# Non-local self energies in pyrochlore iridates from ab-initio TRILEX
calculations, and their relevance for the Weyl semimetal phase

Johannes Graspeuntner  Institute of Theoretical and Computational Physics,
Graz University of Technology, NAWI Graz, Petersgaße 16, Graz, 8010, Austria.
Markus Richter  Institute of Theoretical and Computational Physics, Graz
University of Technology, NAWI Graz, Petersgaße 16, Graz, 8010, Austria.
Markus Aichhorn<EMAIL_ADDRESS>Institute of Theoretical and Computational
Physics, Graz University of Technology, NAWI Graz, Petersgaße 16, Graz, 8010,
Austria.

(August 28, 2024)

###### Abstract

Motivated by recent experiments and computational results on pyrochlore
iridates, we compare single-particle properties of Y2Ir2O7 obtained from
single-site dynamical mean-field calculations with results within the TRILEX
approximation, where the latter takes non-local correlations into account. Our
calculations are all based on ab-initio calculations within density-functional
theory, and take spin-orbit coupling into account. In order to make the
treatment within TRILEX feasible, we first define a single-band
$j_{\textrm{eff}}=1/2$ model, by comparing its spectral features within DMFT
to a three-band model that includes both $j_{\textrm{eff}}=1/2$ and
$j_{\textrm{eff}}=3/2$ orbitals. Our calculations show consistently a
paramagnetic metallic phase at small interaction values, and an insulating
antiferromagnetic phase at larger interaction values. The critical
interactions, however, differ between single-site and TRILEX calculations. The
antiferromagnetic phase shows the already predicted all-in/all-out magnetic
ordering. Different to the single-site results, the TRILEX calculation gives
also evidence for the Weyl-semimetal regime in the vicinity of the metal-
insulator transition.

††preprint: APS/123-QED

##  I Introduction

In recent years, a lot of effort has been devoted to the investigation and
understanding of spin-orbit coupling (SOC) and its interplay with strong
electronic correlations [1]. Among those effects that rise from SOC in
strongly correlated systems, are SOC-driven Mott insulators [2, 3, 4, 5], new
phases such as chiral spin liquids [6, 7], axion insulators [8, 9, 10] and
Weyl semi-metals (WSM) [8, 11]. In particular, pyrochlore-iridates R2Ir2O7
(where R represents rare-earth elements or Yttrium) have been studied very
closely in this respect. In these systems, similar to other iridate compounds,
the presence of SOC splits the t2g orbitals into $j_{\textrm{eff}}=3/2$ and
$j_{\textrm{eff}}=1/2$ orbitals. With the $j_{\textrm{eff}}=1/2$ orbital being
close to half-filled, these systems show particularly strong effects of
electronic correlations, leading to a strongly-correlated spin-orbital
polarized state [3] that eventually leads to insulating behavior in quite a
number of iridate compounds.

First calculations for Y2Ir2O7 within density-functional theory plus mean-
field-like correlations (DFT+U) predicted a WSM with an all-in/all-out (AIAO)
magnetic ordering [8]. Later on, Hartree-Fock-based calculations that use a
tight-binding model description [9, 9], as well as results from cluster
dynamical mean-field (CDMFT) theory [11, 12] were concluding with a similar
picture for the existence of a WSM. The region in parameter space, however,
was very small [12]. In contrast to those works, the combination of DFT with
single site dynamical mean-field (sDMFT) has lead to different conclusions in
one important aspect. In these calculations [13, 14, 15], the system Y2Ir2O7
undergoes a direct transition from a paramagnetic metal to a topologically
trivial AIAO antiferromagnetic insulator, without the appearance of a WSM-
phase in between.

Experimental works have studied the pyrochlore iridates in dependence of the
rare earth element. By exchanging the rare-earth ion along the rare-earth
series, a metal-insulator transition has been found [16, 17, 18, 19, 20, 21,
22]. At the same time, the system exhibits a phase transition from a
paramagnetic to an antiferromagnetic state with AIAO ordering, where the
iridium atoms are occupying a frustrated lattice, and time-reversal symmetry
is broken. From that, a WSM-phase is likely to exist and has shown up
experimentally in optical response measurements [23]. Also the measurement of
the spontaneous Hall effect in pyrochlore iridates under the application of an
external magnetic field confirms the existence of a WSM phase [24, 25].
However, measurements by angle-resolved photoemission spectroscopy (ARPES)
were, not able to detect Weyl points [26].

Coming back to theory, a comparison of the results of sDMFT [13, 14, 15] and
CDMFT [12] calculations suggests that the WSM phase arises due to non-local
self-energies, which are neglected in sDMFT calculation by construction. In
this paper, we take a closer look at the momentum dependence of the self-
energy in Y2Ir2O7. We want to investigate the possible existence of a WSM
phase near the metal-insulator transition within the framework of density
functional theory combined with the triply-irreducible leg extension
(DFT+TRILEX). In contrast to sDMFT, the TRILEX method [27, 28] does not
approximate the self energies as local quantities, but the next higher-order
diagram, namely the triple legged $\Lambda$ diagram that connects the
fermionic and bosonic states. We will show that there is sizeable momentum
dependence in the self-energy, which makes the band structure and also the
phase diagram different from sDMFT results.

##  II Methods and Low-Energy Modelling

In Fig. 1 we show the crystal structure of the pyrochlore-iridate family.
Generally speaking, pyrochlore iridates are described by the chemical formula
R2Ir2O7, where the R stands for the rare-earth elements. In this paper we are
fixing this element to Yttrium. The Ir atoms are arranged in a corner sharing
tetrahedral structure, forming a frustrated lattice. Close to the Fermi energy
only the electrons of the Ir 5d shell are present. Hence, we have five valence
electrons in the t2g orbitals that are affected by strong on-site correlations
as well as strong spin-orbit coupling.

Figure 1: Top: Crystal structure of the pyrochlore iridate, Y2Ir2O7. The large
blue spheres depict the iridium atoms. The large grey spheres stand for the
rare earth elements; Yttrium in this case. The small red spheres are the
oxygen atoms. Bottom: The bandstructure of Y2Ir2O7 obtained from DFT+soc
calculations, where only the energy range of the iridium t2g orbitals is
shown. It can be clearly seen that the the bands split into a
$j_{\textrm{eff}}=1/2$ and a $j_{\textrm{eff}}=3/2$ manifold in the presence
of SOC.

To describe the physics of the system we perform DFT+sDMFT [29] as well as
DFT+TRILEX [27, 28] calculations. The first step is to do the ab-initio
calculation, where we use the Wien2k package [30]. As exchange-correlation
functional the generalized gradient approximation in the parametrization of
the Perdew, Burke, and Ernzerhof (PBE) has been used. In the calculations we
are using a $8\times 8\times 8$ k-point grid. Due to the presence of spin-
orbit coupling in this strongly correlated system the t2g orbitals are further
split into a $j_{\textrm{eff}}=\frac{1}{2}$ and a
$j_{\textrm{eff}}=\frac{3}{2}$ manifold, where the latter is filled up to
90$\%$, while the $j_{\textrm{eff}}=\frac{1}{2}$ manifold takes the remaining
charge-density and shows an occupancy of $1.4$ electrons. Thus, the spin-
orbital polarisation in the non-interacting case is not complete, similar to
other iridate compounds [3].

The bandstructure plot of the DFT calculation in Fig. 1 depicts a clear
separation of the $j_{\textrm{eff}}=\frac{1}{2}$ and the
$j_{\textrm{eff}}=\frac{3}{2}$ manifold in reciprocal space. This separation
suggests that it should also be possible to approximate the system by a half-
filled one-band model. In fact, such an approximation is essential in order to
reduce the complexity of the problem and the computational effort for a
treatment of many-body correlations in the TRILEX approach, which we will use
in this paper. In the close vicinity of the $j_{\textrm{eff}}=\frac{3}{2}$
manifold the physics of the $j_{\textrm{eff}}=\frac{1}{2}$ manifold is
definitely affected. To correctly treat the low energy physics, the system is
projected on maximally localized wannier functions (MLWF) with the wannier90
code [31].

The Wannier-projected Hamiltonian is used to perform single-site DMFT
calculations for both the three band t2g model as well as the
$j_{\textrm{eff}}=\frac{1}{2}$ model, in order to compare the qualitative
behaviour. The calculations of the t2g model have been performed using a
Kanamori Hamiltonian with the Hund’s coupling being $J=0.1U$. For the one band
case of the the $j_{\textrm{eff}}=\frac{1}{2}$ manifold, a Hubbard-type
interaction Hamiltonian with parameter $U$ is sufficient. All calculations are
done using the TRIQS toolkit [29]. For the sDMFT calculations we use the
TRIQS/CTHYB solver [32, 33, 34].

The comparison of the models was performed at higher temperatures, $\beta=40$
eV-1, in order to avoid possible sign problems of the multi-band calculations,
that could arise with the CTHYB solver. Performing multi band calculations
over a range of different interaction values $U$ (with $J=0.1U$) show that the
metal-insulator Mott-transition occurs at an interaction value close to
$U\approx 2.5$ eV , see Fig. 2, which is in line with previous treatments in
literature [15]. This transition is accompanied by a complete spin-orbital
polarization with filled $j_{\textrm{eff}}=\frac{3}{2}$ and half-filled
$j_{\textrm{eff}}=\frac{1}{2}$ orbitals. What the comparison of the multi-band
and the one-band model shows is that the metal insulator transition of the
$j_{\textrm{eff}}=\frac{1}{2}$ model occurs at a reasonable value as well.

Figure 2: Top: Spectral function of the $j_{\textrm{eff}}=\frac{1}{2}$ orbital
in the full t2g-band calculation and for three different interaction values
$U=\\{1.6,2.0,2.5\\}$ eV and $J=0.1U$. The asymmetry up to $U=2.5$ eV stems
from the still incomplete spin-orbital polarization in that case. This
polarization becomes complete for larger interactions, hence, the spectra
become more symmetric. Bottom: The spectral function for the
$j_{\textrm{eff}}=\frac{1}{2}$ one-band model and three interaction values.
All calculations are done at $\beta=40$ eV-1, and spectra are obtained using
the TRIQS/MaxEnt package [35].

To treat the longer-range correlations we are using the triply-irreducible
local expansion (TRILEX) framework [27, 28]. Same as for the sDMFT calculation
we use the Wannier projected Hamiltonian and add on-site interactions to the
Ir atoms. In order to interface the Wannier projection with the TRILEX
calculation, we use again the TRIQS package [29], in particular the lattice
tools application. In comparison to the DMFT approach, TRILEX treats the self-
energy k-dependent and makes the approximation with a local three-index-vertex
$\Lambda_{imp}(i\omega,i\Omega)$. Hence the TRILEX method interpolates between
the fluctuation exchange limit, that treats long-range fluctuations, and the
DMFT limit, that approximates the self-energy as a local frequency dependent
impurity self-energy. For the calculation we have chosen to use $\alpha=0.5$
as the decoupling parameter between the spin and the charge channels of the
Hubbard interaction, based on the fact that the different alpha values had
very little influence on the result. Since we deal with a symmetry-broken
antiferromagnetic system, we use the specific approximation for
antiferromagnetic cases called the $\Lambda^{2}$-TRILEX derived in [36]. For
the TRILEX calculations we are using the CTINT solver [37]. This had the
advantage of the linear error in the tails of the Green’s functions. The
quadratic error of the CTHYB solver would lead to numerical errors in the
Hedin equations.

For the TRILEX calculations the computational efforts become infeasible near
the phase transition, due to the critical slowing down of Monte Carlo
processes near the phase transitions, which impacts the resolution of the
bosonic green’s functions drastically and as a result also impacts the other
observables. If the Green’s functions show to much noise, the calculations
eventually fail when trying to perform Fourier transforms. As a result we were
able to perform converged self-consistent TRILEX calculations up to an
interaction value of $U=1.2$ eV with low enough numerical noise to obtain
useful results. For higher interaction values, in order to probe the
k-dependence and the possible WSM phase, we limited the calculation to one-
shot TRILEX calculations. For that, fully converged DMFT solutions have been
used to calculate the vertex corrections just once in order to obtain insight
into non-local physics.

##  III Results

###  III.1 Magnetic Properties

As stated in the previous section, all calculations that we present in the
following are obtained from the effective $j_{\textrm{eff}}=\frac{1}{2}$ one-
band model. We start with DMFT calculations at a certain interaction value
$U=2.0$ eV in the insulating regime. We already know from the calculations
presented in the previous section, see Fig.2, that this value is well beyond
the phase transition point. Starting out from a rather high temperature
$\beta=40$ eV-1, we perform a series of calculations at lower temperatures to
determine an optimal temperature in where we can see a substantial magnetic
ordering and at the same time keep the computational costs at a reasonable
level.

Based on these considerations we found that doing calculations at $\beta=180$
eV-1 is a good trade off that works for different interaction values. As the
interaction value increases, the system undergoes not only a metal-insulator
transition, but also a paramagnetic to antiferromagnetic transition. It starts
out from a paramagnetic metallic phase and progresses to a all-in/all-out
ordered antiferromagnetic metal phase. By further increasing the interaction
value it transforms to an antiferromagnetic insulator. This result is coherent
with the single-site multi-orbital calculations [13, 5, 15], where the system
undergoes a direct Mott-transition from metal to insulator without any
topological phase in between. In the plot of the bandstructure, which we will
present later, we will see the gradually more compressed bands up to the
critical interaction value, after which the correlations open the Mott gap in
the insulting phase.

In order to determine the critical interaction value where the magnetic
transition occurs, we performed calculations for both the paramagnetic and the
antiferromagnetic solutions and compared the respective ground state energy of
the phases. We calculate the energy using the Migdal formula,

$E=\frac{1}{N_{k}\beta}\sum_{n}\sum_{k}\textrm{Tr}\\{[{H}_{0}(k)+\frac{1}{2}\Sigma(i\omega_{n})]{G}(k,i\omega_{n})\\}\,,$
(1)

where $N_{k}$ is the number of k-points on the grid, $H_{0}(k)$ the non-
interacting Wannier Hamiltonian, $\Sigma(i\omega_{n})$ the local DMFT self
energy, and $G(k,i\omega)$ the interacting DMFT Greens function.

Figure 3: Top: The energy Eq. (1) plotted versus interaction strength for the
sDMFT calculations at $\beta=180$ eV-1. The vertical line denotes the first-
order phase transition at $U\approx 0.6$ eV. Bottom: The expectation value of
the spin operator $\langle S\rangle$ as function of the interaction strength,
where the black line denotes the sDMFT result and the red line gives the
results for the fully converged TRILEX calculations.

With this procedure we can determine the critical interaction for the magnetic
phase transition to $U\approx 0.6$ eV. Knowing where the ordered phase is
starting it is reasonable to determine the expectation value of the spin
operator $\langle\vec{S}\rangle$. The size of the magnetic moment is obtained
via $\sqrt{\langle S_{x}\rangle^{2}+\langle S_{y}\rangle^{2}+\langle
S_{z}\rangle^{2}}$, and we plot it against the interaction values $U$. Figure
3 shows a saturation of the moment for large interaction values.

The magnetisation plot of the TRILEX solution is very similar to the sDMFT
one. As shown in Fig. 3 they are basically aligning on top of each other. This
result is expected, since other works have shown similar magnetic behaviour
regardless whether k-dependent or independent methods were used.

###  III.2 Momentum dependence of the self energy

In this section we want to take a closer look at the k-dependence of the self
energy of Y2Ir2O7. The self energies from sDMFT calculations are obviously
independent of momentum, since that is the approximation we make when using
single-site DMFT. On the other hand, CDMFT takes some non-locality into
account by using clusters of finite size. From the work of Wang and coworkers
[12] we see that the WSM phase occurs in CDMFT. Therefore, we can assume that
the WSM phase can only be described by a momentum dependent self energy, and,
thus, a method that is taking non-locality into account, is needed for a
proper description. The method we use here is the TRILEX approximation, which
approximates a higher-order property as local, and therefore includes non-
local effects terms in the self energy.

For small interaction strength the k-dependence of the self-energy is
negligible, similar to the sDMFT calculations. Up to a value of $U=1.2$ eV the
self energy remains very local and we see little variance of its value over
k-space. In Fig. 4 we show the dependence of the real part of the self energy
at the first Matsubara frequency in k space. The difference between the
maximal and the minimal value is just of the order of $10^{-2}$ eV.

Figure 4: Momentum dependence of the real part of the self energy for $U=1.2$
eV, at the first Matsubara frequency. Figure 5: Momentum dependence of the
real part of the self energy for $U=1.4$ eV, at the first Matsubara frequency.

We now increase the interaction strength in order to approach the phase
transition point. However, for values of $U$ where we would expect the WSM
phase to be, the Monte Carlo steps needed in the CTINT solver to produce
reasonable results with acceptable numerical noise increase dramatically.
Facing such high computational costs, we decided to perform what we call one-
shot TRILEX calculations. Instead of performing the full self-consistent
TRILEX loops, which is still possible for values up to $U=1.2$ eV, we now use
a fully converged sDMFT solution as input to calculate the vertex $\Lambda$
just once, and investigate whether it already has an effect on the k
dependence of the self energy. In addition, the symmetry around $k=[0,0,0]$ of
the self energy is dependent on the smoothness of the polarisation function in
the Hedin equations. Achieving good enough QMC quality to make the
polarization smooth is also getting very costly near the phase transition.

We show the results for a single shot TRILEX calculation at $U=1.4$ eV in Fig.
5, we can see a very strong variance of the self energy in momentum space.
Compared to the k-dependence at $U=1.2$ eV, where the self energy only varies
at third decimal, the variations of the self energy at $U=1.4$ eV are two
orders of magnitude larger.

As stated above, the strong k-dependence in this interaction range is
important for the presence of a WSM phase. To verify the presence of a WSM
phase within the TRILEX framework we will take a look at the bandstructures in
the next section.

###  III.3 Bandstructures

When a system is in a WSM phase, it shows a band crossing at the Fermi level
for a specific k-point, while the bands are fully separated at all other
k-points. At a band crossing like that, the electrons, the Weyl fermions of
our system, have a high mobility. Looking at the momentum-resolved spectral
function obtained from sDMFT, we can only see a trivial Mott-transition, see
Fig. 6. The bandstructures are obtained by analytically continuing the
momentum dependent self energies using TRIQS/MaxEnt [35]. First, the bands get
renormalized and form a quasi-particle peak. Once the system reaches a
critical interaction value, in the case of Y2Ir2O7 at $U=1.6$ eV, the system
transitions from a metal to an insulator and the now incoherent Hubbard bands
are pushed apart quickly as function of $U$. This result is in accordance with
other findings in literature.

Figure 6: Momentum resolved spectral function at $U=1.2$ eV, $U=1.4$ eV, and
$U=1.65$ eV, calculated within sDMFT. Note that these calculations are in the
magnetically ordered phase. Upon increasing $U$, bands get renormalized with a
coherent quasi-particle up to $U=1.6$ eV. At $U=1.65$ eV the Mott gap has
fully opened, see right panel. Figure 7: Momentum resolved spectral function
at $U=1.2$ eV, $U=1.4$ eV, and $U=1.5$ eV, calculated with the TRILEX
approximation. The bandstructure for $U=1.2$ eV is similar to the sDMFT
bandstructure at the same interaction. At $U=1.4$ eV the difference to sDMFT
is very significant.

The TRILEX solutions shown in Fig. 7 behave very similar to sDMFT calculations
up to intermediate interactions. For probing higher interaction values, near
the phase transition, the computational effort becomes exceedingly high.
Therefore, we are forced to rely on one-shot calculations in order to obtain
insight in the physics near the phase transition. When inspecting the
momentum-resolved spectral function, we see the expected band crossing at the
L high-symmetry point. Especially for $U=1.4$ eV, the characteristic band
crossing with massless fermions is appearing. In order to see this better, we
show in Fig. 8 a zoom-in to low energies around the Fermi level. However, the
bands seem to also touch near the K high-symmetry point. This additional
closing for the K high symmetry point could arise due to three different
reasons.

Figure 8: Zoomed-in TRILEX bandstructure for the interaction values $U=1.2$ eV
(left panel) and $U=1.4$ eV (right panel) to better display the Weyl
bandcrossing at the L high-symmetry point.

First, the reason could be the low resolution of the analytic continuation in
combination with statistical noise in the QMC data and therefore the bands
only look like they are touching, when in reality there would be a gap.
However, there is definitely a trend of gap narrowing happening at the K
point, once the solution is allowed to show k-dependence. Even with a possible
low resolution in the analytical continuation, it can not be neglected, that
the bands are narrowing at this high symmetry point. Another possibility is
the fact that at this interaction regime we can only perform one-shot
calculations and these do only display a trend of TRILEX, but not the physics
of a fully converged TRILEX calculation. This is, however, difficult to rule
out since full self-consistent TRILEX solutions at the phase transition point
seem to be out of reach at the moment. The final explanation is that this is
not an artefact but the result one actually gets from TRILEX, and calculating
the triply irreducible vertex indeed leads to the band crossing at the L point
as well as a band touching at the K point, prohibiting a WSM phase and only
giving rise to a Weyl Metal phase. Regardless of the physics happening at the
K high symmetry point, there is definitely a Weyl band crossing to be seen at
the L high symmetry point and, hence, pyrochlore iridates have a Weyl
Semimetal or at least a Weyl metal phase. Further investigation at a larger
interaction value $U=1.5$ eV shows that the phase transition is likely to
occur at a somewhat lower critical value as compared to the sDMFT
calculations. Although we cannot calculate directly in the vicinity of the
transition, by comparisons of sDMFT and TRILEX results for different
interaction values $U$ we estimate the critical transition in TRILEX to be
around $U\approx 1.55$ eV.

##  IV Conclusions

In this work we first investigated to the $j_{\textrm{eff}}=\frac{1}{2}$
model, and its qualitative equivalence to the three-orbital model including
all $t_{2g}$ orbitals. This reduction to a single-orbital model is necessary
in order to make TRILEX calculations computationally manageable. In the
calculations of both single site DMFT as well as TRILEX the magnetic ordering
seen in other works could be reproduced, and the transition to the
antiferromagnetic all-in-all-out ordering confirmed. This symmetry breaking
magnetic ordering is an essential requirement to topological properties in
materials.

With the sDMFT solutions not giving rise the a topological solution, the
k-dependence of the self energy becomes a very important property to
investigate in order to determine the Weyl phase. We could show that the self
energy shows increasing k-dependence for increasing interaction values. In
particular, near the critical interaction value the k-dependence is very
strong. This eventually can give rise to a Weyl-semimetal phase. It also
aligns with the literature results of cluster DMFT where the solutions are
k-dependent as well.

The bandstructures of Y2Ir2O7 show the typical bandcrossing occuring due to
Weyl cones. With the single-shot TRILEX calculations the WSM cannot be
confirmed to highest certainty. This uncertainty is due to the band narrowing
near the K high-symmetry points. Regardless of that, we could show the
presence of Weyl band crossings. Therefore, the pyrochlore iridates host a
Weyl semimetal or at least a Weyl metal phase.

###### Acknowledgements.

We thank Thomas Schäfer and Nils Wentzell for insightful discussions on
TRILEX. This work has been supported by the Austrian Science Fund (FWF), grant
Y746. A part of the calculations have been performed on the Vienna Scientific
Cluster (VSC).

## References

  * Witczak-Krempa  _et al._ [2014] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Correlated Quantum Phenomena in the Strong Spin-Orbit Regime, Annual Review of Condensed Matter Physics 5, 57 (2014).
  * Kim  _et al._ [2008] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G. Cao, and E. Rotenberg, Novel ${J}_{\mathrm{eff}}=1/2$ Mott state induced by relativistic spin-orbit coupling in ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$, Phys. Rev. Lett. 101, 076402 (2008).
  * Martins  _et al._ [2011] C. Martins, M. Aichhorn, L. Vaugier, and S. Biermann, Reduced Effective Spin-Orbital Degeneracy and Spin-Orbital Ordering in Paramagnetic Transition-Metal Oxides: ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$ versus ${\mathrm{Sr}}_{2}{\mathrm{RhO}}_{4}$, Phys. Rev. Lett. 107, 266404 (2011).
  * Martins  _et al._ [2017] C. Martins, M. Aichhorn, and S. Biermann, Coulomb correlations in 4d and 5d oxides from first principles—or how spin–orbit materials choose their effective orbital degeneracies, Journal of Physics: Condensed Matter 29, 263001 (2017).
  * Zhang  _et al._ [2013] H. Zhang, K. Haule, and D. Vanderbilt, Effective $J\mathbf{=}1/2$ Insulating State in Ruddlesden-Popper Iridates: An $\mathrm{LDA}\mathbf{+}\mathrm{DMFT}$ Study, Phys. Rev. Lett. 111, 246402 (2013).
  * Nakatsuji  _et al._ [2006] S. Nakatsuji, Y. Machida, Y. Maeno, T. Tayama, T. Sakakibara, J. v. Duijn, L. Balicas, J. N. Millican, R. T. Macaluso, and J. Y. Chan, Metallic spin-liquid behavior of the geometrically frustrated Kondo lattice ${\mathrm{Pr}}_{2}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. Lett. 96, 087204 (2006).
  * Machida  _et al._ [2010] Y. Machida, S. Nakatsuji, S. Onoda, T. Tayama, and T. Sakakibara, Time-reversal symmetry breaking and spontaneous Hall effect without magnetic dipole order, Nature 463, 210 (2010).
  * Wan  _et al._ [2011] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates, Phys. Rev. B 83, 205101 (2011).
  * Witczak-Krempa and Kim [2012] W. Witczak-Krempa and Y. B. Kim, Topological and magnetic phases of interacting electrons in the pyrochlore iridates, Phys. Rev. B 85, 045124 (2012).
  * Witczak-Krempa  _et al._ [2013] W. Witczak-Krempa, A. Go, and Y. B. Kim, Pyrochlore electrons under pressure, heat, and field: Shedding light on the iridates, Phys. Rev. B 87, 155101 (2013).
  * Go  _et al._ [2012] A. Go, W. Witczak-Krempa, G. S. Jeon, K. Park, and Y. B. Kim, Correlation effects on 3D topological phases: From bulk to boundary, Phys. Rev. Lett. 109, 066401 (2012).
  * Wang  _et al._ [2017a] R. Wang, A. Go, and A. Millis, Weyl rings and enhanced susceptibilities in pyrochlore iridates: $k\cdot{}p$ analysis of cluster dynamical mean-field theory results, Phys. Rev. B 96, 195158 (2017a).
  * Wang  _et al._ [2017b] R. Wang, A. Go, and A. J. Millis, Electron interactions, spin-orbit coupling, and intersite correlations in pyrochlore iridates, Phys. Rev. B 95, 045133 (2017b).
  * Zhang  _et al._ [2017] H. Zhang, K. Haule, and D. Vanderbilt, Metal-insulator transition and topological properties of pyrochlore iridates, Phys. Rev. Lett. 118, 026404 (2017).
  * Shinaoka  _et al._ [2015] H. Shinaoka, S. Hoshino, M. Troyer, and P. Werner, Phase diagram of pyrochlore iridates: All-in–all-out magnetic ordering and non-fermi-liquid properties, Phys. Rev. Lett. 115, 156401 (2015).
  * Yanagishima and Maeno [2001] D. Yanagishima and Y. Maeno, Metal-nonmetal changeover in pyrochlore iridates, Journal of the Physical Society of Japan 70, 2880 (2001), https://doi.org/10.1143/JPSJ.70.2880 .
  * Matsuhira  _et al._ [2007] K. Matsuhira, M. Wakeshima, R. Nakanishi, T. Yamada, A. Nakamura, W. Kawano, S. Takagi, and Y. Hinatsu, Metal–insulator transition in pyrochlore iridates Ln2Ir2O7 (Ln = Nd, Sm, and Eu), Journal of the Physical Society of Japan 76, 043706 (2007), https://doi.org/10.1143/JPSJ.76.043706 .
  * Zhao  _et al._ [2011] S. Zhao, J. M. Mackie, D. E. MacLaughlin, O. O. Bernal, J. J. Ishikawa, Y. Ohta, and S. Nakatsuji, Magnetic transition, long-range order, and moment fluctuations in the pyrochlore iridate Eu2Ir2O7, Phys. Rev. B 83, 180402 (2011).
  * Ishikawa  _et al._ [2012] J. J. Ishikawa, E. C. T. O’Farrell, and S. Nakatsuji, Continuous transition between antiferromagnetic insulator and paramagnetic metal in the pyrochlore iridate Eu2Ir2O7, Phys. Rev. B 85, 245109 (2012).
  * Disseler  _et al._ [2012] S. M. Disseler, C. Dhital, A. Amato, S. R. Giblin, C. de la Cruz, S. D. Wilson, and M. J. Graf, Magnetic order in the pyrochlore iridates ${A}_{2}$Ir2O7 ($A$ = Y, Yb), Phys. Rev. B 86, 014428 (2012).
  * Guo  _et al._ [2013] H. Guo, K. Matsuhira, I. Kawasaki, M. Wakeshima, Y. Hinatsu, I. Watanabe, and Z.-a. Xu, Magnetic order in the pyrochlore iridate Nd2Ir2O7 probed by muon spin relaxation, Phys. Rev. B 88, 060411 (2013).
  * Ueda  _et al._ [2016] K. Ueda, J. Fujioka, and Y. Tokura, Variation of optical conductivity spectra in the course of bandwidth-controlled metal-insulator transitions in pyrochlore iridates, Phys. Rev. B 93, 245120 (2016).
  * Sushkov  _et al._ [2015] A. B. Sushkov, J. B. Hofmann, G. S. Jenkins, J. Ishikawa, S. Nakatsuji, S. Das Sarma, and H. D. Drew, Optical evidence for a Weyl semimetal state in pyrochlore ${\mathrm{Eu}}_{2}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. B 92, 241108 (2015).
  * Ueda  _et al._ [2017] K. Ueda, T. Oh, B.-J. Yang, R. Kaneko, J. Fujioka, N. Nagaosa, and Y. Tokura, Magnetic-field induced multiple topological phases in pyrochlore iridates with mott criticality, Nature Communications 8, 10.1038/ncomms15515 (2017).
  * Ueda  _et al._ [2018] K. Ueda, R. Kaneko, H. Ishizuka, J. Fujioka, N. Nagaosa, and Y. Tokura, Spontaneous hall effect in the Weyl semimetal candidate of all-in all-out pyrochlore iridate, Nature Communications 9, 10.1038/s41467-018-05530-9 (2018).
  * Nakayama  _et al._ [2016] M. Nakayama, T. Kondo, Z. Tian, J. J. Ishikawa, M. Halim, C. Bareille, W. Malaeb, K. Kuroda, T. Tomita, S. Ideta, K. Tanaka, M. Matsunami, S. Kimura, N. Inami, K. Ono, H. Kumigashira, L. Balents, S. Nakatsuji, and S. Shin, Slater to mott crossover in the metal to insulator transition of ${\mathrm{Nd}}_{2}{\mathrm{Ir}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. Lett. 117, 056403 (2016).
  * Ayral and Parcollet [2015] T. Ayral and O. Parcollet, Mott physics and spin fluctuations: A unified framework, Phys. Rev. B 92, 115109 (2015).
  * Ayral and Parcollet [2016] T. Ayral and O. Parcollet, Mott physics and spin fluctuations: A functional viewpoint, Phys. Rev. B 93, 235124 (2016).
  * Parcollet  _et al._ [2015] O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, TRIQS: A toolbox for research on interacting quantum systems, Computer Physics Communications 196, 398 (2015).
  * Blaha  _et al._ [2020] P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H. Madsen, and L. D. Marks, WIEN2k: An APW+lo program for calculating the properties of solids, The Journal of Chemical Physics 152, 074101 (2020).
  * Mostofi  _et al._ [2014] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, An updated version of wannier90: A tool for obtaining maximally-localised Wannier functions, Computer Physics Communications 185, 2309 (2014).
  * Seth  _et al._ [2016] P. Seth, I. Krivenko, M. Ferrero, and O. Parcollet, TRIQS/CTHYB: A continuous-time quantum Monte Carlo hybridisation expansion solver for quantum impurity problems, Computer Physics Communications 200, 274 (2016).
  * Werner  _et al._ [2006] P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Continuous-time solver for quantum impurity models, Phys. Rev. Lett. 97, 076405 (2006).
  * Werner and Millis [2006] P. Werner and A. J. Millis, Hybridization expansion impurity solver: General formulation and application to Kondo lattice and two-orbital models, Phys. Rev. B 74, 155107 (2006).
  * Kraberger  _et al._ [2017] G. J. Kraberger, R. Triebl, M. Zingl, and M. Aichhorn, Maximum entropy formalism for the analytic continuation of matrix-valued Green’s functions, Phys. Rev. B 96, 155128 (2017).
  * Richter  _et al._ [2021] M. Richter, J. Graspeuntner, T. Schäfer, N. Wentzell, and M. Aichhorn, Comparing the effective enhancement of local and nonlocal spin-orbit couplings on honeycomb lattices due to strong electronic correlations, Phys. Rev. B 104, 195107 (2021).
  * Rubtsov  _et al._ [2005] A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein, Continuous-time quantum Monte Carlo method for fermions, Phys. Rev. B 72, 035122 (2005).