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\begin{document}

\title{Supplementary Material for ``Topological nodal-line semimetals in alkaline-earth stannides, germanides and silicides''}

\author{Huaqing Huang}
\affiliation{Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China}
\affiliation{Collaborative Innovation Center of Quantum Matter, Tsinghua University, Beijing 100084, China}
%\affiliation{Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854-0849, USA}

\author{Jianpeng Liu}
\affiliation{Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA}
\affiliation{Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey  08854-8019, USA}

\author{David Vanderbilt}
\affiliation{Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey  08854-8019, USA}

\author{Wenhui Duan\footnote{dwh@phys.tsinghua.edu.cn}}
\affiliation{Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China}
\affiliation{Collaborative Innovation Center of Quantum Matter, Tsinghua University, Beijing 100084, China}
\affiliation{Institute for Advanced Study, Tsinghua University, Beijing 100084, China}
\date{\today}

\date{\today}

\pacs{73.20.-r,73.21.-b,73.43.-f}
\maketitle

\tableofcontents

\section{Computational methods}
The first-principles calculations are carried out within the framework of
density functional theory (DFT) as implemented in the Vienna
\textit{ab initio} simulation package\cite{VASP}, with the Perdew-Burke-Ernzerhof
generalized gradient approximation exchange-correlation functional
and the projector augmented-wave method. \cite{PBE,PAW}
The kinetic-energy cutoff is fixed to 520\,eV and an $8\times 8 \times 8$
$\Gamma$-centered $\mathbf{k}$ mesh is used for the BZ sampling.
The electronic structures are calculated both with and without SOC.
To investigate the topological properties of these materials, we
calculate the surface states and Fermi surfaces using the Wannier
interpolation technique.
More specifically, we use the WANNIER90 package to generate Wannier
function (WFs) from the outputs of standard first-principles calculations\cite{wannier90}.
%WANNIER90 can optionally generate maximally localized WFs,\cite{wannier1, *wannier2}
%and in any case reports the Wannier charge centers, their spreads,
%and the real-space Hamiltonian matrix elements of an effective
%tight-binding (TB) model in the WF basis. This information is often very useful in studying the
%bonding mechanism of materials, as well as for calculating
%topological indices, computing surface and interface states,
%treating disorder, etc.

It should be noted that the tight-binding models constructed from
WANNIER90 are realistic in the sense that the Wannier-interpolated
band structures reproduce the first-principles band
structures essentially exactly within a certain energy window.
This ``frozen window'' is chosen to extend from the lowest eigenvalue
in the system (bottom of the energy range) %-1.5 eV below the Fermi level
up to 1.5 eV above the Fermi level in our calculations.
In addition to the frozen window, there is also an outer
energy window outside which the Bloch eigenstates will not
be included in generating the WFs. However, here we use a default value
of the outer window, i.e., from the lowest to the highest eigenvalues in the system.
%the whole energy range in our calculations.
In the calculation, we project the DFT wavefunctions into
Sn \textit{s}, Sn \textit{p} and Ba \textit{d} orbitals %, which dominate the energy bands around the Fermi level,
to construct the Wannier function without applying a subsequent maximal-localization
procedure.\cite{wannier1,*wannier2}
%construct the Wannier functions \cite{wannier1,*wannier2,wannier90}
%for \textit{s}, \textit{p} orbitals of Sn and \textit{d} orbital of Ba,
We also check the effect of \textit{d} orbitals of Sn by including them
into the initial atomiclike trial orbitals, and got similar results.
Within the Wannier representation, we calculate the surface states
using the iterative Green's function method based on the bulk tight-binding Hamiltonian.\cite{lopez,*lopez2}
%It should be noted that the effects
%of surface reconstructions and charge rearrangements are not included
%in our surface calculations, so the obtained surface band structures
%may not resemble the real ones in detail. However, the essential
%features of the topological surface states are expected to be captured by
%this Wannier-based approach at a greatly reduced computational cost.
%

\section{Symmetry analysis}
%\paragraph*{Symmetry analysis}
The nodal loop in BaSn$_2$ is topologically protected due to the
coexistence of time-reversal ($\mathcal{T}$) and spatial inversion
($\mathcal{P}$) symmetries.\cite{MTC,martin2004electronic}
When SOC
is absent, the system can be considered as a spinless system
for which $\mathcal{T}$ is simply a complex conjugation operator.
With $\mathcal{T}$ symmetry, the Bloch functions at $\k$ become equal to the
complex conjugation of those at $-\k$ up to an arbitrary phase factor:
$\mathcal{T} \psi_{n\k}(\r)\!=\!e^{-i\k\cdot\r}u_{n\k}^{*}(\r)\!=\!e^{-i\k\cdot\r}u_{n-\k}(\r)e^{i\phi}$,
where $n$ is the band index.
Setting $\phi\!=\!0$ leads to $u_{n\k}^{*}(\r)\!=\!u_{n-\k}(\r)$.
On the other hand, inversion symmetry connects  $u_{n\k}(-\r)$ to $u_{n-\k}(\r)$, i.e.,
$\mathcal{P} \psi_{n\k}(\r)\!=\!e^{-i\k\cdot\r}u_{n\k}(-\r)\!=\!e^{-i\k\cdot\r}u_{n-\k}(\r)e^{i\phi}$,
so that $u_{n\k}(-\r)\!=\!u_{n-\k}(\r)$ (setting $\phi\!=\!0$).
Combining the above two equations, one obtains $u_{n\k}^*(\r)\!=\!u_{n\k}(-\r)$.
Then it is straightforward to show that the corresponding effective
Hamiltonian $H(\k)$ has to be real-valued, i.e.,
$H_{mn}(\k)\!=\!H_{nm}(\k)$.

%A BT problem at an arbitrary $\k$ point can be minimally described
%by a two-band effective Hamiltonian, which can be expressed in terms
%of the identity matrix and the three Pauli matrices. According to the
%above argument, the two-band Hamiltonian for a system respecting both
%$\mathcal{T}$ and $\mathcal{P}$ symmetries can be chosen as real valued,
%so the codimension of such a BT problem is 2, one less than the number
%of independent variables (i.e., $k_x$, $k_y$ and $k_z$).  Hence a nodal
%loop is stable in the presence of coexisting $\mathcal{T}$ and
%$\mathcal{P}$ symmetries.

Some other symmetries of the system, such as $C_2$ rotation about the
$A$-$H$ line and $C_3$ rotation about the $k_z$ axis, are crucial in
determining the shape of the nodal loop. For example,
the $C_2$ symmetry about the $A$-$H$ line implies that a BT is allowed
on the $A$-$H$ line as the states with opposite $C_2$ eigenvalues can
cross each other without opening a gap, while the $C_3$ symmetry combined
with $\mathcal{T}$ or $\mathcal{P}$ implies that the nodal loop has to
cross the $k_z\!=\!\pi$ plane at six isolated points on the equivalent
$A$-$H$ lines. Nevertheless, the nodal loop would be robust even in
the absence of these additional crystal symmetries; weakly breaking
them would just distort the loop so that its shape would
no longer respect them.


Now let us consider the symmetries of the systems which impose
constraints on the allowed form of the coefficients $d_i(\mathbf{q})$
($i\!=\!x,y,z$). The little group at $A$ contains a threefold rotation
$C_3$ about $k_z$, a twofold rotation $C_2$ about the $A$-$H$ axis,
inversion $\mathcal{P}$, and time reversal $\mathcal{T}$.
Thus the Hamiltonian has to obey the following symmetry constraints:
\begin{equation}
\begin{split}
C_3h(q_x,q_y,q_z)C_3^{-1}&=h(-q_y,q_x-q_y,q_z),\\
C_2h(q_x,q_y,q_z)C_2^{-1}&=h(q_y,q_x,-q_z),\\
\mathcal{P}h(q_x,q_y,q_z)\mathcal{P}^{-1}&=h(-q_x,-q_y,-q_z),\\
\mathcal{T}h(q_x,q_y,q_z)\mathcal{T}^{-1}&=h(-q_x,-q_y,-q_z).\\
\end{split}
\label{eq_sym}
\end{equation}


\section{Hybrid Wannier functions}
The 1D hybrid Wannier functions (WFs)
localized in the $z$ direction are constructed as
\begin{equation}
|W_{nR_z}(k_x,k_y)\rangle=\frac{1}{2\pi}\int_0^{2\pi} dk_z e^{ik_z(r_z - R_z)}|u_{n,\mathbf{k}}\rangle\;,
\end{equation}
where $R_z$ is the $z$ component of a lattice vector $\mathbf{R}$.
The hybrid WCC $z_n (k_x,k_y)$ is then defined as the expectation value
of $\hat{z}$ for the hybrid WFs in the ``home'' unit cell $R_z=0$, i.e.,
$z_n (k_x,k_y)=\langle W_{n0}|\hat{z}|W_{n0}\rangle $.
The sum of the hybrid WCCs over the occupied subspace,
$z(k_x, k_y)= \sum_{n} \langle z_n(k_x, k_y) \rangle$, is equivalent
(up to a factor of $2\pi$) to the total Berry phase of the occupied Bloch
functions accumulated along the $k_z$ direction.

\section{Surface states}

\begin{figure}
\includegraphics[width =0.7\columnwidth]{FigS1_001surface.eps}%
\caption{\label{figs1} (Color online) Layer-resolved local density of
states of the (001) surface state as a function of the principal layers
index $n$. Here $n$ is the index of the layer counted from the surface.}
\end{figure}

In order to characterize the localization of the drumhead-like surface
state on the (001) surface, we calculate the local density of states
of several principal layers near the surface in the same energy and
\textbf{k}-space range. As shown in Fig.~\ref{figs1}, the surface states
within the projected node loop are localized in the surface region and
decay rapidly away from the surface. In particular, the signature of
surface states almost vanishes within five principal layers.

\begin{figure}
\includegraphics[width =1\columnwidth]{FigS3_difEcut.eps}%
\caption{\label{figs3} (Color online) Fermi surface of the semi-infinite
(010) surface with chemical potential at different values. (c) The
lower surface state spreads both the ellipse area and the exterior of
the projected node loop. (g,h) The upper surface state only exists on
the ellipse area.}
\end{figure}

The Fermi surfaces of the (010) surface with the chemical potential at different
energy values are presented in Fig.~\ref{figs3}. Two surface states
with different energy occur in the BZ. The lower surface state spans
both the ellipse area and the exterior of the projected node loop. The
upper surface state only exists in the ellipse area. Hence there are
two surface states inside the ellipse area of the projected node-line
loop but none in the two crescent areas.


\begin{figure}
\includegraphics[width =0.7\columnwidth]{FigS4_difTerminal001.eps}%
\caption{\label{figs_terminal} (Color online) Surface band structures
of semi-infinite (001) surfaces cut in different terminations.}
\end{figure}
The dispersions of surface band structures are found to be sensitive to
the surface condition. By cutting the surface at different positions, the
dispersion of the surface states can be dramatically different, as shown in
Fig.~\ref{figs_terminal}. %the Sn-terminated (001) surface nestles inside
%the projected node loop. On the contrary, the Ba-terminated (001) surface
%state spreads the whole exterior region of the projected node loop.
It is interesting to note that when the (001) surface is terminated at the
Ba atomic layer, the topological surface states fill the region outside the
projected nodal loop as shown in Fig.~\ref{figs_terminal}(c)-(d),
which is exactly opposite to the Sn-terminated surface.
\Red{To understand such an interesting phenomenon, one may consider the
in-plane wavevectors $k_x$ and $k_y$ as some external parameters of a
fictitious ``1D" chain extending along the $z$ direction with two sublattices
in each unit cell (consider the Sn dimer as one sublattice, the Ba as the other).
As discussed above, the Berry phase of such a 1D chain with both $\mathcal{T}$
and $\mathcal{P}$ symmetries have to be either $0$ or $\pi$. A chain with $\pi$
Berry phase is topologically distinct from that whose Berry phase is $0$, as
they cannot be adiabatically connected to each other without closing the bulk gap.
Since the Berry phase is well-defined modulo $2\pi$, there is a $\mathbb{Z}_2$
classification to such 1D chains, and the one whose Berry phase is $\pi$ (0) is
denoted as $\mathbb{Z}_2$ odd (even). The nodal loop acts exactly as the critical
phase separating the even and odd 1D chains parameterized by $(k_x, k_y)$.
Unlike topological phases in higher dimensions, neither of these two phases have
robust mid-gap end states; instead the existence of the end states is sensitive
to how the chain is truncated. Nevertheless, if one phase, say, the
$\mathbb{Z}_2$-odd chain has mid-gap end state for a given truncation, then there
cannot be topological end state for the $\mathbb{Z}_2$ even chain for the same
truncation, and vice versa.}

\Red{The above argument explains why the topological surface states exist in different
regions for different surface terminations. As shown in Fig.~3(b) in the main text,
the Berry phase along the $z$ direction is $\pi$ (0) within (outside) the projected loop,
so that the fictitious chains inside and outside the projected nodal loop are topologically
distinct. When the system is terminated at the Sn atoms, it turns out that there are mid-gap
end states for the fictitious chains inside the projected nodal loop with $\pi$ Berry
phase, and it is opposite for the Ba-terminated surface.}


\section{Effect of spin-orbit coupling}

\begin{figure}
\includegraphics[width =0.7\columnwidth]{FigS5_TI.eps}%
\caption{\label{figs_TI} (Color online) (a) Calculated band structure with
(red) and without (blue) SOC. (b) Evolution of the Wannier charge center
along $0 < k_z < G_z/2$ on the $k_x = 0$ and $G_x/2$ plane. Red points
mark the midpoint of the largest gap. The (c) Sn-terminated and (d)
Ba-terminated surface band structures of the (001) surface in the presence
of SOC.}
\end{figure}

When SOC is included in the calculation, the node line
is gapped and the system becomes a topological insulator. We calculated
the Z$_2$ invariants [$(\nu _0;\nu _1\nu _2\nu _3)=(1;001)$] and the surface states (Fig.~\ref{figs_TI}) to identify
the nontrivial topological nature of the system.

%
%\hhm{p.14: I will add (a) energy variation around the ring, in meV (point 9 above), and (b) the local SOC-induced splitting in the supplementary materials?}
%
%the node (labeled as P) in ALMG is about 50 meV higher than the one (labeled as Q) in AH line.
%the SOC induced gap at P is about 0.075 eV; while the SOC induced gap at Q is about 0.169 eV. The largest gap opens at Q.
%\iffalse
%\begin {table}%[H]s
%\caption {SOC induced splitting of the node in the $A$-$H$ line. (in unit of eV) }
%\label{tab:soc_gap}
%\begin{center}
%\begin{tabular}{ccccc}
%  \hline\hline
%  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
%   & Sn & Ge & Si \\
%\hline
%  Ba & 0.169 & 0.074 & 0.014 \\
%  Sr & 0.19  & 0.085 & 0.011 \\
%  Ca & 0.170 & 0.077 &  -\\
%  \hline\hline
%\end{tabular}
%\end{center}
%\end{table}
%\fi

\section{Other materials}
We also computed band structures with and without SOC
and surface band structures of other $AX_2$ materials
(Figs.~\ref{figs_srsn}-\ref{figs_casi}), including alkaline-earth stannides
(SrSn$_2$ and CaSn$_2$), alkaline-earth germanides (BaGe$_2$, SrGe$_2$ and
CaGe$_2$) and alkaline-earth silicides (BaSi$_2$ SrSi$_2$ and CaSi$_2$).

\begin{figure}
\includegraphics[width =0.9\columnwidth]{FigS6_SrSn2.eps}%
\caption{\label{figs_srsn} (Color online) (a) Band structure of SrSn$_2$ with (red) and without (blue) SOC. Surface band structures of (b) Sn- and (c) Sr-terminated semi-infinite (001) surface.}
\end{figure}

\begin{figure}
\includegraphics[width =0.9\columnwidth]{FigS7_CaSn2.eps}%
\caption{\label{figs_casn} (Color online) (a) Band structure of CaSn$_2$ with (red) and without (blue) SOC. Surface band structures of (b) Sn- and (c) Ca-terminated semi-infinite (001) surface.}
\end{figure}

\begin{figure}
\includegraphics[width =0.9\columnwidth]{FigS8_BaGe2.eps}%
\caption{\label{figs_bage} (Color online) (a) Band structure of BaGe$_2$ with (red) and without (blue) SOC. Surface band structures of (b) Ge- and (c) Ba-terminated semi-infinite (001) surface.}
\end{figure}


\begin{figure}
\includegraphics[width =0.9\columnwidth]{FigS9_SrGe2.eps}%
\caption{\label{figs_srge} (Color online) (a) Band structure of SrGe$_2$ with (red) and without (blue) SOC. Surface band structures of (b) Ge- and (c) Sr-terminated semi-infinite (001) surface.}
\end{figure}

\begin{figure}
\includegraphics[width =0.9\columnwidth]{FigS10_CaGe2.eps}%
\caption{\label{figs_cage} (Color online) (a) Band structure of CaGe$_2$ with (red) and without (blue) SOC. Surface band structures of (b) Ge- and (c) Ca-terminated semi-infinite (001) surface.}
\end{figure}

\begin{figure}
\includegraphics[width =0.9\columnwidth]{FigS11_BaSi2.eps}%
\caption{\label{figs_basi} (Color online) (a) Band structure of BaSi$_2$ with (red) and without (blue) SOC. Surface band structures of (b) Si- and (c) Ba-terminated semi-infinite (001) surface.}
\end{figure}

\begin{figure}
\includegraphics[width =0.9\columnwidth]{FigS12_SrSi2.eps}%
\caption{\label{figs_srsi} (Color online) (a) Band structure of SrSi$_2$ with (red) and without (blue) SOC. Surface band structures of (b) Si- and (c) Sr-terminated semi-infinite (001) surface.}
\end{figure}

\begin{figure}
\includegraphics[width =0.9\columnwidth]{FigS13_CaSi2.eps}%
\caption{\label{figs_casi} (Color online) (a) Band structure of CaSi$_2$ with (red) and without (blue) SOC. Surface band structures of (b) Si- and (c) Ca-terminated semi-infinite (001) surface.}
\end{figure}

\bibliography{basn2}



\end{document}


