#  Particle-hole asymmetric lifetimes promoted by spin and orbital
fluctuations
in SrVO3 monolayers

Matthias Pickem  0000-0002-0976-845X<EMAIL_ADDRESS>Institute for
Solid State Physics, TU Wien, Vienna, Austria  Jan M. Tomczak
0000-0003-1581-8799 Institute for Solid State Physics, TU Wien, Vienna,
Austria  Karsten Held  0000-0001-5984-8549 Institute for Solid State Physics,
TU Wien, Vienna, Austria

###### Abstract

The two-dimensional nature of engineered transition-metal ultra-thin oxide
films offers a large playground of yet to be fully understood physics. Here,
we study pristine SrVO3 monolayers that have recently been predicted to
display a variety of magnetic and orbital orders. Above all ordering
temperatures, we find that the associated non-local fluctuations lead to a
momentum differentiation in the self-energy, particularly in the scattering
rate. In the one-band 2D Hubbard model, momentum-selectivity on the Fermi
surface (”$k=k_{F}$”) is known to lead to pseudogap physics. Here instead, in
the multi-orbital case, we evidence a differentiation between momenta on the
occupied (”$k<k_{F}$”) and the unoccupied side (”$k>k_{F}$”) of the Fermi
surface. Our work, based on the dynamical vertex approximation, complements
the understanding of spectral signatures of non-local fluctuations, calls to
(re)examine other ultra-thin oxide films and interfaces with methods beyond
dynamical mean-field theory, and may point to correlation-enhanced
thermoelectric effects.

##  I Introduction

In the vicinity of phase transitions and in low-dimensional systems, non-local
long-range fluctuations are known to proliferate. These are not only crucial
for the critical behavior but may also lead to a strong enhancement of the
scattering rate, i.e., a dampening of the quasiparticle lifetime. In three
dimensions it is still debated Vilk and Tremblay (1997); Rohringer  _et al._
(2011); Rohringer and Toschi (2016) whether this scattering rate is actually
diverging at a phase transition or approaches a large but finite value. Even
more peculiar is the situation in two dimensions. There, an actual phase
transition—associated with the breaking of continuous symmetries—is
prohibited. Nonetheless non-local long-range fluctuation may still become huge
and can result in the famous pseudogap that has been experimentally observed
in cuprate superconductors Norman  _et al._ (1998); Timusk and Statt (1999);
Norman  _et al._ (2005); Keimer  _et al._ (2015). The pseudogap arises from a
pronounced momentum differentiation of the scattering rate at low energy. It
is largest in the anti-nodal direction where eventually a gap opens at low
enough temperatures. One possible explanation are long-range antiferromagnetic
spin fluctuations Kampf and Schrieffer (1990a); Vilk and Tremblay (1996); Rost
_et al._ (2012); Kyung  _et al._ (2004); Gull  _et al._ (2013); Cyr-Choinière
_et al._ (2018), with the momentum differentiation originating from the
perfect antiferromagnetic nesting at the hot spots Kampf and Schrieffer
(1990b); Monthoux and Pines (1993); Abanov  _et al._ (2003); Vilk and Tremblay
(1997); Vilk (1997); Wu  _et al._ (2020a), from the vicinity to a van Hove
singularity in the anti-nodal direction González  _et al._ (2000); Halboth and
Metzner (2000); Honerkamp and Salmhofer (2001); Wu  _et al._ (2020b), or from
the spin-fermion vertex turning complex at strong coupling Krien  _et al._
(2021). It has also been suggested, on the basis of model studies, that a
pseudogap phase can be driven by ferromagnetic fluctuations Fay  _et al._
(1988); Monthoux (2003); Hankevych  _et al._ (2003); Katanin  _et al._ (2005);
Katanin (2005).

(a) SrO-terminated monolayer (b) VO2-terminated monolayer

Figure 1: Phase diagrams. (a) SrO- and (b) VO2-terminated SrVO3-monolayer on
top of a SrTiO3 substrate (see insets for crystal structures) exhibit numerous
phases as a function of electrons per site ($n$; lower $x$-axis) in the low-
energy vanadium $t_{2g}$ orbitals or gate voltage ($V_{G}$; upper $x$-axis):
antiferromagnetism (AF: red), ferromagnetism (FM: blue), incommensurate
magnetism (iM: blueish), checkerboard orbital order (cOO: green), stripe
orbital order (sOO: turquoise). The colored domes mark the occurrence of long-
range order within dynamical mean-field theory (DMFT); adapted from Ref.
Pickem  _et al._ , 2021. The “$+$”–marks indicate points for which we present
D$\Gamma$A (and DMFT) data in the present paper.

While pseudogap physics is mostly associated with cuprates, it has also been
evidenced in other layered materialsKlebel-Knobloch  _et al._ (2021): iron
pnictidesXu  _et al._ (2011); Moon  _et al._ (2012); Zhou  _et al._ (2012);
Shimojima  _et al._ (2014) and chalcogenidesLin  _et al._ (2013), (layered)
nickelatesUchida  _et al._ (2011), and iridatesKim  _et al._ (2014). However,
the origin of momentum-differentiated scattering rates is far from understood.

Here, we study a material that is two-dimensional by engineering: a monolayer
of SrVO3. In the bulk, SrVO3 is a correlated paramagnetic metal Inoue  _et
al._ (1998); Mo  _et al._ (2003) with a correlation-induced kink Nekrasov  _et
al._ (2006); Byczuk  _et al._ (2007) linked to the effective Kondo temperature
Held  _et al._ (2013). Grown as a film, it is known to undergo a metal-
insulator transition below a critical thickness when deposited on an SrTiO3
Yoshimatsu  _et al._ (2010); Kobayashi  _et al._ (2017); Zhong  _et al._
(2015) or LSAT Gu  _et al._ (2014) substrate. We focus on a monolayer of SrVO3
grown on the common SrTiO3 substrate and consider two different terminations
of the film to the vacuum: VO2 and SrO, see insets of Fig. 1. Only the former
has been evidenced experimentally Okada  _et al._ (2017), but it could be
preferable to cover the films with a SrTiO3 capping layer which leads to a
structure more akin to the SrO termination. Such a capping layer also prevents
a surface reconstruction with oxygen adatoms, which result in a dead surface
layer Gabel  _et al._ (2021), at least for slightly thicker films. A preceding
study Pickem  _et al._ (2021), based on dynamical mean-field theory (DMFT)
Metzner and Vollhardt (1989); Georges and Kotliar (1992); Georges  _et al._
(1996), revealed a rich variety of orbital ordered and magnetic phases as a
function of doping, see Fig. 1. Experimentally, the phase diagrams could be
perused by applying a gate voltage.

In the present paper, we go beyond DMFT and study the effect of non-local
fluctuations on spectral properties using the dynamical vertex approximation
(D$\Gamma$A) Toschi  _et al._ (2007); Katanin  _et al._ (2009); Galler  _et
al._ (2017); Rohringer  _et al._ (2018). As for cuprates, we find that strong
long-range fluctuations lead to a substantial momentum-dependence in the self-
energy. In stark contrast to the cuprates, however, in ultrathin films of
SrVO3 the momentum differentiation does not distinguish momenta on the Fermi
surface but those perpendicular to it. For example for the SrO-termination and
antiferromagnetic spin fluctuations (above the red dome in Fig. 1a), occupied
states with momenta below the Fermi surface have a long, Fermi liquid-like
lifetime. Instead, unoccupied states above the Fermi surface have short
lifetimes and even kinks (downturns) in the self-energy, signaling a depletion
of states. For orbital-ordering and ferromagnetic fluctuations (above green
and blue regions) it is vice versa. The same is true for the VO2-termination
and the most relevant $xz$/$yz$ orbitals in the regime of orbital and
incommensurate magnetic fluctuations above half-filling ($n>1$ in Fig. 1b,
above the green and blue regions). Below half-filling, instead, non-local
correlations only have a minor impact on the self-energy of this termination.

A pronounced particle-hole asymmetry in the real part of the self-energy is a
common phenomenon, mostly owing to non-local exchange. For example, in $GW$
calculations Godby  _et al._ (1988); Miyake  _et al._ (2013); Tomczak  _et
al._ (2014) it leads to larger semiconductor band gaps than in density
functional theory. For the imaginary part of the self-energy (the scattering
rate), however, such an asymmetry has, to the best of our knowledge, not been
reported so far. The particle-hole asymmetric lifetimes arising here from a
momentum-selectivity of renormalizations can complement more ubiquitous
orbital-selective asymmetries and could drive large thermoelectric effects.

The outline of the paper is as follows: Section II provides information on the
employed electronic structure methods. Section III presents the DMFT Fermi
surfaces and spectral functions and the trends upon doping the SrVO3
monolayer. Non-local fluctuations in AbinitioD$\Gamma$A suppress ordering
instabilities but strong long-range fluctuations persist and affect spectra
and self-energies. An analysis of the AbinitioD$\Gamma$A results and the
evidenced momentum selectivity are presented in Section IV, and further
discussed in Section V. Finally, Section VI summarizes our conclusions.

##  II Method

The crystal structures used are identical to Ref. Pickem  _et al._ , 2021:
Density function theory (DFT) calculations are based on the WIEN2K package
Blaha  _et al._ (2001, 2020) with PBEPerdew  _et al._ (1996) as exchange-
correlation potential. We construct a slab as displayed in the insets of Fig.
1 consisting of one unit cell of SrVO3 on top six unit cells of the SrTiO3
substrate and surrounded (in $z$-direction) by sufficient vacuum of about
$10$Å to both sides. While the in-plane lattice constant of the
heterostructure is locked to the (theoretical) SrTiO3 substrateWang  _et al._
(2019) ($a^{\mathrm{PBE}}_{\mathrm{SrTiO}_{3}}=3.95$Å) all other internal
atomic positions are relaxed, except for the two unit cells of SrTiO3 furthest
away from SrVO3. The WIEN2K band-structure is then projected onto maximally
localized V-t2g Wannier orbitals, using the WIEN2Wannier Kunes  _et al._
(2010) interface to Wannier90 Mostofi  _et al._ (2008). The thus obtained
Wannier Hamiltonian is supplemented by a Kanamori interaction using $U=5$eV,
$J=0.75$eV, $U^{\prime}=3.5$eV, and solved by dynamical mean-field theory
(DMFT) Georges  _et al._ (1996); Held (2007). Doping is modeled by a
posterior-to-DFT adjustment of the chemical potential in DMFT. For DMFT
spectral functions, analytic continuation was performed with the maximum
entropy method implemented in ana_cont Kaufmann (2019); Kaufmann and Held
(2021). There, the hyper parameter $\alpha$ was determined with the chi2kink
method and a preblur window size of $\sigma=0.05$eV was employed.

In this paper we go beyond DMFT Tomczak  _et al._ (2017); Rohringer  _et al._
(2018) and treat non-local correlations in the SrVO3 film with
AbinitioD$\Gamma$A Galler  _et al._ (2017, 2018, 2019). Contrary to finite-
size cluster methods, the D$\Gamma$A approachToschi  _et al._ (2007); Katanin
_et al._ (2009); Rohringer  _et al._ (2018) and other, closely related
diagrammatic extensions of DMFT Kusunose (2006); Rubtsov  _et al._ (2008);
Rohringer  _et al._ (2013); Taranto  _et al._ (2014); Li (2015); Parcollet
_et al._ (2015); Rohringer  _et al._ (2018) are not limited to short-range
fluctuations. It well describes pseudogaps induced by antiferromagnetic
fluctuations in the one-band 2D Hubbard model Gull  _et al._ (2013); Sordi
_et al._ (2011); Sakai  _et al._ (2012); Schäfer  _et al._ (2016, 2015a);
Gunnarsson  _et al._ (2015); Gukelberger  _et al._ (2017); Schäfer  _et al._
(2021) and (quantum) critical behavior Rohringer  _et al._ (2011); Antipov
_et al._ (2014); Schäfer  _et al._ (2017, 2019). Orbital-ordering and
ferromagnetic fluctuations have hitherto not been studied by D$\Gamma$A or
other diagrammatic extensions of DMFT. For the AbinitioD$\Gamma$A, we here
calculate the local particle-hole irreducible vertex at DMFT self-consistency
by continuous-time quantum Monte Carlo simulations in the hybridization
expansion Werner  _et al._ (2006); Gull  _et al._ (2011) using w2dynamics
Wallerberger  _et al._ (2019) with worm sampling Gunacker  _et al._ (2015).
From this local vertex we subsequently calculate the particle-hole and
transversal particle-hole Bethe-Salpeter ladder diagrams, and, through the
Schwinger-Dyson equation, the non-local self-energy. This way we include non-
local correlation effects in the self-energy. The D$\Gamma$A chemical
potential was readjusted to fix the total number of electrons to the
considered doping level. In this study we apply D$\Gamma$A in a one-shot
setting, forgoing non-local self-consistency Kaufmann  _et al._ (2021). For a
review of the method, see Ref. Rohringer  _et al._ , 2018; for computational
details of the AbinitioD$\Gamma$A see Ref. Galler  _et al._ , 2019. D$\Gamma$A
and DMFT Fermi surfaces were obtained from the Green’s function at imaginary
time $\tau=\beta/2$ ($\beta=1/k_{B}T$). This procedure corresponds to a
spectral function $\overline{A(\mathbf{k},\omega=0)}$ that is averaged over a
frequency-interval $\sim k_{B}T$ around the Fermi level.

##  III DMFT: orbital effects

The DMFT phase diagram Fig. 1 shows, as a function of doping and surface
termination, a rich variety of different magnetic and orbitally ordered phases
Pickem  _et al._ (2021). Non-local fluctuations will strongly suppress the
DMFT phase transitions in two dimensions but lead, at the same time, to strong
scattering rates and self-energy corrections. In Sec. IV, we will study these
renormalizations using the D$\Gamma$A at the $(n,T)$-points indicated in Fig.
1. The temperatures have been chosen so that we are close to the respective
phase transitions in DMFT and, thus, can expect pronounced non-local
correlations111Due to the multi-orbital nature and the temperature scaling of
the D$\Gamma$A we are restricted in temperature, i.e. we are not able to
approach the FM instability much further.. Before turning to these D$\Gamma$A
results, in this Section we first analyze the DMFT Fermi surfaces and
${\mathbf{k}}$-integrated spectra at the same fillings (orbitally resolved
occupations, DMFT susceptibilities, and ${\mathbf{k}}$-integrated DMFT spectra
at filling $n=1$ with and without crystal field splitting have already been
presented in Ref. Pickem  _et al._ , 2021). In DMFT, non-local fluctuations
are not included and thus do not affect the self-energy and spectral function.
As a consequence, approaching the ordered states does not result in a
pronounced temperature dependence of the DMFT spectra and self-energy.

Figure 2: SrO-terminated monolayer – DMFT spectral functions $A(\omega)$ for
various fillings $n$ and temperatures $T$: in the Mott insulating state
($n=1.0$), in the vicinity of the AF ($n=0.9,1.1$), FM ($n=1.3$) and cOO
($n=1.5$) phases, resolved into orbital characters ($xy$ and $xz$/$yz$).
Colored boxes around fillings indicate the type of long-range orders realized
at lower $T$, in correspondence to Fig. 1. At nominal filling ($n=1$; orange)
an orbitally polarized Mott insulator forms Pickem  _et al._ (2021); Zhong
_et al._ (2015).

Figure 3: SrO-terminated monolayer – DMFT Fermi surface for the points
highlighted in Fig. 1a: $n=0.9$ ($T=290$K) and $n=1.1$ ($T=290$K) order AF at
low $T$ (red box indicating the color code of Fig. 1), FM at $n=1.3$
($T=190$K; blue box), and cOO at $n=1.5$ ($T=230$K; green box).

###  III.1 SrO termination

Fig. 3 shows the DMFT Fermi surface for the SrO-terminated SrVO3 monolayer at
four different dopings (left to right). The upper panels display the
contribution of the $xy$ orbital and the lower panels the $xz$ orbital (the
$yz$ orbital is equivalent to the latter if rotated by 90∘).

Figure 4: VO2-terminated monolayer – DMFT spectral functions $A(\omega)$ for
various fillings $n$ and temperature $T$: $n=0.85$: sOO; $n=0.94$: cOO;
$n=1.0$: Mott insulating cOO; $n=1.23$: cOO; $n=1.3$: iM. Otherwise identical
to Fig. 3.

Figure 5: VO2-terminated monolayer – DMFT Fermi surface for the points
highlighted in Fig. 1b: $n=0.94$ ($T=290$K) and $n=1.23$ ($T=290$K) order cOO
at low $T$ (green box indicating the color code of Fig. 1), $n=0.85$ sOO
($T=210$K; turquoise box), and $n=1.3$ iM ($T=190$K; blue box).

We find the stoichiometric sample ($n=1$) to be an orbitally polarized
insulatorZhong  _et al._ (2015) with a gap of about 1 eV, see Fig. 3. That is,
the in-plane ${xy}$ orbital is essentially half-filled, while the ${xz}$,
${yz}$ orbitals are almost completely depleted. Hence, the undoped SrO-
terminated SrVO3 monolayer is an effective one-orbital system. The reduced
orbital-degeneracy (with respect to the bulks threefold $t_{2g}$ orbitals)
leads to a smaller critical interaction for the Mott stateGunnarsson  _et al._
(1996); Pavarini  _et al._ (2004). This turns the undoped SrVO3 monolayer Mott
insulating with strong antiferromagnetic (AF) fluctuations.

Doping with 10% electrons or holes, we obtain a metal, see the panels with
$n=0.9$ and $n=1.1$, respectively, in Fig. 3 and Fig. 3. The ${xz}$ and ${yz}$
orbitals are now slightly filled, pushing the AF phase transition to lower
$T$, see Fig. 1, while strong AF spin fluctuations persist. The
${\mathbf{k}}$-integrated spectral function in Fig. 3 further shows that the
${xz}$ and ${yz}$ orbitals, while only slightly filled, already contribute a
sizable amount to the quasi-particle peak at the Fermi level.

Similarities of this system to high-$T_{c}$ cuprates are uncanny: While, here,
the low-energy physics is dominated by a half-filled ${xy}$ orbital instead of
the ${x^{2}-y^{2}}$ orbital in cuprates, we find a ratio of nearest to next-
neighbor in-plane hopping $t^{\prime}/t=+0.31$, ($t=-0.237\,$eV,
$t^{\prime}=-0.073\,$eV) which is essentially the same as found for YBa2Cu3O7
and Bi2Sr2CaCu2O8 Pavarini  _et al._ (2001), but $t^{\prime}$ has the opposite
sign. In a one-band picture, one can compensate for the opposite sign by
making a particle-hole transformation and we obtain an electron-like Fermi
surface instead of a hole like one in cuprates. The decisive difference is
however that, upon doping, the $xz$/$yz$ orbitals become partially filled,
leading to a different, multi-orbital kind of physics.

Indeed, at larger doping, $n=1.5$, Fig. 1 indicates a checkerboard orbital-
order (cOO) in DMFT with a spatially alternating occupation of the $xz$ and
$yz$ orbitals, whereas the $xy$ orbital does not participate in the cOO. Here,
the $xy$ and $yz$ orbitals are already almost as much filled as the $xy$
orbital as is evident from Fig. 3 and also from the Fermi surfaces in Fig. 3.
As the $xz$ ($yz$) lobes point in the $x-$ ($y-$) and $z$-direction, their
Fermi surface in Fig. 3 is highly asymmetric, whereas their
${\mathbf{k}}$-integrated spectrum in Fig. 3 is similar to that of the $xy$
orbital. In-between, around $n=1.3$, the $xz$ and $yz$ orbitals are still
significantly less filled, however the spectral function at the Fermi level
$A(\omega=0)$ is strongly enhanced, see Fig. 3. Ferromagnetic (FM) order
therefore develops in Fig. 1 from the interplay of the Hund’s coupling $J$ and
the hopping $t$ Pickem  _et al._ (2021).

###  III.2 VO2 termination

We now turn to the DMFT electronic structure of the VO2-terminated surface.
Again, we show Fermi surfaces (Fig. 5) and ${\mathbf{k}}$-integrated spectra
(Fig. 5) for varying doping. For the VO2\- instead of the SrO-termination to
the vacuum, the crystal-field splitting between the $xz$/$yz$ and the $xy$
orbital flips its sign Pickem  _et al._ (2021). That is, the $xy$ orbital now
lies above the $xz$/$yz$ orbitals. At $n=1$, the latter accommodate all of the
charge and their spectrum is split into upper and lower Hubbard bands, see
Fig. 5, whereas the $xy$-orbital is unoccupied. The two degenerate $xz$ and
$yz$ orbitals are at or near quarter filling around $n=1$. This gives rise to
checkerboard orbital fluctuations and, at low enough temperature, ordering
(cOO) in DMFT, see Fig. 1. For slight hole doping and substantial electron
doping, the cOO tendencies remain intact, but the SrVO3 layer turns metallic.
Inverting the role of the $xz$/$yz$ and the $xy$ orbitals compared to the SrO-
termination, we now observe a small hole pocket for the $xy$ orbital in Fig.
5, in agreement with their small filling in Fig. 5.

Reducing the filling from $n=0.94$ to $n=0.85$, this $xy$ Fermi-surface pocket
becomes slightly enhanced, albeit it remains small in Fig. 5. As for the
fluctuations: because of their reduced filling, the $xz$/$yz$ orbitals are no
longer quarter-filled. Thus cOO gives way, first, to stripe orbital order
(sOO) at $n=0.85$ and, eventually, at lower fillings to FM, similar as for the
two-band Hubbard model Held and Vollhardt (1998).

Further electron doping from $n=1.23$ to $n=1.3$ instead changes the DMFT
ordered state from cOO to incommensurate magnetism (iM) with a small
${\mathbf{q}}$-vector in Fig. 1, see Ref. Pickem  _et al._ , 2021. It has,
however, little effect on the spectral function and the Fermi surface in Fig.
5 and Fig. 5, respectively. The sharper Fermi surface for $n=1.3$ can be
explained by the slight decrease in the temperature and the fact that local
DMFT correlations get reduced the farther we are away from half-filling, see
Fig. 9 below.

Figure 6: SrO-terminated monolayer – Momentum differentiation of the
D$\Gamma$A self-energy. Top 4 rows: real and imaginary parts for the $xy$ and
$xz$ orbital at 4 different momenta, compared to DMFT, for the dopings and
temperatures indicated in Fig. 1a. Bottom row: Slope of the imaginary part of
the D$\Gamma$A self-energy for a path through the Brillouin zone. Negative
values correspond to a Fermi-liquid like self-energy, positive values indicate
the formation of a (pseudo)gap. Figure 7: SrO-terminated monolayer –
D$\Gamma$A Fermi surfaces at the same dopings and $T$ as in Fig. 3. The white
contour lines represent isolines of the slope between the first two Matsubara
frequencies of the imaginary part of $\Sigma$: Solid, fat lines indicate a
_positive_ slope, i.e., a kink in the self-energy, dashed, thin lines a
_negative_ value, suggestive of a Fermi-liquid-like state.

##  IV D$\Gamma$A: momentum differentiation

On the dynamical mean-field level, many-body renormalizations are assumed to
be isotropic (i.e., independent of momentum). In 3D this is mostly a good
approximation (see, however, Ref. Schäfer  _et al._ , 2015b). Yet, when the
effective dimensionality is reduced, as in our ultrathin film,
renormalizations become increasingly non-local Klebel-Knobloch  _et al._
(2021). The major question we will answer here is: To what extent do the non-
local critical fluctuations—in the vicinity of the associated ordered
states—lead to momentum-selective renormalizations? To elucidate this
question, we use the AbinitioD$\Gamma$AGaller  _et al._ (2017, 2018, 2019)
methodology and scrutinize the electron self-energy $\Sigma(\hbox{{k}},i\nu)$
in the vicinity of the DMFT ordering instabilities summarized above.

###  IV.1 SrO termination

Fig. 6 shows the real and imaginary part of the AbinitioD$\Gamma$A self-energy
in the vicinity of the DMFT phase transitions where non-local correlations
become strong. Shown are the two inequivalent orbitals, $xy$ (top) and $xz$
(middle panel) as a function of Matsubara frequency $\nu_{n}$. The $yz$
orbital is equivalent to the $xz$ orbital if the momenta are rotated by 90∘
rotated; the DMFT self-energy is shown for comparison.

In the vicinity of half-filling, $n=0.9$ and $n=1.1$, AF spin fluctuations
prevail with leading eigenvalue $\lambda_{M}(\pi,\pi)=0.95$ and $0.79$,
respectively, in the magnetic (M) channel at ${\mathbf{q}}=(\pi,\pi)$. Note,
$\lambda=1$ indicates a divergence of the susceptibility, i.e., an ordering
instability. These AF spin fluctuations are driven by the $xy$ orbital that is
close to half filling, whereas the $xz$ and $yz$ orbitals rather act as
passive bystanders Pickem  _et al._ (2021). Consequently, we see for $n=0.9$
and $n=1.1$ in Fig. 6 a pronounced momentum differentiation only for the $xy$
orbital.

The Matsubara frequency self-energy has the advantage that it does not require
the ill-conditioned analytic continuation. Nonetheless, we can gain valuable
information: The momentum differentiation of the real part of the self-energy
in Fig. 6 between unoccupied [${\mathbf{k}}=(0,0)$, red] and occupied states
[${\mathbf{k}}=(\pi,\pi)$, green] signals that the quasi-particle poles at
$\omega+\mu=\hbox{Re}\Sigma+\epsilon_{\mathbf{k}}$ are pushed further away
from the Fermi energy, causing an overall enhancement of the bandwidth. The
momentum differentiation between ${\mathbf{k}}=(0,\pi)$ (blue) and
${\mathbf{k}}=(\pi/2,\pi/2)$ (orange) that are closer to the Fermi level,
indicates a deformation of the Fermi surface for $n=0.9$, but not for $n=1.1$
which has a similar self-energy for these two ${\mathbf{k}}$-points. Indeed a
deformation is observed in Fig. 7, where the electron-like DMFT Fermi surface
(Fig. 3) turns into a hole-like one in D$\Gamma$A for $n=0.9$. For $n=1.3$
with strong FM fluctuations ($\lambda_{M}(0,0)=0.78$) and $n=1.5$ with strong
cOO fluctuations in the density (D) channel ($\lambda_{D}(\pi,\pi)=0.98$), the
momentum differentiation of $\hbox{Re}\Sigma$ is less pronounced.

Let us now turn to $\hbox{Im}\Sigma$ from which we can read off the scattering
rate, as the $\nu_{n}\rightarrow 0$-extrapolated value. Further, from its
slope the quasi-particle renormalization
$Z_{\mathbf{k}}=[1-\partial\hbox{Im}\Sigma({\mathbf{k}},i\nu)/\partial\nu|_{\nu\rightarrow
0}]^{-1}$ is accessible for a Fermi liquid phase. A positive slope of
$\hbox{Im}\Sigma(i\nu\rightarrow 0)$ indicates the crossover to a diverging
(Mott-like) self-energy, which splits the spectrum and leads to an insulating
gap.

Clearly, for all four fillings shown in Fig. 6, there are momenta for which
the system exhibits non-Fermi liquid behavior, identifiable by a kink and a
downturn in $\hbox{Im}\Sigma$ at low energies. In case of AF fluctuations
($n=0.9$ and $n=1.1$) this downturn is in the $xy$ orbital, whereas it occurs
in the $xz$ (and $yz$) orbital which dominates the FM ($n=1.3$) and cOO
($n=1.5$) fluctuations. These kinks are salient indicators for the occurrence
of a pseudogap state, and they get more pronounced when cooling the system
toward the respective phase transition.

Interestingly, in the vicinity of the AF phase, the structure of the
scattering rate is opposite to the cuprates: It is larger for the diagonal
($\pi$,$\pi$) direction than for the (0,$\pi$) direction. This momentum
differentiation on the Fermi surface is, however, much less pronounced than
the momentum dependence perpendicular to the Fermi surface, i.e., comparing
occupied vs. unoccupied states.

This can be seen in Fig. 6 (bottom), where we plot the slope between the first
two positive Matsubara frequencies, i.e.,
$slope={(\hbox{Im}\Sigma(i\nu_{1})-\hbox{Im}\Sigma(i\nu_{0}))}\beta/(2\pi)$,
along the indicated k-path. Isolines of this slope are superimposed on the
D$\Gamma$A Fermi surfaces in Fig. 7, with the sign indicated by solid, fat
(positive) and dashed, thin (negative) lines. In the electron doped regime,
the slope in $\hbox{Im}\Sigma$ is always negative on the Fermi surface, i.e.,
Fermi liquid-like. However, when moving away from the Fermi energy, we observe
positive slopes, which corresponds to the kinks in Fig. 6;: at $n=1.1$ for the
unoccupied $xy$ states above the Fermi level; and at $n=1.3$ and $n=1.5$ for
the occupied $xz$ states. In the hole doped regime, at $n=0.9$, we find
$\hbox{Im}\Sigma$ isoline patterns similar to $n=1.1$. However, owing to the
larger $xy$-occupation in combination with the equally strong reconstruction
through $\hbox{Re}\Sigma$, negative slopes of $\hbox{Im}\Sigma$ instead appear
across the transformed $xy$ Fermi surface. This insulating-like behavior is
found only in the most relevant orbitals, i.e., the $xy$ orbital for the AF
fluctuations around $n=1$, and the $xz/yz$ orbitals where FM and cOO long-
range fluctuations are dominant. The ancillary orbitals ($xz/yz$ for $n=1.1$
and $xy$ for $n=1.3,1.5$) on the other hand exhibit only a comparatively minor
momentum differentiation (see Fig. 6)—implying also a stark orbital
differentiation.

Figure 8: VO2-terminated monolayer – momentum differentiation of the
D$\Gamma$A self-energy. Top 4 rows: real and imaginary parts for the $xy$ and
$xz$ orbital for 4 different momenta (colors) and, for comparison, the DMFT
self-energy (dashed) at the four dopings and $T$ indicated by the “+” in Fig.
1b. Bottom row: Slope of the imaginary part of the D$\Gamma$A self-energy for
a momentum path through the Brillouin zone. Figure 9: VO2-terminated monolayer
– D$\Gamma$A Fermi surfaces at the same dopings and $T$ as in Fig. 5. Isolines
are again the low-energy slope of $\hbox{Im}\Sigma$, where the solid, fat
lines represent a positive, and the dashed, thin lines represent a negative
value.

###  IV.2 VO2 termination

The corresponding AbinitioD$\Gamma$A results for the VO2-terminated SrVO3
monolayer for the self-energy and the Fermi surface are presented in Fig. 8
and Fig. 9, respectively. For cOO fluctuations at $n=1.23$
($\lambda_{D}(\pi,\pi)=0.97$), the momentum differentiation of the self-energy
and Fermi surface are qualitatively similar to the cOO results at $n=1.5$ for
the SrO-terminated layer. But for the cOO at $n=0.94$ and sOO at $n=0.85$, we
only find a minor momentum differentiation of the self-energy, see Fig. 8.
Correspondingly, the Fermi surface in Fig. 9 is similar to that of DMFT in
Fig. 5, and there are no positive non-Fermi-liquid like slopes (solid lines in
Fig. 9). This is surprising since the leading eigenvalue
$\lambda_{D}(0,\pi)=\lambda_{D}(\pi,0)=0.985$ at $n=0.85$ and
$\lambda_{D}(\pi,\pi)=0.91$ at $n=0.94$ is similarly close to 1 as for
$n=1.23$ or the SrO-termination, indicating that strong orbital ordering
fluctuations are present.

On the contrary, at $n=1.3$, above iM order, we observe the by far strongest
momentum differentiation in Fig. 8, even though
$\lambda_{M}(\delta,\delta)=0.97$ with $\delta\approx\pm\pi/4$ is again
comparable to the strength of other fluctuations. A clear pole develops in the
vicinity of the Fermi level not only for the $xz$ and $yz$ orbitals, that
drive the iM ordering, but also for the $xy$ orbital. This pole is so large
that the spectrum splits into two parts, akin to the splitting into upper and
lower Hubbard band; and it pushes the Fermi surface to
${\mathbf{k}}=(0,\pm\pi)$. However, the divergence occurs only for a region of
the Brillouin zone that does not account for the Fermi surface of the
respective orbital character.

##  V Discussion and Perspective

Recapitulating, we have studied a SrVO3 monolayer on a SrTiO3 substrate with
two different surface terminations, SrO and VO2, to vacuum within
AbinitioD$\Gamma$A. Depending on the termination and filling, there are strong
non-local fluctuations of various kinds: antiferromagnetic, ferromagnetic,
incommensurate magnetic, striped or checkerboard orbital. These non-local
fluctuations will suppress the mean-field DMFT ordering but also have
pronounced effects on the self-energy—the focus of the present paper. They can
deform the Fermi surface, as observed for antiferromagnetic ordering with
$n=0.9$ for SrO-terminated SrVO3, and quite generally can lead to a strong
enhancement of $\hbox{Im}\Sigma$. Strong non-local fluctuations can even cause
the development of a pole in the self-energy, signaling the splitting of the
spectrum into two parts—here not because of Mott-Hubbard physics but because
of large non-local fluctuations. The latter is particularly strong for the
incommensurate ferromagnetic phase of the VO2-terminated SrVO3 monolayer at
$n=1.3$ filling. First indications, i.e., downturns of the self-energy at the
lowest Matsubara frequency are however ubiquitous for various dopings and both
terminations. While such non-local physics have been investigated quite
intensively for antiferromagnetic fluctuations in the Hubbard model in the
context of the cuprates, to the best of our knowledge it has not been analyzed
before for orbital fluctuations.

The undoped ($n=1$) SrVO3 monolayer is Mott insulating and, for SrO-
termination, appears to be akin to the cuprates with the $xy$ orbital playing
the role of the high-$T_{c}$’s $x^{2}-y^{2}$ orbital. However, electron-doping
reveals the multi-orbital physics of the SrVO3 system: The $xy$ orbital is
depopulated when adding electrons to the system, and all three orbitals, $xy$,
$xz$ and $yz$, participate in developing a quasi-particle resonance at the
Fermi level.

For the cuprates, AF fluctuations lead to pseudogap physics with a momentum
differentiation distinguishing between a Fermi liquid-like self-energy in the
nodal direction on the Fermi surface, and a kink in the self-energy signaling
the opening of a gap in the anti-nodal direction. Here, we also observe the
joint presence of these two behaviors in the self-energy. However, the
momentum differentiation is not realized on the Fermi surface but
perpendicular to it: For the SrO-termination in the electron-doped regime, AF
fluctuations lead to a Fermi liquid-like behavior for momenta on the occupied
side of the Fermi surface (”$k<k_{F}$”) and a kink-like insulating behavior in
the imaginary part of the self-energy on the unoccupied side (”$k>k_{F}$”). In
case of FM and OO fluctuations, the momentum differentiation between occupied
and unoccupied momenta is reversed.

For the VO2 termination, iM fluctuations at $n=1.3$ lead to massive non-local
correlations and a pole in the self-energy. In contrast to all other cases not
only the $xz$/$yz$ orbitals—driving the iM fluctuations—are affected but also
the ancillary $xy$ orbital. Below half filling, on the other hand, cOO and sOO
fluctuations only result in a minor momentum differentiation of the self-
energy for the VO2-terminated monolayer.

The imaginary part of the self-energy corresponds to the lifetimes and the
broadening of the spectral function. Our results hence show that the lifetimes
of an added hole or electron are extremely different. The hole-lifetime can be
measured by angular resolved photoemission spectroscopy (ARPES); the electron
lifetime by inverse photoemission spectroscopy, by ARPES at elevated
temperatures, or in non-equilibrium situations (e.g., pump-probe measurements)
in which states above the Fermi level become populated.

The differentiation between states above and below the Fermi surface that we
observe is quite extreme. Technologically this might be exploited for
thermoelectrics which rely on a strong electron-hole asymmetry Mahan and Sofo
(1996); Held  _et al._ (2009); Zlatic and Monnier (2014); Tomczak (2018).
Particularly beneficial are sharp peaks in the spectral function on only one
side of the Fermi level Mahan and Sofo (1996), as found for the SrO-terminated
monolayer, see Fig. 3, within DMFT. There, local electronic correlations can
enhance thermoelectricity through energy-dependent renormalizations that are
different for electrons ($\omega<0$) and holes ($\omega<0)$Haule and Kotliar
(2009); Held  _et al._ (2009). Our finding of a momentum-selectivity in the
scattering rate may provide an additional route: A particle-hole asymmetry
that is driven (or enhanced) by non-local renormalizations. Indeed, looking
again at the SrO-terminated monolayer, OO and FM fluctuations at $n=1.5$ and
$n=1.3$, respectively, drive a dispersive scattering rateSun  _et al._ (2013)
that is larger for occupied momenta (”$k<k_{F}$”) then for empty states
(”$k>k_{F}$”): Specifically, the downward kinks in the $xz$-component of
$\hbox{Im}\Sigma$, see Fig. 6, occur for $\mathbf{k}=(0,0)$ and
$\mathbf{k}=(0,\pi)$ which are inside the (DMFT) $xz$ Fermi surface, see Fig.
3. For $\mathbf{k}=(\pi,\pi)$ and $\mathbf{k}=(\pi/2,\pi/2)$, which are
outside the Fermi surface, the $xz$ scattering rate instead decreases when
approaching zero frequency. This electron-hole asymmetry of the scattering
time will make the already electron-like DMFT thermopower even more negative,
thus increasing its magnitude.

##  VI Conclusions

Based on simulations of oxide ultrathin films, we demonstrated that, in
(quasi-) two-dimensional systems, strong long-range fluctuations are quite
generically reflected in a momentum differentiation of the self-energy,
irrespective of the dominant fluctuation channel (spin/charge/orbital).
Further, we demonstrated that this momentum differentiation has a much richer
structure than the current focus on cuprates and pseudogap physics suggests:
Strong variations in renormalizations may not only occur on the Fermi surface
but also perpendicular to it. Our results call for a (re)examining—with
beyond-DMFT methods—of correlated electron systems that host strong non-local
fluctuations: layered materials as well as ultra-thin oxide films and
heterostructures.

###### Acknowledgements.

We thank M. Fuchs, A. Galler, J. Kaufmann, G. Sangiovanni, P. Thunström and Z.
Zhong for fruitful discussions. The authors acknowledge support from the
Austrian Science Fund (FWF) through grants P 30819, P 32044, and P 30213.
Calculations were performed on the Vienna Scientific Cluster (VSC).

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