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\begin{document}

\title{Supplemental Material for \\
``Metal-insulator transition and topological properties of pyrochlore iridates"}

\author{Hongbin~Zhang}
\email[corresp.\ author: ]{hzhang@physics.rutgers.edu}
\author{Kristjan Haule}
\author{David Vanderbilt}
\affiliation{Department of Physics and Astronomy, Rutgers University, Piscataway, USA}

\begin{abstract}
In this Supplement we expand on the computational details and provide
additional results concerning the spectral functions, entropies,
structural information, and the evolution of the $X$-point eigenvalues
with mixing fraction of the DMFT self-energy.
\end{abstract}

\maketitle

\begin{figure}
\includegraphics[width=8.0cm]{./Figs1.eps}
\caption{(Color online)
Spectral functions of six pyrochlore compounds $R_2$Ir$_2$O$_7$ ($R$=Y, Eu, Sm, Nd, Pr, and Bi) 
studied in this work. The spectral functions are shown for those compounds at corresponding
ground states, that is, all-in-all-out magnetic insulating state for the Y, Eu, Sm, and Nd cases, and
paramagnetic metallic state for the Pr and Bi compounds. Dashed white horizontal lines denote 
the Fermi energy.
}\label{fig:akw}
\end{figure}

\section{Computational details}
Our charge self-consistent density functional theory (DFT) with dynamical mean-field theory (DMFT)
calculations are done using the projection/embedding implementation~\cite{Haule:2010} based on 
the WIEN2k package.~\cite{wien2k}
The auxiliary impurity problem is solved using the continuous-time quantum Monte Carlo method,~\cite{Haule:CTQMC} 
where a proper local basis adapted to the IrO$_6$ 
octahedra are chosen to minimize the sign problem.
The exchange-correlation potential is approximated using the Perdew-Wang LDA functionals.~\cite{Perdew:1992}
A set of system-independent local Coulomb interaction parameters $U\!=\!4.5\,$eV and
$J\!=\!0.8\,$eV is used, which is determined to be good for Ruddlesden-Popper iridates.~\cite{Zhang:2013}
The reason for using a larger $U$ compared to the typical value
in LDA+U calculations is
that the hybridizations and self-energies are computed in a large energy window of $\pm10\,$eV around the 
Fermi energy ($E_\text{F}$) in our DFT+DMFT calculations.
We note that the 4f-electrons (when present) are treated using the open-core approximation, which is
justified because the magnetic $4f$ moments become ordered at a temperature much
lower than the typical metal-insulator transition temperature in the pyrochlore iridates 
(see for instance Ref.~\onlinecite{Guo:2013}).
The AIAO magnetic configuration is considered by transforming the local self-energies
with site-dependent spin quantization axis to the global frame using 
proper Wigner rotations.~\cite{Zhang:2013}
The free energies are evaluated following the methodology presented in Ref.~\onlinecite{Haule:2015}.
All calculations are carried out at the experimental lattice
parameters.~\cite{Lee:2013, Clancy:2011, Millican:2007, Disseler:2012b, Matsuhira:2007}
%For comparison, we also carried out LDA+U calculations using the
%VASP code~\cite{vasp} with $U\!=\!2.5\,$eV by fitting to the DMFT results.

The LDA+DMFT spectral functions for all six compounds are shown in Fig.~\ref{fig:akw}.
For the Y, Eu, Sm, and Nd compounds, the ground state is an all-in-all-out (AIAO) magnetic insulating
state, with an average band gap of $0.3$ eV, while both the Pr and Bi compounds are 
paramagnetic metals.
%The free energies shown in Fig.~1 of the main text are obtained without 
%considering the electronic entropy of impurities (Ref.~\onlinecite{Haule:2015}).
Following Ref.~\onlinecite{Haule:2015}, the free energies shown in Fig.~1 of the main text 
are obtained by evaluating the Luttinger-Ward functional using the self-consistent LDA+DMFT
Green's functions. As given by Eq.~4 of Ref.~\onlinecite{Haule:2015}, an accurate evaluation of 
the free energy of the LDA+DMFT quantum impurity model $F_\text{imp}\!=\!E_\text{imp}-TS_\text{imp}$ 
is required, where $E_\text{imp}$ ($S_\text{imp}$) denotes the internal energy (entropy) of the
corresponding impurity model.
In Fig.~\ref{fig:entropy}, $S_\text{imp}$ for Nd$_2$Ir$_2$O$_7$ in both the
AIAO and PM states are shown, and the difference at 50~K is about 0.033~$k_B$ per Ir atom. 
Thus, it is justified
to neglect the $TS_\text{imp}$ contribution to the total free energies and their difference for the
paramagnetic and AIAO magnetic states.


\begin{figure}
\includegraphics[width=7.0cm]{./Figs2.eps}
\caption{(Color online)
Entropy of the LDA+DMFT quantum impurity model 
for Nd$_2$Ir$_2$O$_7$ in the all-in-all-out (AIAO) magnetic state and
in the paramagnetic (PM) state. Lines are guides for the eyes.
}\label{fig:entropy}
\end{figure}


\section{structural analysis}
%
%bond angle and distance
%

\begin{figure}
\includegraphics[width=7.0cm]{./Figs3.eps}
\caption{(Color online)
Variation of the (a) $t_{2g}$ bandwidth and (b) averaged hybridization function replotted together
with (c) Ir-O bond length and (d) Ir-O-Ir bond angle with respect to the A-cation radius for the pyrochlore
iridates studied in this work. Hollow (solid) symbols indicate that the corresponding ground state is
insulating (metallic). Lines are guides for the eyes.
}\label{fig:struct}
\end{figure}

For pyrochlore iridates, there are two distinct Wyckoff positions for oxygen atoms: 48f$(x,\frac{1}{8},\frac{1}{8})$
and 8b$(\frac{3}{8},\frac{3}{8},\frac{3}{8})$. The oxygen atoms occupying the 48f position form the
IrO$_6$ octahedra, hence the Ir-O bond length and the Ir-O-Ir bond angle (Fig.~\ref{fig:struct}) are
determined by a single parameter $x$.
The variation of the $t_{2g}$ band-width and the averaged hybridization function are replotted in
Fig.~\ref{fig:struct} together with the Ir-O bond length and the Ir-O-Ir bond angle,
showing a strong correlation between the former two quantities with the latter two parameters.
It is observed that the Bi compound is different from the other five compounds, 
due to the fact that for Bi$^{3+}$ ions, their extended $6s$ orbitals have enhanced hybridization
with the O-$2p$ orbitals, leading to significant changes in the band structure comparing to 
those of the other compounds (cf. Fig.~\ref{fig:akw}).
For all the other compounds, the bandwidth and hybridization function are determined by the 
Ir-O-Ir bond angle and the Ir-O bond length, which are both controlled by $x$.
On the one hand, a $180^\circ$ Ir-O-Ir bond angle would correspond to a situation where two 
corner sharing IrO$_6$ octahedra have two IrO$_4$ plaquettes on the same plane, favoring 
electrons hopping between two iridium atoms via the oxygen atoms in between.
Bending of the Ir-O-Ir bond leads to smaller hopping parameters of electrons between Ir atoms,
and hence smaller Ir-O-Ir bond angle giving rise to a reduced band-width, as shown in
Fig.~\ref{fig:struct}.
On the other hand, a shorter Ir-O distance can enhance the hybridization between Ir~$d$ and 
O~$p$ orbitals, leading to larger hopping and hence larger band-width.
This is also true for the rare-earth compounds, as shown in Fig.~\ref{fig:struct}.
Therefore, the internal oxygen coordinate is the most important structural parameter, which
determines the bandwidth and hence the ground state.

%
% trigonal crystal fields and distorted J=1/2 states
%
\section{Distorted $J\!=\!1/2$ wave functions}

For all pyrochlore compounds considered in this work, the IrO$_6$ octahedra are
compressed along one of the {111} directions. For the one compressed along
(111), the resulting trigonal crystal field 
in the \{$d_{yz}, d_{zx}, d_{xy}$\} basis can be expressed as
\begin{equation}
H_\text{trigonal}=\frac{\delta}{3}\left(\begin{array}{ccc} 0 & 1 & 1 \\
                                                                1 & 0 & 1 \\
                                                                1 & 1 & 0 \end{array}\right).
\end{equation}
where $\delta$ denotes the strength of the trigonal crystal-field splitting.
We observed that $\delta$ has a significant value,
as listed in Table.~\ref{table:trigonal}.
It is comparable to the strength of the atomic SOC of Ir atoms (about $0.5$ eV),
in good agreement with resonant x-ray scattering measurements.~\cite{Hozoi:2012}
Correspondingly, the $J\!=\!1/2$ wave functions in pyrochlore iridates
are strongly distorted, leading to an orbital-to-spin ratio of magnetic moments
that deviates significantly from the $SU(2)$ limit (Table~\ref{table:trigonal}).
For instance, in Y$_2$Ir$_2$O$_7$ the ratio of orbital to spin moment is about $1.27$,
which deviates significantly from the ideal value of $2$.
This can be verified in future experiments using non-resonant magnetic x-ray diffraction, 
as done for Sr$_2$IrO$_4$.~\cite{Fujiyama:2014}
Furthermore, we observed that the ordered orbital and spin moments are not 
pointing along the same direction.
In our calculations, the total magnetic moments are fixed along the (111) direction 
following the AIAO pattern, whereas the spin and orbital moments deviate
from this direction.
The angle between these two moments can be as large as 42$^\circ$ as listed in
Table~\ref{table:trigonal}, which is much larger than that in Sr$_2$IrO$_4$ (about 5$^\circ$).
%------------------------------
%Table1: trigonal crystal field, orbital-spin-moment ratio, and angle
%------------------------------
\begin{table}[h]
\caption{Trigonal crystal field parameter $\delta$, ratio of orbital to spin moments $\mu_L/\mu_S$, 
and the angle between
the orbital and spin moments $\alpha_{LS}$ in $R_2$Ir$_2$O$_7$}\label{table:trigonal}
\begin{tabular}{c|c|c|c|c|c|c}
\hline\hline
                               &       Y         &          Eu             &     Sm           &     Nd         &        Pr           & Bi          \\ \hline
$\delta$ (eV)        &    0.35      &          0.42           &     0.43         &   0.43        &        0.30       &  0.42    \\ \hline
$\mu_L/\mu_S$   &    1.27    &  1.28         &    1.28     &  1.25      &   N.A.       &    N.A.      \\ \hline
$\alpha_{LS}\;(^\circ) $    &   31        &   41       &     41     &  42      &   N.A.       &    N.A.     \\ \hline
\end{tabular}
\end{table}

%
% Evolution of the eigenvalues at the X point
%
\section{Evolution of the eigenvalues at the X point}

As stated in the main text, at the $X$ point in the Brillouin zone,
the fourfold degeneracy is guaranteed by symmetry, with each degenerate group comprising
two even-parity and two odd-parity states.
Unlike at the $L$ ($L^\prime$) point, where one occupied eigenstate exchanges its parity
with one unoccupied eigenstate when the DMFT self-energies are considered,
no band inversion or other change in the
ordering of parities between occupied and unoccupied eigenstates occurs at the $X$ point, as shown in Fig.~\ref{fig:x}.
Apparently, each fourfold-degenerate state is split into two doubly degenerate states when the DMFT self energies
are considered, due to the breaking of time-reversal symmetry. 
Therefore, the number of occupied states with odd
parities does not change after considering the DMFT self energies, as summarized in Table.~I in the Main Text.


\begin{figure}[h]
\includegraphics[width=7.0cm]{./Figs4.eps}
\caption{(Color online)
Evolution of the eigenvalues of the
effective topological Hamiltonian at $X$ with
mixing fraction of the DMFT self-energy.
Lines are guides for the eyes.
}\label{fig:x}
\end{figure}


\section{Spectral function with smaller $U$}

\begin{figure}[h]
\includegraphics[width=7.0cm]{./Figs5.eps}
\caption{(Color online)
LDA+DMFT spectral function of Y$_2$Ir$_2$O$_7$ with on-site $U\!=\!4.0$ eV for Ir atoms.
}\label{fig:u40}
\end{figure}

To check the possibility of achieving topologically a nontrivial semi-metal or 
axion-insulator phase by reducing the value of the $U$ parameter on 
the Ir atoms,~\cite{Wan:2011, Go:2012}  we performed LDA+DMFT calculations for Y$_2$Ir$_2$O$_7$ with
the on-site $U$ reduced to $4.0$ eV. The corresponding spectral function is shown
in Fig.~\ref{fig:u40}. In this case, the AIAO magnetic state is not stable and it spontaneously
converges to the paramagnetic solution.


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\end{document}


