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\begin{document}

\title{Supplementary Material for``Flux states and topological phases from spontaneous time-reversal symmetry breaking 
in CrSi(Ge)Te$_3$-based systems''}

\date{May 23, 2016}

\maketitle


\section{Projected density of states and orbital character}
%%
\begin{figure}[bth]
\subfigure{
\includegraphics[width =0.4\columnwidth]{lu_projdos.pdf}}
\subfigure{
\includegraphics[width =0.4\columnwidth]{la_projdos.pdf}}
\caption{\label{figs1} Local density of states  projected onto
various orbitals in the majority-spin channel, (a) for
1 ML Lu deposited on single-layer CrGeTe$_3$, and (b) for
1 ML La deposited on single-layer CrSiTe$_3$. The Fermi level is at $E=0$.}
\end{figure}
%%



In this section, we make a detailed analysis of the orbital character of 1 mono-layer (ML) La 
deposited on single-layer (SL) CrSiTe$_3$ and 1 ML Lu deposited on SL CrGeTe$_3$. Following the convention
introduced in the main text, we call these two systems ``Si-La" and ``Ge-Lu" for simplicity.
We first  review the electronic structure of bulk CrSiTe$_3$ (CrGeTe$_3$). 
In the ionic crystal picture, 
the on-site Coulomb repulsion between  the Cr $3d$ electrons ($\sim$3.5\,eV) 
pushes the $3d$ orbitals in the minority spin channel 
far above the Fermi level. The important physics of the system is 
thus dominated by the $3d$ orbitals in the majority spin channel.
The five-fold degenerate majority-spin $3d$ orbitals 
 are then split into $a_{1g}$ 
($d_{z^2}$), $e_g$ ($d_{yz}$ and $d_{xz}$) and $e_g'$ ($d_{x^2-y^2}$ and $d_{xy}$) levels
in the rhombohedral crystal field environment. 
The magnetic moment of CrSiTe$_3$ (CrGeTe$_3$) was reported to be 
$\sim2.73$ (2.80)~$\mu_{\textrm{B}}$ per Cr atom,
\cite{crsite3-exp, crgete3-exp} suggesting that Cr is a 3+ cation in these compounds.
The remaining three electrons on the Cr ion  then occupy the majority-spin
$a_{1g}$ and $e_{g}'$ orbitals,  giving rise 
to $\sim$3\,$\mu_{\textrm{B}}$ for each ion. 
We assume a $2-$ oxidation state for Te, with the  Te $p$ orbitals fully occupied
and lying somewhat
below the Fermi level.  The four empty $e_g$ states (from the two Cr sites) 
in the majority spin channel are located immediately above the Fermi level, leading
to an indirect gap $\sim$0.4\,eV for CrSiTe$_3$ as reported in Ref.~\onlinecite{spin-lattice-crsite3}. 


If 1 ML La (Lu) is deposited on SL CrSiTe$_3$ (CrGeTe$_3$), 
the La (Lu) atom tends to donate two
of its three valence electrons to the unoccupied Cr $e_g$ ($d_{yz}$ and $d_{xz}$) orbitals, making
the four majority-spin $e_g$ states  half filled,
while the remaining electron occupies the $5d$ or $6s$ orbitals of the adatom also in 
the majority-spin channel. Therefore, with adatoms deposited, there are five physically important
states around the Fermi level, namely, the two occupied Cr $e_g$ states, the one occupied 
$s$ or $d$ state from the adatom, and the two unoccupied Cr $e_g$ states, all in the 
majority spin channel.  

However, as discussed in the main text, CrSiTe$_3$ and CrGeTe$_3$ have a
strongly covalent character,
as the hybridization between Cr $3d$ and Te $4p$ orbitals is fairly strong. The above
analysis based on the ionic crystal picture can
be carried over, except that the states around the Fermi level no longer have simple orbital
character.  In Fig.~\ref{figs1} we show the
local density of states (DOS) for the Ge-Lu (Fig.~\ref{figs1}(a)) and Si-La (Fig.~\ref{figs1}(b)) 
systems projected onto various atomic orbitals in the majority-spin
channel. The states around the Fermi level have significant contributions from
both  Cr $d$ and Te $p$ orbitals, and the Lu (La) $s$ and $d$ orbitals also play important roles.
It follows that the hybridization effects in these two systems are strong, so the influence
of nonlocal Coulomb interactions is expected to be significant.



\section{Wannierized tight-binding models, surface states, anomalous Hall conductivity, and inter-site currents}

We use the Wannier90 code package \cite{wannier90,MLWF-rmp} to 
generate realistic tight-binding (TB) models for the Si-La and Ge-Lu systems.
These TB models are ``realistic" in the sense that the  
bandstructures from the TB models exactly reproduce those
from first-principles calculations within an energy window that extends from $-$1.5\,eV
below to +1.5\,eV above the Fermi level. 
The Bloch functions obtained from the first-principles calculations are projected onto 
Cr $s$, Cr $d$, Te $p$, Si (Ge) $p$, and La (Lu) $s$ and $d$ orbitals to construct 
the corresponding Wannier functions in each spin channel. 
Then one can extract the hopping parameters of 
the 42-band TB model in each spin channel for each system, and calculate
various physical quantities using such TB models.

In order to calculate the edge states as shown in Fig.~5(a) and (c) in the main text,
we  truncate the Wannierized TB models in the (010) direction, 
and the semi-infinite spectral functions are calculated
using the iterative Green's-function method reported in Ref.~\onlinecite{surface-gf}.  
The anomalous Hall conductivities shown in Fig.~5(b) and (d) of the main text 
are calculated using the Wannier-interpolation method proposed by Wang \textit{et al.}\cite{wang-prb06}

The inter-site current is also calculated in the Wannier basis.  The
net current flowing onto
site $j$ can be expressed  as 
\begin{eqnarray}
\frac{d n_j} {dt}&=&\textrm{Tr}\Big[ \frac{d\hat{n}_j}{dt}\,\hat{\rho} \Big]\;\nn
&=&\sum_{j',m} \langle W_{m j'}\vert \,\frac{i}{\hbar}[H, \hat{n}_j]\,\hat{\rho} \,\vert W_{m j'} \rangle \;\nn
&=&\frac{i}{\hbar}\sum_{m}\langle W_{m j}\vert\,[\hat{\rho}, H]  \,\vert W_{m j} \rangle\;\nn
&=&\frac{i}{\hbar}\sum_{j'}\sum_{m,m'} ( \rho_{mj,m'j'}H_{m'j',mj}-H_{mj,m'j'}\rho_{m'j',mj})\;\nn
&=&\sum_{j'}F_{j'j}\;,
\label{eqs1}
\end{eqnarray}
%
where $j$ and $j'$ run over atomic sites
and $m$ is the index of Wannier functions on each site, with
$\vert W_{mj} \rangle$ representing
the $m$th Wannier function located at site $j$. $H$ is the Hamiltonian,
and $\hat{n}_j=\sum_{m}\vert W_{mj}\rangle\langle W_{mj}\vert$ is the number operator on site $j$.
$\hat{\rho}$ is the density operator; at zero temperature, $\hat{\rho}$ can be expressed in terms of
the occupied Bloch states as
$\hat{\rho}=A_0\int\,d^2k/(2\pi^2)\sum_{n} \vert\psi_{n\mathbf{k}}\rangle\langle\psi_{n\mathbf{k}}\vert\,\theta(\epsilon_f-\epsilon_{n\mathbf{k}})$, 
where $\vert\psi_{n\mathbf{k}}\rangle$ and $\epsilon_{n\mathbf{k}}$ are the Bloch state and the corresponding band energy,
 $\epsilon_f$ is the Fermi level ($\theta(x)=1$ for $x>0$, and $\theta(x)=0$ otherwise), and $A_0$ is the area
 of the primitive cell. $\rho_{mj,m'j'}$ and 
$H_{mj,m'j'}$ are the matrix elements of the density operator and the Hamiltonian in the Wannier basis.
In the last line of Eq.~(\ref{eqs1}), $F_{j'j}=(-2/\hbar)\sum_{m,m'}\textrm{Im}[\rho_{mj,m'j'}H_{m'j',mj}]$ is interpreted as the current flowing between site $j$ and $j'$.


\section{Energy bands, anomalous Hall conductivities, and orbital moments  including spin-orbit coupling}
%%
\begin{figure}[bth]
\subfigure{
\includegraphics[width =0.4\columnwidth]{lu_soband.pdf}}
\subfigure{
\includegraphics[width =0.4\columnwidth]{la_soband.pdf}}
\caption{\label{figs2} Bandstructures when spin-orbit coupling is included, 
(a) for Ge-Lu, and (b) for  Si-la. The Fermi levels
are at $E=0$.}
\end{figure}
%%

%%
\begin{figure}[bth]
\subfigure{
\includegraphics[width =0.4\columnwidth]{lu_ahc_yx_so.pdf}}
\subfigure{
\includegraphics[width =0.4\columnwidth]{la_ahc_yx_so.pdf}}
\caption{\label{figs3} The anomalous Hall conductivities including spin-orbit coupling for
(a) Ge-Lu system, and (b) Si-La.}
\end{figure}
%%

The spin-orbit-coupled bandstructures of Ge-Lu and Si-La are shown
in Fig.~\ref{figs2}(a) and (b) respectively. With
spin-orbit coupling (SOC) included, the indirect gap for Ge-Lu
decreases from $\sim$70\,meV to $\sim$60\,meV, while it increases
from $\sim$130\,meV to $\sim$160\,meV for the Si-La system. The
Chern numbers for both systems are $+$1.  The anomalous Hall
conductivities ($\sigma_{yx}$) for the Ge-Lu and Si-La
systems are shown in Fig.~\ref{figs3}(a) and Fig.~\ref{figs3}(b)
respectively. Clearly $\sigma_{yx}$ in the gap is quantized at
$e^2/h$ for both systems. One may also notice that there is an
abrupt sign change of $\sigma_{yx}$ for the Ge-Lu system when the
the Fermi level is placed in the conduction band. This results
from the SOC-induced avoided crossings at $\sim$0.1\,eV around
$K$ and $\sim$0.15\,eV around $\overline{K}$ as shown in
Fig.~\ref{figs2}(a).  Such avoided crossings would lead to peaks of
Berry curvature with opposite signs around  $K$ and $\overline{K}$,
which are not at the same energy due to broken TR symmetry. This
gives rise to the abrupt sign change of $\sigma_{yx}$ shown in
Fig.~\ref{figs3}(a).
   
%\Red{[DV: Of course, you can always apply TR to the entire system,
%reverse all spin and orbital moments, and reverse $C$.  So I think
%the green part below is a bit misleading, since it seems to suggest
%that the system choses its Chern number, as though +1 is intrinsically
%preferred over $-1$.  I think you have to make explicit your implicit
%assumption, namely, that the discussion applies to the case where
%the spin moment is assumed not to reverse.  (Probably in the calculation
%there is a lower barrier for the orbital moment to reverse than for
%the spin moment to do so, which is why you observe the orbital
%moments as the flipping ones, but I don't think we should think of
%this as fundamental.)  Can you somehow modify the next few (green)
%sentences to avoid this possible confusion?]}
%
If we assume that the directions of the spin moments are not reversed after including
SOC, then the choice of the Chern
number becomes unique due to SOC. 
In other words, one of the two otherwise time-reversal-equivalent 
ground states with opposite Chern numbers  
becomes lower in energy.
To be specific, we list the $z$ components 
of the spin and orbital moments (denoted by $M_S$ and $M_L$ respectively) of 
different atomic sites for the two 
systems in the following two tables. 

The spin and orbital moments for the Ge-Lu system
are shown in Table~\ref{table:gelu}. When SOC is turned off, the ground state is a flux state
with orbital moments \{$-$0.126\,\ub, 0.113\,\ub\} for the two Cr sites, where
the first moment refers to that of the adatom-covered Cr site. This is equivalent
to the state with all the orbital moments flipped. When SOC is turned on,
the two degenerate ground states become nondegenerate, and the orbital moments on the two 
Cr sites become   \{$-$0.120\,\ub, 0.115\,\ub\}. Moreover, SOC tends to induce weak orbital moments
on the three bottom Te sites, which sum to 0.018\,\ub, and are parallel to the corresponding on-site spin moments
(which sum to 0.291\,\ub). As the $p$ shell of a Te atom is more than half-filled, the orbital and spin
moments tend to be parallel to maximize the energy gain from SOC according to Hund's third rule. 
%
\begin{table}
\setlength{\tabcolsep}{3pt}
\caption{Spin and orbital moments for the Ge-Lu system, in \ub.}
\begin{ruledtabular}
\begin{tabular}{lcdddddd}
  &&\multicolumn{1}{c}{Cr1}
   &\multicolumn{1}{c}{Cr2}
   &\multicolumn{1}{c}{Ge dimer}
   &\multicolumn{1}{c}{bottom Te}
   &\multicolumn{1}{c}{top Te}
   &\multicolumn{1}{c}{Lu}\\
\hline
Without SOC& $M_S$ & -3.767 & -3.755 & -0.094 & 0.282 & 0.041 & -0.253 \\
           & $M_L$ & -0.126 &0.113  & 0.001  &0.004  & -0.003 & -0.016  \\
With SOC& $M_S$ & -3.771 & -3.762 & -0.093 & 0.291 & 0.044  & -0.249   \\
        & $M_L$ & -0.120 & 0.115 & 0.001 & 0.018 & 0.003 &  0.000\\
\end{tabular}
\end{ruledtabular}
\label{table:gelu}
\end{table}
%

%
\begin{table}
\caption{Spin and orbital moments for the Si-La system, in \ub.}
\begin{ruledtabular}
\begin{tabular}{lcdddddd}
  &&\multicolumn{1}{c}{Cr1}
   &\multicolumn{1}{c}{Cr2}
   &\multicolumn{1}{c}{Ge dimer}
   &\multicolumn{1}{c}{bottom Te}
   &\multicolumn{1}{c}{top Te}
   &\multicolumn{1}{c}{La}\\
\hline
Without SOC& $M_S$ & -3.804 & -3.729 & -0.084 & 0.374 & 0.178 & -0.636 \\
           & $M_L$ & -0.066 & 0.082 & 0.000 & 0.001 & -0.003  & -0.180 \\
With SOC& $M_S$ & -3.805 & -3.728 & -0.081 & 0.369  & 0.177  & -0.644 \\
        & $M_L$ & 0.071 & -0.080 & 0.000 & 0.015 & 0.006 & 0.294 \\
\end{tabular}
\end{ruledtabular}
\label{table:sila}
\end{table} 
%

%\Red{[DV: Formatting of tables modified a bit.]}

The spin and orbital moments for the Si-La system are listed
in Table~\ref{table:sila}.  Without SOC, the reported orbital moments on the two Cr sites are
\{$-$0.066\,\ub, 0.082\,\ub\}, and the orbital moment on the La site is $-$0.180\,\ub\ and is
associated with the La $5d$ electrons. Again, such a flux 
state is degenerate with its time-reversal (TR) partner with flipped orbital moments.
When SOC is turned on, the signs of the orbital moments  become
reversed: the two Cr sites
change from \{$-$0.066\,\ub , 0.082\,\ub\} to \{0.071\,\ub, $-$0.080\,\ub\}, 
and the orbital moment on the La site changes from $-$0.180\,\ub\ to 0.294\,\ub.  
This results from the fact that the La $5d$  shell is less than half filled,
so that the energy gain from SOC
is maximized if the La on-site orbital moment (0.294\,\ub) and spin moment ($-$0.644\,\ub)
are anti-parallel. Since the SOC strength of La is much larger than that of Cr, the system
tends to choose the state with anti-parallel spin and orbital moments on the La site.



\section{Flux states with trivial band topology}

As discussed in the main text, a flux state is not necessarily
topologically nontrivial. For example, TR symmetry is
broken in the majority-spin subspace in both 1 ML Lu deposited on
SL CrSiTe$_3$ and 1 ML La deposited on SL CrGeTe$_3$
(denoted as ``Si-Lu" and ``Ge-La" for simplicity),
and there are spontaneously generated current loops surrounding the
Cr sites leading to antiferromagnetically ordered orbital moments.
However, both of these systems are topologically trivial.  
Fig.~\ref{figs4}(a) and (b) show the bandstructures in the
absence of SOC for Si-Lu and Ge-La respectively. The blue (red)
lines indicate the energy bands in the majority (minority) spin
channel.  It is straightforward to see that orbital TR symmetry is
broken only in the majority spin channel, suggesting that the ground
states of the systems are flux states. However, orbital-character
analysis (not shown) indicates that there is no band inversion
in these systems, and that their computed Chern numbers remain zero.

%%
\begin{figure}[bth]
\subfigure{
\includegraphics[width =0.4\columnwidth]{SiLu_figband.pdf}}
\subfigure{
\includegraphics[width =0.4\columnwidth]{GeLa_figband.pdf}}
\caption{\label{figs4} Bandstructures for (a) 1 ML Lu deposited on
single-layer CrSiTe$_3$, and (b) 1 ML La deposited on single-layer
CrGeTe$_3$. The blue and red curves represent energy bands in the
majority-spin and minority-spin channels.}
\end{figure}
%%



\bibliography{crsite3}



\end{document}


