou-Cheng Zhang, and Claudia Felser, “Topological insulators in ternary compounds with a honeycomb lattice,” Phys. Rev. Lett. 106, 156402 (2011).
  * Yan  _et al._ (2012) Binghai Yan, Lukas Müchler, and Claudia Felser, “Prediction of weak topological insulators in layered semiconductors,” Phys. Rev. Lett. 109, 116406 (2012).
  * Zhang  _et al._ (2018) Xiuwen Zhang, Qihang Liu, Qiunan Xu, Xi Dai, and Alex Zunger, “Topological insulators versus topological dirac semimetals in honeycomb compounds,” J. Am. Chem. Soc. 140, 13687–13694 (2018).
  * Gao  _et al._ (2018) Heng Gao, Youngkuk Kim, Jörn W. F. Venderbos, C. L. Kane, E. J. Mele, Andrew M. Rappe, and Wei Ren, “Dirac-weyl semimetal: Coexistence of dirac and weyl fermions in polar hexagonal ABC crystals,” Phys. Rev. Lett. 121, 106404 (2018).
  * Bennett  _et al._ (2012) Joseph W. Bennett, Kevin F. Garrity, Karin M. Rabe, and David Vanderbilt, “Hexagonal ABC semiconductors as ferroelectrics,” Phys. Rev. Lett. 109, 167602 (2012).
  * Kuo  _et al._ (2019) Chin-Shen Kuo, Tay-Rong Chang, Su-Yang Xu, and Horng-Tay Jeng, “Unconventional topological phase transition in non-symmorphic material KHgX (X = As, Sb, Bi),” npj Comput. Mater. 5, 65 (2019).
  * Song  _et al._ (2021a) Junseong Song, Sunghun Kim, Youngkuk Kim, Huixia Fu, Jahyun Koo, Zhen Wang, Gyubin Lee, Jouhahn Lee, Sang Ho Oh, Joonho Bang, Taku Matsushita, Nobuo Wada, Hiroki Ikegami, Jonathan D. Denlinger, Young Hee Lee, Binghai Yan, Yeongkwan Kim, and Sung Wng Kim, “Coexistence of surface superconducting and three-dimensional topological dirac states in semimetal KZnBi,” Phys. Rev. X 11, 021065 (2021a).
  * Song  _et al._ (2021b) Junseong Song, Byung Cheol Park, Kyung Ik Sim, Joonho Bang, Sunghun Kim, Zhuo Yang, Yoshimitsu Kohama, Yeongkwan Kim, and Sung Wng Kim, “Tunable berry curvature and transport crossover in topological dirac semimetal KZnBi,” npj Quantum Mater. 6, 77 (2021b).
  * Park  _et al._ (2022) Byung Cheol Park, Taewoo Ha, Kyung Ik Sim, Taek Sun Jung, Jae Hoon Kim, Yeongkwan Kim, Young Hee Lee, Teun-Teun Kim, and Sung Wng Kim, “Real-space imaging and control of chiral anomaly induced current at room temperature in topological dirac semimetal,” Sci. Adv. 8, eabq2479 (2022).
  * Kohn and Sham (1965) W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev. 140, A1133–A1138 (1965).
  * Blöchl (1994) P. E. Blöchl, “Projector augmented-wave method,” Phys. Rev. B 50, 17953–17979 (1994).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B 54, 11169–11186 (1996).
  * Perdew and Schmidt (2001) John P. Perdew and Karla Schmidt, “Jacob’s ladder of density functional approximations for the exchange-correlation energy,” AIP Conference Proceedings 577, 1–20 (2001).
  * Marzari  _et al._ (2021) Nicola Marzari, Andrea Ferretti, and Chris Wolverton, “Electronic-structure methods for materials design,” Nat. Mater. 20, 736–749 (2021).
  * Perdew and Zunger (1981) J. P. Perdew and Alex Zunger, “Self-interaction correction to density-functional approximations for many-electron systems,” Phys. Rev. B 23, 5048–5079 (1981).
  * Ceperley and Alder (1980) D. M. Ceperley and B. J. Alder, “Ground state of the electron gas by a stochastic method,” Phys. Rev. Lett. 45, 566–569 (1980).
  * Perdew  _et al._ (1996a) John P. Perdew, Kieron Burke, and Matthias Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77, 3865–3868 (1996a).
  * Sun  _et al._ (2015) Jianwei Sun, Adrienn Ruzsinszky, and John P. Perdew, “Strongly constrained and appropriately normed semilocal density functional,” Phys. Rev. Lett. 115, 036402 (2015).
  * Perdew  _et al._ (1996b) John P. Perdew, Matthias Ernzerhof, and Kieron Burke, “Rationale for mixing exact exchange with density functional approximations,” J. Chem. Phys. 105, 9982–9985 (1996b).
  * Heyd  _et al._ (2003) Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof, “Hybrid functionals based on a screened Coulomb potential,” J. Chem. Phys. 118, 8207–8215 (2003).
  * Talman and Shadwick (1976) James D. Talman and William F. Shadwick, “Optimized effective atomic central potential,” Phys. Rev. A 14, 36–40 (1976).
  * Becke and Johnson (2006) Axel D. Becke and Erin R. Johnson, “A simple effective potential for exchange,” J. Chem. Phys. 124, 221101 (2006).
  * Tran and Blaha (2009) Fabien Tran and Peter Blaha, “Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential,” Phys. Rev. Lett. 102, 226401 (2009).
  * Grimme  _et al._ (2010) Stefan Grimme, Jens Antony, Stephan Ehrlich, and Helge Krieg, “A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu,” J. Chem. Phys. 132, 154104 (2010).
  * Mostofi  _et al._ (2008) Arash A. Mostofi, Jonathan R. Yates, Young-Su Lee, Ivo Souza, David Vanderbilt, and Nicola Marzari, Comput. Phys. Commun. 178, 685–699 (2008).
  * Sancho  _et al._ (1985) M P Lopez Sancho, J M Lopez Sancho, J M L Sancho, and J Rubio, “Highly convergent schemes for the calculation of bulk and surface green functions,” J. Phys. F: Met. Phys. 15, 851 (1985).
  * Wu  _et al._ (2018) QuanSheng Wu, ShengNan Zhang, Hai-Feng Song, Matthias Troyer, and Alexey A. Soluyanov, “Wanniertools : An open-source software package for novel topological materials,” Comput. Phys. Commun. 224, 405 – 416 (2018).
  * Papoian and Hoffmann (2000) Garegin A. Papoian and Roald Hoffmann, “Hypervalent bonding in one, two, and three dimensions: Extending the zintl-klemm concept to nonclassical electron-rich networks,” Angew. Chem., Int. Ed. 39, 2408–2448 (2000).
  * Wieder  _et al._ (2018) Benjamin J. Wieder, Barry Bradlyn, Zhijun Wang, Jennifer Cano, Youngkuk Kim, Hyeong-Seok D. Kim, Andrew M. Rappe, C. L. Kane, and B. Andrei Bernevig, “Wallpaper fermions and the nonsymmorphic dirac insulator,” Science 361, 246–251 (2018).
  * Koller  _et al._ (2011) David Koller, Fabien Tran, and Peter Blaha, “Merits and limits of the modified becke-johnson exchange potential,” Phys. Rev. B 83, 195134 (2011).