# Orbitally-selective Breakdown of the Fermi Liquid and Simultaneous
Enhancement of Metallic and Insulating States in Correlated Multi-band Systems
with Spin-orbit Coupling

Ze-Yi Song  Shanghai Key Laboratory of Special Artificial Microstructure
Materials and Technology, School of Physics Science and engineering, Tongji
University, Shanghai 200092, P.R. China  Xiu-Cai Jiang  Shanghai Key
Laboratory of Special Artificial Microstructure Materials and Technology,
School of Physics Science and engineering, Tongji University, Shanghai 200092,
P.R. China  Yu-Zhong Zhang  Email<EMAIL_ADDRESS>Shanghai Key
Laboratory of Special Artificial Microstructure Materials and Technology,
School of Physics Science and engineering, Tongji University, Shanghai 200092,
P.R. China

###### Abstract

We show that spin-orbit coupling (SOC) plays Janus-faced roles on the
orbitally-selective Mott transitions in a three-orbital Hubbard model with
crystal field splitting at a specific filling of $2/3$, which is a minimal
Hamiltonian for ruthenates. While the SOC favors metallic state due to
enhancement of orbital hybridization at smaller on-site Coulomb repulsions, it
stabilizes the Mott insulating state ascribed to lifting of orbital
degeneracies and enhancement of band polarizations at larger electronic
interaction. Moreover, an orbitally-selective non-Fermi liquid (OSnFL), where
breakdown and retention of the Fermi liquid coexist in different orbitals,
emerges between the orbitally-selective Mott phase and the Fermi-liquid state.
This novel state can be used to account for the exotic metallic behavior
observed in 4$d$ materials, such as Ca1.8Sr0.2RuO4, Ba2RuO4 under strain and
Sr2RuO4 under uniaxial pressure. We propose that orbitally-selective Kondo
breakdown may account for the OSnFL.

##  I introduction

Multi-orbital correlated electronic systems have been extensively investigated
for decades since the discoveries of iron-based superconductors Kamihara  _et
al._ (2006, 2008), and 4$d$, 5$d$ materials like ruthenates and iridates
Witczak-Krempa  _et al._ (2014); Rau  _et al._ (2016); Martins  _et al._
(2017), etc. Interplay of the following factors like the kinetic energy,
crystal field splitting, spin-orbit coupling (SOC), Hund’s rule coupling and
the Hubbard interaction dominates various exotic properties of these systems.
For example, sizable SOC and Coulomb repulsion in 4$d$ and 5$d$ materials
Martins  _et al._ (2017) lead to unconventional superconductivity Meng  _et
al._ (2014); Chaloupka and Khaliullin (2016), SOC-assisted Mott transition Kim
_et al._ (2009); Chaloupka  _et al._ (2010); Comin  _et al._ (2012); Plumb
_et al._ (2014); Winter  _et al._ (2016); Lang  _et al._ (2016), quantum spin
liquid Zhou  _et al._ (2017), spin-orbit exciton condensation Khaliullin
(2013); Sato  _et al._ (2019), and exotic magnetic order Khaliullin (2013);
Chen  _et al._ (2010); Chen and Balents (2011); Meetei  _et al._ (2015). Among
all the phenomena mentioned above, SOC-assisted Mott transition is of
particular interest, where the SOC lifts orbital degeneracy, resulting in an
effective half-filled system. Then, the Mott transition occurs at intermediate
Coulomb interaction, as observed in Sr2IrO4 Kim  _et al._ (2009).

Meanwhile, the Hund’s rule coupling served as a band decoupler de’ Medici
(2011), together with the crystal field splitting which lowers the orbital
degeneracy, open a promising way to generate orbital selectivity which is
believed to widely exist in ruthenates Georges  _et al._ (2013); Gorelov  _et
al._ (2010); Mravlje  _et al._ (2011); Stricker  _et al._ (2014); Dang  _et
al._ (2015a, b); Sutter  _et al._ (2019); Kugler  _et al._ (2020) and iron-
based superconductors Georges  _et al._ (2013); Haule and Kotliar (2009); Yin
_et al._ (2011); Lanatà  _et al._ (2013); Yi  _et al._ (2013); Sprau  _et al._
(2017). The corresponding metallic state, so-called Hund’s metal, with
electronic correlations dominated by Hund’s coupling, rather than the Hubbard
interaction, are of current intensive interest since it may be responsible for
exotic metallic behavior Georges  _et al._ (2013) and unconventional
superconductivity Haule and Kotliar (2009). And under certain circumstances
Koga  _et al._ (2004); Werner and Millis (2007); de’ Medici  _et al._ (2009);
Lee  _et al._ (2011); Song  _et al._ (2015), the orbitally-selective Mott
(OSM) transition Anisimov  _et al._ (2002) takes place, where partial bands
become Mott-insulating and the others remain metallic.

However, comprehensive understanding of quantum phase transitions in real
materials with multiple active orbitals is still missing Anisimov  _et al._
(2002); Shimoyamada  _et al._ (2009); Neupane  _et al._ (2009); Gorelov  _et
al._ (2010); Sutter  _et al._ (2019) since the SOC and the crystal field
splitting are always separately taken into account Huang  _et al._ (2012); Du
_et al._ (2013); Kim  _et al._ (2017); Triebl  _et al._ (2018); Piefke and
Lechermann (2018) when the electronic correlations are theoretically treated
within some reliable approximations like the dynamical mean field theory
(DMFT) Georges  _et al._ (1996). Such an unrealistic modelling may result in
misleading of the effect of SOC on the Hund’s metals, the orbital selectivity,
and the Mott insulators.

Figure 1: (color online) The schematic phase diagram for the three-orbital
Hubbard model in the plane of $\lambda$ and $U$. FL, OSM, OSnFL, and Mott
denote the Fermi liquid, orbitally-selective Mott phase, orbitally-selective
non-Fermi liquid, and Mott insulator, respectively.

In this article, we have studied a three-orbital Hubbard model to investigate
the influence of SOC on the orbital-differentiated correlations induced by
both the lifted degeneracies due to the crystal field splitting and the
orbital decoupling ascribed to the Hund’s coupling at zero temperature. The
main results are schematically summarized in Fig. 1. In the absence of SOC,
two phases are identified within the interacting range we studied, including
the Fermi-liquid state and the OSM state. The SOC is found to effectively
suppress the electronic correlations at small Coulomb interaction and favor
the Fermi-liquid state. The competition between SOC and electronic
correlations leads to the emergence of an exotic metallic state Cooper  _et
al._ (2009), characterized by the coexistence of the Fermi-liquid state and
non-Fermi-liquid state in different orbitals, which can account for orbitally-
selective breakdown of the Fermi liquid observed in Ca1.8Sr0.2RuO4 by recent
angle-resolved photoemission spectroscopy (ARPES) experiments Sutter  _et al._
(2019) and early transport measurements Nakatsuji  _et al._ (2003).
Contrastly, the SOC enhances electronic correlations at large Coulomb
repulsion and results in reduced critical values of phase transitions from the
OSM state to the Mott insulator.

##  II Model and Method

To obtain above phase diagram, the three-orbital Hubbard model on the bethe
lattice is considered, which is defined as

$\displaystyle H$
$\displaystyle=t\sum_{\langle{ij}\rangle\alpha\sigma}C^{{\dagger}}_{i\alpha\sigma}C_{j\alpha\sigma}+\sum_{i\alpha\sigma}(\Delta_{\alpha}-\mu)n_{i\alpha\sigma}$
$\displaystyle+U\sum_{i\alpha}n_{i\alpha\uparrow}n_{i\alpha\downarrow}+(U^{\prime}-J_{z})\sum_{i\alpha>\beta\sigma}n_{i\alpha\sigma}n_{i\beta\sigma}$
$\displaystyle+U^{\prime}\sum_{i\alpha>\beta\sigma}n_{i\alpha\sigma}n_{i\beta\bar{\sigma}}-J_{f}\sum_{i\alpha>\beta}\left[S^{+}_{i\alpha}S^{-}_{i\beta}+S^{-}_{i\alpha}S^{+}_{i\beta}\right]$
$\displaystyle+J_{p}\sum_{i\alpha\neq\beta}C^{{\dagger}}_{i\alpha\uparrow}C^{{\dagger}}_{i\alpha\downarrow}C_{i\beta\downarrow}C_{i\beta\uparrow}+H_{\text{SOC}},$
(1)

where $t$ denotes the nearest-neighbor hopping, $\Delta_{\alpha}$ represents
crystal field splitting for $t_{2g}$ basis with orbital indices
$\alpha=\\{yz,xz,xy\\}$, and $\mu$ is the chemical potential. $U$ and
$U^{\prime}$ are the onsite intraorbital and interorbital Coulomb repulsions,
respectively. The Hund’s rule couplings consist of the Ising-type coupling
$J_{z}$, the spin-flip term $J_{f}$ and pair-hopping term $J_{p}$. The
relationship $U=U^{\prime}+2J_{z}$ is employed to assure the electronic
interaction rotationally invariant.
$C^{{\dagger}}_{i\alpha\sigma}(C_{i\alpha\sigma})$ creates (annihilates) an
electron with spin $\sigma$ in orbital $\alpha$ of lattice site i. $S$ and $n$
represent the spin and particle number operators, respectively. The
relativistic SOC reads

$\displaystyle
H_{\text{SOC}}=\lambda\sum_{i\alpha\beta}\sum_{\sigma_{1}\sigma_{2}}\langle\alpha|{\mathbf{L}}_{i}|\beta\rangle\langle\sigma_{1}|{\bf{}S}_{i}|\sigma_{2}\rangle{}C^{{\dagger}}_{i\alpha\sigma_{1}}C_{i\beta\sigma_{2}},$
(2)

where $\lambda$ is the strength of SOC, $\bf L$ is the local orbital angular
momentum operator. The matrix representations of $L=2$ in the $t_{2g}$ basis
are the same as the ones for $L=1$ in cubic basis except for a sign in
accordance with the T-P correspondence Martins  _et al._ (2017); S. Sugano, Y.
Tanabe, and H. Kamimura (1970).

We employed the DMFT in combination with exact diagonalization (ED) Georges
_et al._ (1996) as impurity solver to solve the model (II) on the bethe
lattice with infinite coordinates at a filling of $n=2/3$, namely $4$
electrons in $3$ orbitals, and at zero temperature. The noninteracting density
of states is
$\rho_{\alpha}(\omega)=\frac{2}{\pi{D}^{2}}\sqrt{D^{2}-\omega^{2}}$ and the
half bandwidth $D$ is used as the energy unit. The effective inverse
temperature was set to $\beta{D}=200$ which serves as a low-frequency cutoff.
On the bethe lattice, the DMFT self-consistent conditions simply read
$\hat{\Delta}(\omega)=\frac{D^{2}}{4}\hat{G}(\omega)$, where
$\hat{\Delta}(\omega)$ is a matrix for hybrid functions, $\hat{G}(\omega)$ for
local lattice Green’s functions. Totally 6 bathes were used to fit the hybrid
function $\hat{\Delta}(\omega)$. In our calculations, the orbital degeneracy
is lifted by crystal field splitting (i.e.,
$\Delta_{yz}=\Delta_{xz}\neq\Delta_{xy}$), leading to a nondegenerate $d_{xy}$
orbital and doubly degenerate $d_{yz/xz}$ orbitals. We fix electronic
populations to be $(1.5,1.5,1.0)$ in accordance with that of Ca1.8Sr0.2RuO4
Sutter  _et al._ (2019) in the absence of SOC, which can be realized by tuning
the orbitally-dependent potential $\Delta_{\alpha}$ de’ Medici  _et al._
(2009). The calculations were performed in the paramagnetic state with
isotropic Hund’s coupling where $J_{z}=J_{f}=J_{p}$. The broadening factor
$\eta=0.02D$ is used to calculate real-frequency dynamical quantities
including the Green’s functions, self-energies, and dynamical susceptibilities
for spin, orbital, and total angular momentums.

##  III results

Figure 2: (color online) (a) The quasiparticle residue $Z$ of $t_{2g}$
orbitals as a function of $U$ at $\lambda=0.0$ and $\lambda=0.14$, (b) a
blowup for $Z$ of $d_{xy}$ orbital near the critical point
$U^{c}_{xy}(\lambda=0.0)$ of about 2.3, those for $d_{yz/xz}$ orbitals are
shown in the inset.

In order to explore the influence of SOC on the electronic correlations in
multi-band systems, we have calculated the quasiparticle residue
$Z_{\alpha}=(1-\frac{\partial{\text{Re}}\Sigma_{\alpha}(\omega)}{\partial\omega}|_{\omega\to
0})^{-1}$, as shown in Fig. 2 (a), where ${\text{Re}}\Sigma_{\alpha}(\omega)$
is the real part of self-energies of $t_{2g}$ orbitals. In the absence of SOC,
$Z_{xy}$ is rapidly suppressed as $U$ increases and vanishes at a critical
value of $U^{c}_{xy}(\lambda=0.0)\approx 2.3$, indicating a Mott-Hubbard gap
opens in $d_{xy}$ band. In contrast, although $Z_{yz/xz}$ is drastically
reduced at the beginning, it remains finite in a wide region of $U$ with
$U^{c}_{yz/xz}(\lambda=0.0)$ much larger than $U^{c}_{xy}(\lambda=0.0)$, which
suggests an occurrence of the OSM phase where electrons in $d_{xy}$ band
become Mott-localised and those in $d_{yz/xz}$ bands remain itinerant. This is
in excellent agreement with previous results de’ Medici  _et al._ (2009).

When the SOC is taken into account, e.g. $\lambda=0.14$, the quasiparticle
residues behave distinctly at small and large Coulomb repulsions in comparison
to those at $\lambda=0.0$. At large $U$, the SOC enhances the electronic
correlations in $d_{yz/xz}$ bands as indicated by suppression of $Z_{yz/xz}$,
leading to a smaller critical value of Mott transition
$U^{c}_{yz/xz}(\lambda=0.14)$ than that of $\lambda=0.0$ case. Since the Mott-
insulating state with a vanishing $Z_{xy}$ preserves in $d_{xy}$ band, it
suggests that the SOC cooperates with the Coulomb interaction and stablizes
the Mott-insulating ground state. Conversely, at small $U$, the SOC suppresses
the electronic correlations in all bands as inferred by larger values of $Z$
in comparison to those of $\lambda=0$ case as shown in Fig. 2 (b) which is a
blowup of Fig. 2 (a) at small $U$ region. This gives rise to an intersection
of $Z_{yz/xz}$ between $\lambda=0.14$ and $\lambda=0.0$ cases at $U\approx
2.5$ as seen in the inset of Fig. 2 (b). Furthermore, the quasiparticle
residue $Z_{xy}$ of $\lambda=0.14$ case remains nonzero for a wide range of
$U>U^{c}_{xy}(\lambda=0.0)$, suggesting that a transition from Mott insulator
to metal may take place in $d_{xy}$ band when SOC is turned on. This indicates
that the SOC competes with the on-site Coulomb repulsion and favors a metallic
ground state at small $U$. Obviously, the SOC shows opposite effects on the
Mott transitions in multi-band systems.

Figure 3: (Color online) (a-b) Density of states projected onto the $t_{2g}$
orbitals and (c-d) the imaginary part of self-energy on the Matsubara axis at
$U=2.58$ and $U=8.78$ for $\lambda=0.0$ and $\lambda=0.14$.

We further demonstrate the pronounced effects of the SOC in the vicinity of
phase transitions in Fig. 3 where the calculated density of states projected
onto the $t_{2g}$ basis and corresponding imaginary part of the Matsubara
self-energy Im$\Sigma_{\alpha}(i\omega_{n})$ are exhibited. In the absence of
the SOC, at both $U=2.58$ and $U=8.78$, the density of states at the Fermi
level vanishes in $d_{xy}$ orbital and Im$\Sigma_{xy}(i\omega_{n})$ diverges
in proximity to zero frequency, which is a typical character of Mott
insulator. Meanwhile, the $d_{yz/xz}$ orbitals behave as Fermi liquids since
the density of states is finite at $\omega=0$ and
Im$\Sigma_{yz/xz}(i\omega_{n})$ approches zero at low frequency. It suggests
that the system is in the OSM phase. When $\lambda$ increases to 0.14, the SOC
exhibits distinct effects at small and large $U$, respectively. At $U=8.78$,
while the SOC barely influences the nature of $d_{xy}$ orbital, the complete
suppression of density of states of $d_{yz/xz}$ orbitals at the Fermi level
and the divergence of Im$\Sigma_{yz/xz}(i\omega_{n})$ as $\omega_{n}$ goes to
zero, suggests an appearance of the SOC-assisted Mott phase. On the other
hand, at $U=2.58$, the SOC drastically affects the properties of $d_{xy}$
orbital but hardly influences that of $d_{yz/xz}$ orbitals. The appearance of
central peak of density of states for $d_{xy}$ orbital at the Fermi level and
Im$\Sigma_{xy}(i\omega_{n})$ extrapolating to zero as $\omega_{n}\rightarrow
0$ suggests a SOC-induced Fermi liquid behavior.

The occurrence of SOC-assisted Mott phase at large $U$ can be easily
understood within $|J,\pm{m}\rangle$ basis where the local Hamiltonian (2) can
be diagonalized. Here $J$ denotes the total angular momentum and $\pm{m}$
represents its projection in $z$ direction. It is found that the SOC enhances
band polarizations and leads to a full occupation of the
$|\frac{3}{2},\pm\frac{3}{2}\rangle$ bands at $\lambda_{c}$ of about $0.1$.
Then, the rest two electrons reside in $|\frac{3}{2},\pm\frac{1}{2}\rangle$
and $|\frac{1}{2},\pm\frac{1}{2}\rangle$ bands, resulting in an effective
half-filled system, rather than originally $4$ electrons per $3$ orbitals.
Therefore, the SOC favors Mott transition at large $U$ since the effective
filling is changed.

On the contrary, in the small $U$ region, an OSM phase requires decoupling
between Mott-insulating $d_{xy}$ orbital and metallic $d_{yz/xz}$ orbitals,
which is originally fulfilled by Hund’s rule interaction in the absence of
SOC. However, the SOC introduces coupling between the $d_{xy}$ and $d_{yz/xz}$
orbitals, leading to the enhancements of both orbital fluctuations and the
kinetic energies of all orbitals. Therefore, the SOC tends to destroy the Mott
phase of $d_{xy}$ orbital and favors metallic state due to the increase of
bandwidth and decrease of band decoupling at small $U$.

Figure 4: (color online) The influence of spin-orbit coupling on the imaginary
part of Matsubara self-energy for $d_{xy}$ band, those for $d_{yz/xz}$ bands
are shown in the inset. (b) Density of states for $d_{xy}$ band in the Mott
($\lambda=0.0$), non-Fermi-liquid ($\lambda=0.12$) and the Fermi-liquid
($\lambda=0.14$) states. (c) Real and (d) imaginary part of self-energy of
$d_{xy}$ band on the real frequency axis. Here, the electron repulsion is
fixed at $U=2.58$.

Besides the opposite effect of SOC on electronic correlations, it is also
interesting to find an intermediate phase, called orbitally-selective non-
Fermi liquid (OSnFL) where breakdown of the Fermi liquid happens only in
$d_{xy}$ orbitals, emerging between the Fermi-liquid state and the OSM phase
in the small $U$ and intermediate $\lambda$ region, as presented in Fig. 1.
This exotic metallic state can be clearly identified by the imaginary part of
the Matsubara self-energy Im$\Sigma_{\alpha}(i\omega_{n})$ as shown in Fig. 4
(a). For example, at $U=2.58$, while Im$\Sigma_{yz/xz}(i\omega_{n})$ always
goes to zero as $\omega_{n}\rightarrow 0$, indicating the Fermi liquids in
$d_{yz/xz}$ bands, (see the inset of Fig. 4 (a)), Im$\Sigma_{xy}(i\omega_{n})$
extrapolates to a finite value as $\omega_{n}\rightarrow 0$ around
$\lambda\approx 0.12$, which is in sharp contrast to the divergent behavior
for $\lambda\leq 0.08$ and the tendency to approach zero for $\lambda\geq
0.14$. The finite scattering rate suggests a finite lifetime of quasiparticles
and a breakdown of the Fermi liquid at zero temperature.

The self-energy $\Sigma_{xy}(\omega)$ on the real frequency axis again reveals
the SOC-induced non-Fermi-liquid behavior in $d_{xy}$ band. Fig. 4 (c) and (d)
shows the real part of self-energy Re$\Sigma_{xy}(\omega)$ and imaginary part
of self-energy Im$\Sigma_{xy}(\omega)$ at $U=2.58$ for $\lambda=0.0$,
$\lambda=0.12$ and $\lambda=0.14$. When $\lambda=0.12$, the positive slop of
Re$\Sigma_{xy}(\omega)$ at the Fermi level and the finite
Im$\Sigma_{xy}(\omega)$ suggests the breakdown of quasiparticle picture. The
development of additional low-energy poles of
$\omega+\mu-\varepsilon-\text{Re}\Sigma_{xy}(\omega)=0$ close to the Fermi
level and the finite scattering rate at $\omega=0$ lead to the appearance of a
pseudogap in $d_{xy}$ band, as depicted in Fig. 4 (b), reminiscent of that
observed in Hubbard model within cluster DMFT Zhang and Imada (2007). This is
strikingly different from those for $\lambda=0.0$ and $0.14$. For the former,
the divergent Im$\Sigma_{xy}(\omega)$ at $\omega=0$ and the steep positive
slop of Re$\Sigma_{xy}(\omega)$ suggest a Mott-insulating state. In contrast,
for the latter, the Im$\Sigma_{xy}(\omega)$ can be fitted by $\omega^{2}$ and
the Re$\Sigma_{xy}(\omega)$ is linearly $\omega$-dependent, indicating a
typical Fermi-liquid behavior at zero temperature. Fig. 4 (b) shows the
density of states for $d_{xy}$ band in the Mott, pseudogap, and the Fermi-
liquid states at $\lambda=0.00$, $0.12$ and $0.14$, respectively.

Figure 5: (color online) (a) Dynamical susceptibilities for spin ($\bf S$),
orbital ($\bf L$) and total ($\bf J$) angular momentums at $U=2.58$ and
$\lambda=0.12$. (b) Dynamical spin susceptibilities at $U=2.58$ for
$\lambda=0.0$, $\lambda=0.12$ and $\lambda=0.14$ cases, the inset shows
corresponding orbitally-resolved dynamical spin susceptibilities, where solid
lines denote $d_{yz/xz}$ orbitals and dashed lines $d_{xy}$ orbital.

Now we discuss the underlying physics for the OSnFL state. Since the nature of
correlated metals is controlled by the low-energy excitations in the absence
of SOC for multi-band systems Song  _et al._ (2017); Kugler  _et al._ (2019),
we performed similar calculations for the dynamical susceptibilities
$\chi^{ii}(\omega)$ of total, orbital, and spin angular momentums with $i=J$,
$L$, and $S$, respectively, at $U=2.58$ and $\lambda=0.12$, as shown in Fig. 5
(a). The dynamical spin susceptibilities are defined as

$\displaystyle\chi^{SS}(t)=-i\theta(t)\langle|[S(t),S]_{-}|\rangle,$ (3)

where $\theta(t)$ is a step function, $|\rangle$ is the ground state and
$[]_{-}$ denotes the commutator of two operators, $S$ is the operator of total
spin angular momentums. After performing a Fourier transformation, it yields

$\displaystyle\chi^{SS}(z=\omega+i\eta)=-\text{Im}\int^{\infty}_{-\infty}dte^{izt}\chi^{SS}(t).$
(4)

We only display the imaginary parts since the real parts can be reproduced by
the Kramers-Kronig relations. The dynamical susceptibilities
$\chi^{LL}(\omega)$ and $\chi^{JJ}(\omega)$ for the corresponding orbital and
total angular momentums can be obtained similarly. The calculation details
regarding dynamical correlations are given in Appendix B. From Fig. (5) (a),
it is found that the low-energy excitations mainly come from spin
fluctuations, while the high-energy parts are ascribed to the excitations of
orbital momentum $\bf L$. Thus we focus on analyses of dynamical spin
susceptibilities $\chi^{SS}(\omega)$ at $U=2.58$ for $\lambda=0.0$, $0.12$,
and $0.14$. At $\lambda=0.0$, the system is in the OSM phase. The low-energy
spin fluctuations consist of two parts, one is attributed to the Kondo
screening of spins in $d_{yz/xz}$ bands by itinerant electrons, and the other
is ascribed to the existence of spin triplets formed by $4$ electrons in $3$
$t_{2g}$ orbitals Song  _et al._ (2017); Kugler  _et al._ (2019). At
$\lambda=0.14$, the system becomes a Fermi liquid. The intensity of low-energy
spin excitations is drastically enhanced. Since the low-energy spin
excitations of $d_{yz/xz}$ bands remains almost unchanged for different
$\lambda$ as seen in the inset of Fig. 5 (b), the abrupt increase should be
ascribed to additional Kondo resonances between spins in $d_{xy}$ band and
itinerant electrons. At $\lambda=0.12$, while electrons in $d_{xy}$ band
become itinerant, the corresponding low-energy spin excitations are similar to
those for $\lambda=0.0$, indicating that the spins in $d_{xy}$ band do not
take part in the Kondo screening. The lack of Kondo screening in $d_{xy}$ band
leads to the emergence of non-Fermi liquid. Hence we conclude that the OSnFL
state is a result of orbitally-selective Kondo breakdown.

##  IV discussion

The calculated results show the Janus-faced influence of SOC on the Mott
transitions in the presence of crystal field splitting at a filling of
$n=2/3$. Owing to the orbital degeneracy lifted by crystal field splitting,
the Mott transitions in different orbitals take place separately as $U$
increases. The opposite effect of SOC on the electronic correlations leads to
the increase of $U_{xy}^{c}$ and decrease of $U_{yz/xz}^{c}$, which is in
sharp contrast to previous theoretical results in the absence of crystal field
splitting, where a common Mott transition occurs and the critical value of
$U_{c}$ can only be increased or decreased by the SOC for given integer
fillings Kim  _et al._ (2017); Triebl  _et al._ (2018); Piefke and Lechermann
(2018). Furthermore, the competition between SOC and Coulomb repulsion results
in the OSnFL state, characterized by the coexistence of the Fermi liquid and
non-Fermi liquid in different orbitals, which is fundamentally distinct from
the case of $\lambda=0.0$, where the metallic ground state is always a Fermi
liquid, as displayed in Fig. 1. The coexisting region is of particular
importance as it provides a unique platform to study the breakdown of the
Fermi-liquid picture and the nature of non-Fermi-liquid state. The appearance
of OSnFL state requires remarkable orbital-differentiated correlations and
sizable orbital hybridizations, reminiscent of non-Fermi liquid obtained
within cluster DMFT Zhang and Imada (2007); Park  _et al._ (2008) where
multiple sites are corresponding to multiple orbitals. In addition, we suppose
that non-Fermi liquid might be the collective excitations of spin-charge
separation.

Next, we discuss the relevance of our results to real materials.
Ca1.8Sr0.2RuO4 satisfies the conditions for the OSnFL state. On one hand, the
intermediate SOC has been observed in Sr2RuO4 Veenstra  _et al._ (2014) and
Ca2RuO4 Fatuzzo  _et al._ (2015); Gretarsson  _et al._ (2019). On the other
band, Ca1.8Sr0.2RuO4 shows significant orbital-differentiated correlations
with effective mass for $d_{xy}$ band much larger than that for $d_{yz/xz}$
bands Shimoyamada  _et al._ (2009); Sutter  _et al._ (2019). Furthermore,
enhanced crystal field splitting induced by lattice distortion is detected in
Ca1.8Sr0.2RuO4 Friedt  _et al._ (2001). Although the material-realistic DMFT
calculations claimed that SOC hardly affects the correlations in Sr2RuO4 Kim
_et al._ (2018) and Ca2RuO4 Zhang and Pavarini (2017), we still suppose that
Ca1.8Sr0.2RuO4 should be susceptible to the SOC due to its proximity to the
Mott transition. Hence orbitally-selective breakdown of the Fermi liquid
observed experimentally in Ca1.8Sr0.2RuO4 Sutter  _et al._ (2019) may be a
result of competition between SOC and orbital-differentiated correlations.
Moreover, the non-Fermi liquid caused by strain in Ba2RuO4 Burganov  _et al._
(2016) and uniaxial pressure in Sr2RuO4 Barber  _et al._ (2018) may also be
attributed to the competition, since both strain and uniaxial pressure
slightly affect $d_{yz/xz}$ bands but significantly increase the
renormalization mass of $d_{xy}$ band due to the band flattening, resulting in
enhanced orbital differentiation. In addition, the OSnFL picture may have
important implication for understanding the low-temperature non-Fermi liquid
in CaRuO3 Lee  _et al._ (2002); Kamal  _et al._ (2006); Schneider  _et al._
(2014); Yang  _et al._ (2016); Liu  _et al._ (2018).

##  V conclusion

In conclusion, we have investigated the three-orbital Hubbard model with both
SOC and crystal field splitting using the DMFT combined with ED at $2/3$
filling. The OSM transitions take place as the orbital degeneracy is lifted by
crystal field splitting in the absence of SOC. It is found that the SOC plays
Janus-faced roles on the OSM transitions. While it suppresses the OSM
transition at smaller $U$, it favors the OSM transition at larger $U$. The
competition between the SOC and electronic correlations leads to the emergence
of an OSnFL state, where the Fermi liquid coexits with non-Fermi liquid. The
OSnFL state is originated from the orbitally-selective Kondo breakdown and can
be applied to understand the exotic metals in $4d$ materials.

###### Acknowledgements.

This work is financially supported by the National Natural Science Foundation
of China (Grant No. 11774258, 12004283) and Postgraduate Education Reform
Project of Tongji University (Grant No. GH1905), Z. Y. Song acknowledges the
financial support by China Postdoctoral Science Foundation (Grant No.
2019M651563).

##  Appendix A Dynamical Mean Field Theory for Multi-orbital Hubbard Models

Figure 6: (color online) Comparison between the Weiss fields
$\hat{g}(i\omega_{n})$ and the noninteracting impurity Green’s functions
$\hat{g}_{\text{imp}}(i\omega_{n})$ obtained from a discretized Anderson
impurity model with 6 optimal bath sites. $\hat{g}(i\omega_{n})$ can be well
reproduced by $\hat{g}_{\text{imp}}(i\omega_{n})$ in the OSM ((a)-(c)), the
OSnFL ((d)-(f)), and the Fermi-liquid (FL) ((h)-(i)) phases for all $(J,m)$
orbitals including $p=(\frac{1}{2},\frac{1}{2})$,
$p=(\frac{3}{2},\frac{1}{2})$, and $p=(\frac{3}{2},\frac{3}{2})$ orbitals.

The DMFT Georges  _et al._ (1996) was employed to investigated the multi-
orbital Hubbard model (II) on the bethe lattice with infinite coordination
number, on which the DMFT has already been proved to be exact and the model
(II) can be exactly mapped onto an Anderson impurity model with self-
consistent conditions $\hat{\Delta}(\omega)=\frac{D^{2}}{4}\hat{G}(\omega)$,
where $\hat{G}(\omega)$ is the matrix of local lattice Green’s functions,
$\hat{\Delta}(\omega)$ denotes the matrix of hybridization functions for the
Anderson impurity model, and $D$ represents the half bandwidth. Note that the
hat symbol is used to denote a matrix. When the spin-orbit coupling is
considered, it is convenient to do the calculations in the $|J,\tau{m}\rangle$
basis, where $J=\\{\frac{1}{2},\frac{3}{2}\\}$,
$m=\\{\frac{1}{2},\frac{3}{2}\\}$, $m\leq{J}$ and $\tau=\pm$ specifying a pair
of Kramers doublet $m$, which are the eigenvectors of the Hamiltonian (2). In
this representation, the Anderson impurity model reads

$\displaystyle H_{\text{imp}}$
$\displaystyle=\sum_{k\tau{m}}\epsilon_{km}C^{{\dagger}}_{k\tau{m}}C_{k\tau{m}}+\sum_{p}\hat{E}_{p}d^{{\dagger}}_{p}d_{p}+\sum_{pq}\hat{M}_{pq}d^{{\dagger}}_{p}d_{q}$
$\displaystyle+\sum_{kJ\tau{m}}V_{kJ\tau{m}}\left(C^{{\dagger}}_{k\tau{m}}d_{J\tau{m}}+d^{{\dagger}}_{J\tau{m}}C_{k\tau{m}}\right)$
$\displaystyle+\frac{1}{4}\sum_{pqst}\tilde{U}^{pq}_{st}d^{{\dagger}}_{p}d^{{\dagger}}_{q}d_{s}d_{t},$
(5)

where $\epsilon_{km}$ denotes the dispersion relationship of bath electrons,
$p,q,s,t$ are the orbital indices of $(J,\tau{m})$, $\hat{E}_{p}$ is the
eigenvalue of Hamiltonian (2), $\hat{M}_{pq}$ is the matrix element of crystal
field spliting in the $|J,\tau{m}\rangle$ basis, $V_{kJ\tau{m}}$ denotes the
hybridization between local orbitals on the impurity and bath, and
$\tilde{U}^{pq}_{st}=\sum_{\alpha\beta\gamma\delta}U_{\alpha\beta\gamma\delta}\hat{A}^{*}_{p\alpha}\hat{A}^{*}_{q\beta}\hat{A}_{s\gamma}\hat{A}_{t\delta}$
is the Coulomb interaction tensor element in the $|J,\tau{m}\rangle$
representation, where $U_{\alpha\beta\gamma\delta}$ is the Coulomb interaction
tensor element in the $t_{2g}$ basis and $\hat{A}$ the unitary transformation
Du  _et al._ (2013) between the $t_{2g}$ and $|J,\tau{m}\rangle$ basis.

The main tasks of present DMFT calculations are to solve the Anderson impurity
model (A). We took ED as the impurity solver Georges  _et al._ (1996). In
order to solve model (A) with ED, we have to use a finite number of bath sites
to optimally fit the continuous bath. In this paper, totally six discrete bath
sites were adopted to couple to three Kramers doublets, i.e. two bath sites
per Kramers doublet. Owing to the presence of orbital mixing between
$|\frac{1}{2},\frac{1}{2}\rangle$ ($|\frac{1}{2},-\frac{1}{2}\rangle$) and
$|\frac{3}{2},\frac{1}{2}\rangle$ ($|\frac{3}{2},-\frac{1}{2}\rangle$)
orbitals, the corresponding cross-orbital hybridization functions, like
$\hat{\Delta}_{\frac{1}{2},\frac{1}{2};\frac{3}{2},\frac{1}{2}}$ and
$\hat{\Delta}_{\frac{1}{2},-\frac{1}{2};\frac{3}{2},-\frac{1}{2}}$, are
considered. From Fig. 6, it is found that the Weiss field
$\hat{g}(i\omega_{n})=(\hat{\Sigma}(i\omega_{n})+\hat{G}^{-1}(i\omega_{n}))^{-1}$
with $\hat{\Sigma}(i\omega)$ the matrix of self-energy can be well reproduced
by the noninteracting impurity Green’s function
$\hat{g}_{\text{imp}}(i\omega_{n})$ defined by (7) in the OSM ((a)-(c)), the
OSnFL ((d)-(f)), and the Fermi-liquid ((g)-(i)) phases, suggesting that the
finite size effects are negligible and the results presented in the paper are
convincing. The procedures of the DMFT calculations in combination with ED as
the impurity solver are described below.

Starting from an initial set of bath parameters
$\\{\epsilon_{km},V_{kJ\tau{m}}\\}$, we construct the Anderson model (A). The
hybridization function of the corresponding impurity model (A) is written as

$\displaystyle\hat{\Delta}_{J,\tau{m};J^{\prime},\tau{m}}(i\omega_{n})=\sum_{k}\frac{V_{kJ\tau{m}}V_{kJ^{\prime}\tau{m}}}{i\omega_{n}-\epsilon_{km}},$
(6)

where $\omega_{n}=\frac{(2n-1)\pi}{\beta}$ is the Matsubara frequency with a
fictitious temperature $\beta{D}=200$, which serves as a low frequency cutoff.
The noninteracting impurity Green’s function of the Anderson model (A) is

$\displaystyle\hat{g}_{\text{imp}}(i\omega_{n})=i\omega_{n}+\mu-\hat{E}-\hat{M}-\hat{\Delta}^{-1}(i\omega_{n}).$
(7)

After the impurity Green’s function $\hat{G}_{\text{imp}}(i\omega_{n})$ is
obtained by solving model (A) with ED, the self-energy
$\hat{\Sigma}(i\omega_{n})$ can be calculated based on the Dyson’s equation

$\displaystyle\hat{\Sigma}(i\omega_{n})=\hat{g}^{-1}_{\text{imp}}(i\omega_{n})-\hat{G}^{-1}_{\text{imp}}(i\omega_{n}).$
(8)

Finally, the local Green’s function of the three-orbital Hubbard model (II) in
the $|J,\tau{m}\rangle$ basis is calculated as

$\displaystyle\hat{G}(i\omega_{n})=\int^{+\infty}_{-\infty}\frac{\rho(\epsilon)d\epsilon}{i\omega_{n}+\mu-\hat{E}-\hat{M}-\hat{\Sigma}(i\omega_{n})-\epsilon},$
(9)

where $\rho(\omega)=\frac{2}{\pi{D}^{2}}\sqrt{D^{2}-\omega^{2}}$ is the
density of states for three Kramers doublets, which is the same as the
counterpart for the $t_{2g}$ orbitals. Considering the self-consistent
conditions $\hat{\Delta}(i\omega_{n})=\frac{D^{2}}{4}\hat{G}(i\omega_{n})$, we
can iteratively calculate the local Green’s function $\hat{G}(i\omega_{n})$
through equations (A)—(9).

In our calculations, the parameters $\\{\epsilon_{km},V_{kJ\tau{m}}\\}$ to
build the model (A) are obtained by using the conjugate gradient method to
minimizing the cost function

$\displaystyle\chi=\frac{1}{N_{\text{max}}}\sum^{N_{\text{max}}}_{n=1}\frac{1}{\omega^{2}_{n}}\sum_{pq}\left|(\hat{g}(i\omega_{n})-\hat{g}_{\text{imp}}(i\omega_{n}))_{pq}\right|,$
(10)

where $N_{\text{max}}=256$ is the upper limit of the summation. Starting from
a guessed Weiss field $\hat{g}(i\omega_{n})$, we can self-consistently obtain
the convergent results when the difference $\Delta_{g}$ between the new Weiss
field $\hat{g}^{\text{new}}(i\omega_{n})$ and the old Weiss field
$\hat{g}^{\text{old}}(i\omega_{n})$ is less than 10-6. The difference
$\Delta_{g}$ is defined as

$\displaystyle\Delta_{g}=\text{max}\left\\{\left|\hat{g}^{\text{new}}_{pq}(i\omega_{n})-\hat{g}^{\text{old}}_{pq}(i\omega_{n})\right|\right\\}$
(11)

Although the present DMFT calculations were performed in the
$|J,\tau{m}\rangle$ basis, the dynamical quantities, like local lattice
Green’s function $\hat{G}_{t_{2g}}(i\omega_{n})$ and self-energy
$\hat{\Sigma}_{t_{2g}}(\omega_{n})$, in the $t_{2g}$ basis can be readly
obtained via a unitary transformation

$\displaystyle\hat{G}_{t_{2g}}(i\omega_{n})=\hat{A}\hat{G}(i\omega_{n})\hat{A}^{{\dagger}},$
(12)

and

$\displaystyle\hat{\Sigma}_{t_{2g}}(i\omega_{n})=\hat{A}\hat{\Sigma}(i\omega_{n})\hat{A}^{{\dagger}}.$
(13)

Since ED has direct access to the real-frequency dynamical correlations as
discribed in Appendix B, we can calculate the local lattice Green’s function
$\hat{G}(i\omega+i\eta)$ after the self-energy is obtained via the Dyson
equation
$\hat{\Sigma}(\omega+i\eta)=\hat{g}^{-1}_{\text{imp}}(\omega+i\eta)-\hat{G}^{-1}_{\text{imp}}(\omega+i\eta)$,
where the impurity Green’s function $\hat{G}_{\text{imp}}(\omega+i\eta)$ is
directly produced by ED. Similar to the Matsubara Green’s function
$\hat{G}_{t_{2g}}(i\omega_{n})$, the local Green’s function
$\hat{G}_{t_{2g}}(\omega+i\eta)$ and self-energy
$\hat{\Sigma}_{t_{2g}}(\omega+i\eta)$ in the $t_{2g}$ basis are calculated as

$\displaystyle\hat{G}_{t_{2g}}(\omega+i\eta)=A\hat{G}(\omega+i\eta)A^{{\dagger}},$
(14)

and

$\displaystyle\hat{\Sigma}_{t_{2g}}(\omega+i\eta)=A\hat{\Sigma}(\omega+i\eta)A^{{\dagger}}.$
(15)

On the basis of $\hat{G}_{t_{2g}}(\omega+i\eta)$, the projected density of
states, as shown in the main text, is defined as

$\displaystyle\rho_{\alpha\sigma}(\omega)=-\frac{1}{\pi}\text{Im}\left(\hat{G}_{t_{2g}}(\omega+i\eta)\right)_{\alpha\sigma,\alpha\sigma},$
(16)

where $\alpha$ is the orbital index for $t_{2g}$ orbitals and $\sigma$ denotes
electron spin.

##  Appendix B Exact Diagonalization

When ED is employed as impurity solver of the DMFT, it needs two steps to
obtain the dynamical correlation functions. The first step is to calculate the
ground-state energy $E_{g}$ and corresponding eigenvector $|\rangle$ of the
Anderson impurity model (A) by the Lanczos method. On the basis of the Lanczos
method, the $E_{g}$ and $|\rangle$ can be obtained by iteratively constructing
a Krylov space $\\{|\phi_{n}\rangle\\}$ from an arbitrary initial
configuration $|\phi_{1}\rangle$ via

$\displaystyle|\tilde{\phi}_{n+1}\rangle=H_{\text{imp}}|\phi_{n}\rangle-
a_{n}|\phi_{n}\rangle-b^{2}_{n}|\phi_{n-1}\rangle,$ (17)

where n= 2, 3, 4, $\cdots$,
$a_{n}=\langle\phi_{n}|H_{\text{imp}}|\phi_{n}\rangle$,
$b^{2}_{n}=\langle\tilde{\phi}_{n+1}|\tilde{\phi}_{n+1}\rangle$. Note
$|\phi_{n}\rangle$ denotes a normalized vector, $b_{1}=0$,
$|\phi_{0}\rangle=0$,
$|\tilde{\phi}_{2}\rangle=H_{\text{imp}}|\phi_{1}\rangle-
a_{1}|\phi_{1}\rangle$ and
$a_{1}=\langle\phi_{1}|H_{\text{imp}}|\phi_{1}\rangle$. The iteration (17)
continues until $b_{n}$ is less than a threshold. In this basis, the
Hamiltonian for the Anderson impurity model (A) is a tridiagonal matrix and
simply reads

$\displaystyle\begin{bmatrix}a_{1}&b_{2}&0&0&\cdots\\\
b_{2}&a_{2}&b_{3}&0&\cdots\\\ 0&b_{3}&a_{3}&b_{4}&\cdots\\\
0&0&b_{4}&a_{4}&\cdots\\\ \vdots&\vdots&\vdots&\vdots\end{bmatrix},$ (18)

which can be diagonalized by the modern standard library subroutines.

The second step is to calculate the dynamical correlation function based on
the ground-state energy $E_{g}$ and ground-state eigenvector $|\rangle$
obtained in the first step. For the given operator $O$, the real-time
dynamical correlation function is defined as

$\displaystyle
C_{\alpha\beta}(t)=-i\theta(t)\langle|\left[O_{\alpha}(t),O_{\beta}\right]_{\xi}|\rangle$
(19)

where $\theta(t)$ is a step function of time $t$,
$O_{\alpha}(t)=e^{iH_{\text{imp}}t}O_{\alpha}e^{-iH_{\text{imp}}t}$,
$[]_{\xi}$ denotes the commutator of two operators, $\xi=1$ if $O$ is a Fermi
operator and $\xi=-1$ otherwise. After performing a Fourier transformation,
the dynamical correlation function on the real frequency axis can be written
as

$\displaystyle
C_{\alpha\beta}(\omega+i\eta)=C^{>}_{\alpha\beta}(\omega+i\eta)+\xi{}C^{<}_{\alpha\beta}(\omega+i\eta),$
(20)

where $\eta$ is a broadening factor and

$\displaystyle C^{>}_{\alpha\beta}(\omega+i\eta)$
$\displaystyle=\langle|O_{\alpha}\frac{1}{\omega-
H_{\text{imp}}+E_{g}+i\eta}O_{\beta}|\rangle,$ (21) $\displaystyle
C^{<}_{\alpha\beta}(\omega+i\eta)$
$\displaystyle=\langle|O_{\beta}\frac{1}{\omega+H_{\text{imp}}-E_{g}+i\eta}O_{\alpha}|\rangle.$
(22)

Similar to calculate the ground-state energy $E_{g}$ and eigenvector
$|\rangle$, the method of the Krylov space can be applied to calculate both
dynamical correlation functions above.

In order to calculate $C^{>}_{\alpha\beta}(\omega+i\eta)$, we start the
Lanczos iteratons with the initial vector
$|\phi_{1}^{\beta}\rangle=O_{\beta}|\rangle/\langle|O^{{\dagger}}_{\beta}O_{\beta}|\rangle$
to construct the new basis $\\{|\phi^{\beta}_{n}\rangle\\}$. By inserting the
completeness $\sum_{n}|\phi^{\beta}_{n}\rangle\langle\phi^{\beta}_{n}|=1$ into
equation (21), it yields

$\displaystyle C^{>}_{\alpha\beta}(\omega+i\eta)$
$\displaystyle=\sqrt{\langle|O_{\alpha}O^{{\dagger}}_{\alpha}|\rangle}\sqrt{\langle|O^{{\dagger}}_{\beta}O_{\beta}|\rangle}$
$\displaystyle\times\sum_{n}U^{\alpha\beta}_{n}V^{\beta}_{n}(\omega+i\eta)$
(23)

where

$\displaystyle
V^{\beta}_{n}(\omega+i\eta)=\langle\phi^{\beta}_{n}|\frac{1}{\omega-
H_{\text{imp}}+E_{g}+i\eta}|\phi^{\beta}_{1}\rangle,$ (24)

and $U^{\alpha\beta}_{n}=\langle\phi^{\alpha}_{1}|\phi^{\beta}_{n}\rangle$
with
$\langle\phi^{\alpha}_{1}|=\langle|O_{\alpha}/\sqrt{\langle|O_{\alpha}O^{{\dagger}}_{\alpha}|\rangle}$.
It is obvious that the main difficulties are to calculate
$V^{\beta}_{n}(\omega+i\eta)$ in the new basis. By taking advantage of the
completeness of $\sum_{n}|\phi^{\beta}_{n}\rangle\langle\phi^{\beta}_{n}|=1$
and the identity

$\displaystyle\langle\phi^{\beta}_{m}|(\omega-
H_{\text{imp}}+E_{g})\frac{1}{\omega-
H_{\text{imp}}+E_{g}}|\phi^{\beta}_{1}\rangle=\delta_{m,1},$ (25)

$V^{\beta}_{n}(\omega+i\eta)$ can be obtained through

$\displaystyle S^{\beta}_{mn}(\omega+i\eta)V^{\beta}_{n}(\omega+i\eta)=E_{m},$
(26)

where $E_{m}=\delta_{m,1}$ and $S^{\beta}_{mn}(\omega+i\eta)$ is a tridiagonal
matrix

$\displaystyle S^{\beta}_{mn}(\omega+i\eta)=\begin{bmatrix}\omega-
a_{1}+E_{g}+i\eta&-b_{2}&0&0&\cdots\\\ -b_{2}&\omega-
a_{2}+E_{g}+i\eta&-b_{3}&0&\cdots\\\ 0&-b_{3}&\omega-
a_{3}+E_{g}+i\eta&-b_{4}&\cdots\\\ 0&0&-b_{4}&\omega-
a_{4}+E_{g}+i\eta&\cdots\\\ \vdots&\vdots&\vdots&\vdots\end{bmatrix},$ (27)

where $a_{n}$ and $b_{n}$ are the coefficients generated by the Lanczos
iterations with the initial configuration $|\phi^{\beta}_{1}\rangle$. The
linear equations (26) can be easily solved by the standard library
subroutines. Finally, we can obtain the dynamical correlation function
$C^{>}_{\alpha\beta}(\omega+i\eta)$ by solving equations (23) and (26). As for
$C^{<}_{\alpha\beta}(\omega+i\eta)$, it can be obtained similarly.

When substituting the Matsubara frequency $i\omega_{n}$ for the real frequency
$\omega+i\eta$ in equation (20), the dynamical correlation functions
$C_{\alpha\beta}(i\omega_{n})$ on the Matsubara frequency axis can be obtained
by following the above two-step procedures. It suggests ED has direct access
to the dynamical quantities on both real and Matsubara frequency axes. This is
in sharp contrast to the quantum Monte Carlo, which cannot directly sample the
real-frequency dynamical correlations.

##  Appendix C Results with Eight Bath Sites Coupled to Three Kramers Doublets

Figure 7: (color online) The influence of spin-orbit coupling on the imaginary
part of Matsubara self-energy for $d_{xy}$ band, those for $d_{yz/xz}$ bands
are shown in the inset. (b) Density of states for $d_{xy}$ band in the Mott
($\lambda=0.0$), non-Fermi-liquid ($\lambda=0.12$) and the Fermi-liquid
($\lambda=0.16$) states. (c) Real and (d) imaginary part of self-energy of
$d_{xy}$ band on the real frequency axis. Here, the electron repulsion is
fixed at $U=2.51$.

In order to further investigate the effect of bath discretization on the OSnFL
phase, we have done the DMFT calculations with eight bath sites at zero
temperature using ED as the impurity solver, where four bath sites couple to
$|\frac{1}{2},\pm\frac{1}{2}\rangle$ and $|\frac{3}{2},\pm\frac{1}{2}\rangle$
Kramers doublets and the remained four bath sites couple to
$|\frac{3}{2},\pm\frac{3}{2}\rangle$ doublet. The calculated results at
$U=2.51$ are displayed in Fig. 7, which are similar to those obtained with six
bath sites. The OSnFL state can be clearly identified by the imaginary part of
self-energy $\textrm{Im}\Sigma(i\omega_{n})$ on the Matsubara frequency axis.
As shown in the inset of Fig. 7 (a), $\textrm{Im}\Sigma_{yz/xz}(i\omega_{n})$
for $d_{yz/xz}$ orbitals always extrapolates to zero when $\omega_{n}$ goes to
zero, suggesting the typical Fermi-liquid behavior in $d_{yz/xz}$ orbitals. In
contrast, at around $\lambda\approx 0.12$,
$\textrm{Im}\Sigma_{xy}(i\omega_{n})$ for $d_{xy}$ orbital approaches a finite
value at low frequencies, which is distinctly different from the divergent
behavior for $\lambda<0.8$ and the tendency to approach zero for
$\lambda>0.14$, indicating the occurrence of the Fermi-liquid breakdown in
$d_{xy}$ orbital due to the finite scattering rate at zero temperature.

The self-energy $\Sigma_{xy}(\omega)$ on the real frequency axis further
manifest the SOC-induced non-Fermi-liquid nature of $d_{xy}$ orbital. Fig. 7
(c) and (d) shows the real part of self-energy Re$\Sigma_{xy}(\omega)$ and the
imaginary part of self-energy Im$\Sigma_{xy}(\omega)$ at $U=2.51$ for
$\lambda=0.0$, $\lambda=0.12$ and $\lambda=0.16$. For the case of
$\lambda=0.12$, the positive slop of Re$\Sigma_{xy}(\omega)$ and finite value
of Im$\Sigma_{xy}(\omega)$ at the Fermi level indicate the breakdown of the
Fermi liquid at zero temperature. In contrast, the sharp slop of
Re$\Sigma_{xy}(\omega)$ and divergent Im$\Sigma_{xy}(\omega)$ at $\omega=0$
suggest a Mott-insulating state for $\lambda=0.0$ case, and the linearly
$\omega-$dependent Re$\Sigma_{xy}(\omega)$ and quadratically $\omega^{2}-$
dependent Im$\Sigma_{xy}(\omega)$ in the vicinity of the Fermi level indicate
a typical Fermi-liquid behavior for $\lambda=0.16$ case. Fig. 7 (c) displays
the density of states for $d_{xy}$ orbital in the Mott, non-Fermi-liquid and
Fermi-liquid states at $\lambda=0.0$, $0.12$ and $0.16$, respectively.

Figure 8: (color online) (a) Dynamical susceptibilities for spin ($\bf S$),
orbital ($\bf L$) and total ($\bf J$) angular momentums at $U=2.51$ and
$\lambda=0.12$. (b) Dynamical spin susceptibilities at $U=2.51$ for
$\lambda=0.0$, $\lambda=0.12$ and $\lambda=0.16$ cases, the inset shows
corresponding orbitally-resolved dynamical spin susceptibilities, where solid
lines denote $d_{yz/xz}$ orbitals and dashed lines $d_{xy}$ orbital.

In order to reveal the mechanism which is responsible for the appearance of
the OSnFL state, we have calculated the dynamical susceptibilities
$\chi^{ii}(\omega)$, defined by Eq (3) and (4), for total spin momentum
$\mathbf{S}$, total orbital angular momentum $\mathbf{L}$ and total angular
momentum $\mathbf{J}$ at $U=2.51$ and $\lambda=0.12$, as displayed in Fig. 8
(a). This result is similar to that obtained with totally six bath sites. It
is obvious that the low-energy excitations of the total angular momentum
$\mathbf{J}$ are mainly contributed by the spin fluctuations and the high-
energy ones are ascribed to the excitations of the total orbital angular
momentum $\mathbf{L}$. Since the low-energy excitations were found to dominate
the physical properties of the correlated multiorbital systems Song  _et al._
(2017); Kugler  _et al._ (2019), we now pay attention on the low-energy spin
fluctuations. Fig. 8 (b) depicts the spin susceptibilities at $U=2.51$ for
$\lambda=0.0$, $0.12$ and $0.16$ cases. As explained in the maintext, at
$\lambda=0.0$, the system is in the OSM state. The appearance of low-energy
spin excitations is due to the spins in $d_{yz/xz}$ orbitals screened by their
itinerant electrons and the formation of the local triplets with three
$t_{2g}$ orbitals filled by four electrons. At $\lambda=0.16$, the abrupt jump
in spin excitations is attributed to the presence of the additional Kondo
resonances in $d_{xy}$ orbital because the spin susceptibilities for
$d_{yz/xz}$ orbitals keep almost unchange, as displayed in the inset of Fig. 8
(b). In contrast, at $\lambda=0.12$, the spin susceptibilities are almost the
same with those at $\lambda=0.0$, indicating the spins in $d_{xy}$ band do not
participate in the Kondo screening. Obviously, the lack of the Kondo
resonances in $d_{xy}$ orbital leads to the non-Fermi liquid behavior.

In summary, the results calculated with eight bath sites are qualitatively
consistent with those obtained with six bath sites, suggesting the finite-size
effect is negligible.

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