# L-hole Pockets of the Palladium Fermi Surface Revealed by Positron
Annihilation Spectroscopy

Michael Sekania  Institut für Physik, Martin-Luther Universität Halle-
Wittenberg, 06120 Halle/Saale, Germany  Theoretical Physics III, Center for
Electronic Correlations and Magnetism, Institute of Physics, University of
Augsburg, 86135 Augsburg, Germany  Andronikashvili Institute of Physics,
Javakhishvili Tbilisi State University, 0177 Tbilisi, Georgia
and Faculty of Natural Sciences and Medicine, Ilia State University, 0162
Tbilisi, Georgia  Andreas Östlin  Theoretical Physics III, Center for
Electronic Correlations and Magnetism, Institute of Physics, University of
Augsburg, 86135 Augsburg, Germany  Augsburg Center for Innovative
Technologies, University of Augsburg, 86135 Augsburg, Germany  Wilhelm H.
Appelt  Theoretical Physics III, Center for Electronic Correlations and
Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg,
Germany  S. B. Dugdale  H. H. Wills Physics Laboratory, University of Bristol,
Tyndall Avenue, Bristol, BS8 1TL, United Kingdom  Liviu Chioncel  Theoretical
Physics III, Center for Electronic Correlations and Magnetism, Institute of
Physics, University of Augsburg, 86135 Augsburg, Germany  Augsburg Center for
Innovative Technologies, University of Augsburg, 86135 Augsburg, Germany

###### Abstract

Using the combined Density Functional and Dynamical Mean Field theory we study
relativistic corrections to the Fermi surface of palladium. We find indeed
that relativistic corrections create a small hole pockets at the $L$-symmetry
points. Furthermore we show that the computed two dimensional Angular
Correlation of Electron Positron Annihilation Radiation (the so called
$2D$-ACAR) clearly demonstrates the existence of these $L$-hole pockets, which
remains robust against electronic correlations. A $2D$-ACAR experiment should
therefore provide the “smoking-gun” proof for the existence of the $L$-hole
pockets in the palladium Fermi surface.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Despite its partially filled $d$-shell, elemental palladium (Pd) is not
ferromagnetic. It belongs, however, to a class of materials called a nearly
ferromagnetic metals Moriya (1985); Lonzarich and Taillefer (1985). At low
temperatures, Pd possesses the highest density of states (DOS) at the Fermi
energy ($E_{\mathrm{F}}$) among all transition metals. The Stoner criterion is
almost fulfilled van Leeuwen et al. (1992) and therefore Pd is considered to
be situated at the verge of ferromagnetism. Nearly ferromagnetic metals
exhibits strong magnetic fluctuations which are believed to have a great
impact on macroscopic quantities such as the heat capacity and magnetic
susceptibility Moriya (1985).

Early band structure calculations predicted that the Fermi surface consists of
four sheets. An electron sheet centered at the $\Gamma$-point, open-hole
sheets and hole pockets centered at $X$, and very small hole pockets at $L$
Andersen (1970); Mueller et al. (1970). Experimental de Haas-van Alphen (dHvA)
studies Windmiller et al. (1971) identified the $\Gamma$ and $X$-centered
sheets but failed to capture the small $L$-pockets. The detailed examination
of the data did not allowed for an adequate fit of the ellipsoid centered at
$L$ Windmiller et al. (1971). A possible explanation was provided by Andersen
Andersen (1970) who noted the existence of a saddle point (shallow band) in
the dispersion (band structure) near the $L$-point which leads to a large
effective mass and low velocities, unusual for such a small size pockets.
Unlike dHvA studies, ultrasonic attenuation experiments found indications of
existence Brown et al. (1972) of $L$-hole pockets. Subsequent analysis of dHvA
measurements taken at higher fields and lower temperatures in combination with
parametrization of the Fermi surface was able to observe signals also from the
small $L$ hole pockets Dye et al. (1981); Joss and Crabtree (1984). Most of
the theoretical studies modeling the Fermi surface of Pd use Density
Functional Theory (DFT) Jones and Gunnarsson (1989); Kohn (1999); Jones
(2015). Electronic interaction effects are included through approximate
exchange-correlation potentials such as the Local Density Approximation (LDA)
or its gradient corrected version (GGA). For paramagnetic correlated ($3d$ and
$4f$) electron systems it is well known that LDA/GGA calculations fail to
provide the correct ground state properties. A quantitative theory for the
explanation of the electronic structure and the physical properties of such
systems has been consistently developed during the last two decades in the
form of a combination of density functional theory and Dynamical-Mean Field
Theory (DMFT) Metzner and Vollhardt (1989); Georges et al. (1996); Kotliar and
Vollhardt (2004); Kotliar et al. (2006), which is generally referred to as
LDA+DMFT Kotliar et al. (2006); Held (2007). In the LDA+DMFT scheme, the LDA
provides the ab initio, material dependent input (orbitals and hopping
parameters), while the DMFT solves the many-body problem for the local
interactions. Previously Östlin et al. (2016) we have applied the LDA+DMFT
method to Pd and showed that the agreement between the experimentally
determined and the theoretical lattice constant and bulk modulus is improved
when correlation effects are included. Furthemore we found that correlations
modify the Fermi surface around the neck at the $L$-point while small
correlation effects are seen on the tube structures of the Fermi surface. For
relatively weak Coulomb interactions the spectral functions, obtained within
the LDA+DMFT and GW van Schilfgaarde et al. (2006); Kotani et al. (2007)
methods, show no major differences. Although in our studies a perturbative
DMFT impurity solver was employed in the LDA+DMFT computations, our results
agree well also with those obtained using non-local many-body impurity solvers
Savrasov et al. (2018). Using the same LDA+DMFT method we computed the changes
in the phonon dispersions and phonon density of states of Pd under a hydro-
static pressure. We found that electronic correlations at finite temperatures
strongly renormalize the phonon frequencies and are the source of the so-
called Kohn anomaly Appelt et al. (2020). The wave-vectors at which the Kohn
anomaly arise are the nesting vectors of the Fermi surface which for Pd are
the tube structures along $X-W$ direction. In contrast to DFT results, for
which the phonon frequency remains essentially constant in a large temperature
range, correlation effects reduce the restoring force of the ionic
displacements at low temperatures, leading to a mode softening Appelt et al.
(2020).

In this paper we report results of the Two Photon Momentum Density [TPMD or
$\rho^{2\gamma}({\bf p})$] emerging from the process of annihilation of
electron-positron pairs in bulk Pd. The annihilation process of electron-
positron pairs in solids is well understood West (1973); Karshenboim (2005):
As a consequence of momentum and energy conservation electron-positron pairs
decay predominantly into two $\gamma$-quanta (photons). The resulting
$\rho^{2\gamma}({\bf p})$ carries valuable information about the electron
momentum density sampled by the positron. Early electron-positron annihilation
studies in metals already observed a strong electron-positron attraction
superposed on the many-body correlations between the electrons DeBenedetti et
al. (1950). Recently, the methodology to compute the spin-polarized two-
dimensional angular correlation of annihilation radiation ($2D$-ACAR) in
combination with LDA+DMFT was developed Ceeh et al. (2016). This made it
possible, for example, to experimentally pinpoint the strength of the local
electronic interaction in Ni Ceeh et al. (2016). In a related study we
performed the full 3D-electron momentum density reconstruction from the
measured $2D$-ACAR in vanadium which allowed us to identify specific
signatures of the Fermi surface Weber et al. (2017). The presence of the Fermi
Surface (FS) is signaled by the cusps in $2D$-ACAR spectra projected along
specific directions. By the Lock-Crisp-West (LCW) back-folding method the
momentum densities may be represented within the first Brillouin zone and
features of the Fermi surface can be identified.

The paper is organized as follows: After the theoretical description of
computational methods Sec. II, we describe in Sec. III the spectral function
and the Fermi surface modeled by the LDA(+DMFT) calculations in which we
compared relativistic effects by artificially tuning the speed of light. Sec.
IV starts with a brief description of the method to compute the momentum
densities and $2D$-ACAR spectra using Green’s functions. In the corresponding
subsections (Sec. IV.1) we discuss the changes in the electronic densities
induced by the presence of positrons and compare the theoretical and
experimental spectra along some particular direction (Sec. IV.2). The summary
section (Sec. V) concludes the paper.

##  II Theoretical techniques

Electronic structure calculations were performed with the Spin-Polarized
Relativistic Korringa-Kohn-Rostoker (SPR-KKR) code Ebert et al. (2011). In the
LDA computations the exchange-correlation potentials parametrized by Vosko,
Wilk, and Nusair Vosko et al. (1980) were employed. The experimental lattice
parameter for face-centered cubic (fcc) Pd is $3.8907\,$Å, and a BZ-mesh of
${57\times 57\times 57}$ was used throughout the calculations.

In order to study the possible correlation effects within the framework of
LDA+DMFT Kotliar et al. (2006); Held (2007) one adds the multi-orbital on-site
interaction term,
${\mathcal{H}}_{U}=\frac{1}{2}\sum_{{i\\{m,\sigma\\}}}U^{\vphantom{\dagger}}_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}c^{{\dagger}}_{im\sigma}c^{{\dagger}}_{im^{\prime}\sigma^{\prime}}c^{\vphantom{\dagger}}_{im^{\prime\prime\prime}\sigma^{\prime}}c^{\vphantom{\dagger}}_{im^{\prime\prime}\sigma}$,
to the LDA Hamiltonian ${\mathcal{H}}_{\text{LDA}}$. The corresponding many-
body problem described by the total Hamiltonian
${\mathcal{H}}={\mathcal{H}}_{\text{LDA}}+{\mathcal{H}}_{U}-{\mathcal{H}}_{\text{DC}}$
is solved using the LDA+DMFT method, where ${\mathcal{H}}_{\text{DC}}$ serves
to eliminate double counting of the interactions already included in
${\mathcal{H}}_{\text{LDA}}$. Here,
$c^{\vphantom{\dagger}}_{im\sigma}$($c^{\dagger}_{im\sigma}$) destroys
(creates) an electron with spin $\sigma$ on orbital $m$ at the site $i$. The
Coulomb matrix elements
$U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}$ are expressed in the
standard way Imada et al. (1998) in terms of three Kanamori parameters $U$,
$U^{\prime}$ and $J$. Charge and self-energy self-consistent LDA+DMFT
computations are performed using the KKR approach Minár et al. (2005).
Contrary to the Hamiltonian formulation, the KKR implementation of the
LDA+DMFT uses the concept of the multiple-scattering theory. Namely, the
solution for the single-site problem includes the local self-energy of the
many-body problem, and allows for the evaluation of the scattering path-
operators and the real-space DMFT corrected Green’s function. The many-body
self-energy is constructed using the spin-polarized $T$-matrix fluctuation
exchange (SPT-FLEX) Lichtenstein and Katsnelson (1998); Pourovskii et al.
(2005) impurity solver. This impurity solver is fully rotationally invariant
even in the multi-orbital version and is reliable when the interaction
strength is smaller than the bandwidth, a condition which is fulfilled in the
case of Pd. The LDA+DMFT computations require the parametrization of the
interaction matrix $U_{mm^{\prime}m^{\prime\prime}m^{\prime\prime\prime}}$ in
terms of the average local Coulomb $U$ and the exchange parameter $J$. The
value of $U$ is sometimes used as a fitting parameter. However, recent
developments made it possible, in principle, to compute the dynamical
electron-electron interaction matrix elements with a good accuracy
Aryasetiawan et al. (2004), but with substantial variations associated with
the choice of the local orbitals Miyake and Aryasetiawan (2008). Since the
parameter $J$ is not affected by screening it can be calculated directly
within the LDA and is approximately the same for all 3d elements, i.e
$J\approx 0.9$ eV.

Figure 1: Relativistic band structure of Pd along high-symmetry directions in
the BZ and the total density of states.

##  III Band structure and Fermi surface

Fig. 1 presents the LDA+DMFT spectral function overlaid on the band structure.
These results do not differ qualitatively from the previously reported
results, see Ref. Östlin et al., 2016 and references therein. The Fermi level
is situated near the top of the spectra in a relatively flat portion of the
almost completely occupied $d$-bands. The density of states (DOS) is thus
fairly large at the Fermi level, as is shown on the rightmost plot in Fig. 1.
Various details of the DOS have been recently discussed in Ref. Östlin et al.
(2016). We focus our discussion on the hole pockets around the $L$-point which
lies just above the Fermi energy. This particular band is very shallow, and as
we move away from this specific point, it drops below the $E_{\mathrm{F}}$.
Its existence was predicted by relativistic Augmented Plane Wave (RAPW)
calculation Krogh Andersen and Mackintosh (1968); Andersen (1970); Mueller et
al. (1970). The initial dHvA experimental results reported its absence
Windmiller et al. (1971), the data reporting the position of the corresponding
band as located below $E_{\mathrm{F}}$. Note that the $L$-pockets have small
areas (extent in $k$-space) therefore according to the usual dHvA
interpretation (that the angular variation of the effective mass should follow
the angular variation in the area) holes in the $L$-pocket should have small
effective masses. The fact that effective masses on the $L$-pockets are large
reflects the non-parabolic nature of the bands that are actually very shallow.
Later on more precise experiments Dye et al. (1981); Joss et al. (1984); Joss
and Crabtree (1984) accompanied by theoretical parametrization of Fermi
surface sheets together with various other experimental techniques such as
magnetoacustics Brown et al. (1972) and PES Himpsel and Eastman (1978) finally
revealed its existence. Thus, from the gathered information at the $L$
symmetry points a hole-like closed surface is formed and because it lies near
a band extremum, it should be nearly ellipsoidal in shape.

Another ellipsoidal hole pocket is found around the $X$-point. Along the
${\Gamma-X}$ direction its size is larger than along the ${X-W}$. At a
slightly higher energies moving from $X$ to $W$ we follow a band situated
above $E_{\mathrm{F}}$. This band passes below the Fermi level as we move form
$W$ to $K$. The hole surface associated with this band is open in the ${X-W}$
direction forming the cylinders seen in Fig. 2. The flatness of these bands
makes them very sensitive to changes in the parameters.

The largest surface is that situated around the $\Gamma$-point. The volume of
this surface is equal to the sum of the previously described hole surfaces.
This surface has extensions along the ${\langle 100\rangle}$, ${\langle
110\rangle}$, and ${\langle 111\rangle}$ directions. The open hole surfaces
are particularly interesting as they are associated with large effective
masses, and contribute significantly to the density of state at the Fermi
level. As expected, the correlation effects broaden the one-particle bands,
see Figs. 1 and 2, but nonetheless the overall shapes are not considerably
changed with respect to the DFT-LDA results.

Figure 2: Fermi-surface cuts for ${k_{[010]}=0}$ (upper part) and
${k_{[1\bar{1}0]}}$ (lower part) planes.

The change in the Pd Fermi surface cuts in the $(100)$ (${k_{[010]}=0}$) and
$(1\bar{1}0)$ planes (${k_{[1\bar{1}0]}=0}$) are shown in Fig. 3 for various
strengths of relativistic effects and in the presence of electronic
correlations. Within the KKR approach the relativistic effects can be “tuned”
by artificially modifying the magnitude of parameter describing the speed of
light $c$ (the ratio $c_{0}/c$). Because the leading relativistic effects vary
as $1/c^{2}$ they should all be affected accordingly. Reducing the ratio
$c_{0}/c$, i.e. increasing the speed of light, all relativistic effects are
switched off and the description of the electronic structure is changed from
the Dirac- to the Schrödinger-Kohn-Sham type equation. If on the other hand
the speed of light is reduced, one enters into the so-called “super-
relativistic” regime. Calculations are performed within the ”fully
relativistic” setup with the exception $c_{0}=0$ which corresponds to the non-
relativistic limit.

For the non-relativistic case of (${c_{0}/c=0}$) no $L$-hole pockets are seen,
while for all other ${c_{0}/c=0.9}$ and ${1.1}$ the Fermi surface consists of
the same sheets with the same connectivity as was discussed in the previous
LDA+DMFT computations Östlin et al. (2016) (and references therein) with the
scalar relativistic approach. Nevertheless, relativistic corrections change
some details of the Fermi surface sheets. The large “jungle-gym” hole surface
(tubes) are enlarged (Fig. 3 upper panel) as well as the $\Gamma$ centered
electron sheet, while the $X$ centered hole pockets are somewhat reduced in
size (Fig. 3). In the non-relativistic case at the $X$-point the “jungle-gym”
and the hole pockets touch. For a non-zero value of the $c_{0}$ parameter
these two sheets become distinct and their separation increases along with
increasing $c_{0}$. As the magnitude of the spin-orbit interaction is
proportional to a $1/c^{2}$ factor the separation of the “jungle-gym” and the
hole pockets may also be associated to spin-orbit splitting.

Although the $L$ hole pockets are absent in the non-relativistic case (see the
lower plot in Fig. 3), it is interesting to note that the presence of
electronic correlations effects (LDA+DMFT results) indicate the possible
formation of this extra Fermi Surface pockets by the residual weight of the
“spikes” (“jungle-gym” spikes) along ${X-L}$ direction. Notably, the size of
spikes is also increasing with increasing $c_{0}$.

Figure 3: Fermi surface cuts for ${k_{[010]}=0}$ (upper plot) and ${k_{[1\bar{1}0]}=0}$ (lower plot) planes for different $c_{0}/c$ parameters.  Table 1: Fermi surface radii for $X$\- and $L$-centered hole pockets (cf. Fig. 2).  | LDA | LDA+DMFT
---|---|---
Directions | ($\pi/a$) | ($\pi/a$)
$X-\Gamma$ | $0.1290$ | $0.1240(5)$
$X-U$ | $0.0846$ | $0.0815(4)$
$X-W$ | $0.0821$ | $0.0791(4)$
$L-\Gamma$ | $0.0903$ | $0.081(1)\phantom{0}$
$L-U$ | $0.0398$ | $0.0317(5)$
$L-W$ | $0.0398$ | $0.0316(5)$
$L-K$ | $0.0399$ | $0.0317(8)$

To quantify the correlation effects on the Fermi surface hole pockets, we
compared the radii of the $X$\- and $L$-centered hole pockets and found a
reduction of about $4\%$ for $X$\- and about $20\%$ for $L$ in the linear
dimensions of the Fermi sheets (see Table 1 and Fig. 1). In table 1 we list
the symmetry plane radii of the $X$\- and $L$-centered hole pockets. The two
columns correspond to the relativistic LDA and LDA+DMFT calculations. Early
theories to elucidate the role of electronic interactions were based on the
paramagnon model Berk and Schrieffer (1966); Doniach and Engelsberg (1966) and
were focused on the effective mass enhancement computed from the self-energy
which was taken as frequency dependent but not momentum dependent, i.e.
retarded in time but local in space. Previously we have computed the quasi-
particle weights ${Z=(1-\partial\real[\Sigma(E)]/\partial
E|_{E_{\mathrm{F}}})^{-1}}$ for the different $U$ values in the range from
${U=1-4}$ eV. The corresponding mass enhancement factor was found to be in the
range ${Z=0.975-0.916}$ which provides for the effective mass ratios
${m^{*}/m_{\text{LDA}}=Z^{-1}=1.03-1.09}$, where $m_{\text{LDA}}$ is the LDA
band mass. The experimental estimated value from the specific heat measurement
is larger than the mass enhancement due to correlation effects even when the
electron-phonon and the paramagnon contribution is added Östlin et al. (2016).

Figure 4: 2D Fermi-surface projection along $[001]$ direction from LDA
calculations (upper-left plot). Normal regions are blank. $X$-hole fins shown
here separately are part of the $X$-hole tubes. The bottom-right plot: the
Brillouin zone for the fcc crystal structure (upper part) and $2D$ projection
of it along the $[001]$ direction (lower part). High-symmetry points and
directions, as well as extra images of high-symmetry points on the projection
plane (blue crosses) are also shown. In the 2D FS projection (upper-left
plot), the $\Sigma$-direction in ${\Gamma-K-W-X-\Gamma}$-path is along the
$[1\bar{1}1]$ direction.

In Fig. 4 we show the projected Fermi surface along $[001]$ direction. Note
that the small $L$-pockets that are situated on the hexagonal faces remain
clearly visible also in the $2D$ projected FS, and are not overlapped by other
FS sheets in the projection along this direction. By the projection, the hole
pockets centered at the $X$-point (square faces of fcc BZ) two out of the six,
when projected down into the $\Gamma$-point, have a circular shape (cut across
minor axis of revolution ellipsoid) with the remaining four projected into the
$W$-point are identified as two crossing ellipses with major and minor axis
exchanges (one of ellipses comes from the neighboring BZ). The “jungle-gym”
structure is also easily recognizable by (anti-)diagonals along projected the
$\Gamma$-points. $X$-hole fins which are the part of the “jungle-gym”
structure (shown as a separate FS sheet in Fig. 4) can be identified as cross
like features centered at $(X,W)$-points on projection plane. The “jungle-gym”
tubes, perpendicular to the projection plane, are identified around the
$(W,X)$-points with the projected $X$-hole pockets inside. Projections of the
high-symmetry points $K$ and $U$ coincide with some of the $X$\- and
$W$-points. In Fig. 4 we also show the fcc BZ with high-symmetry points and
their projection on ${k_{[100]}=0}$ plane together with extra images of high-
symmetry points (blue crosses and letters in projection plane). This projected
FS sheets image turns to be useful when considering the LCW back-folded EMD
and $2D$-ACAR which will be discussed in the following sections.

##  IV Electron momentum densities, $2D$-ACAR and Fermi surface

The theoretical analysis of the $2D$-ACAR spectra requires the knowledge of
the two-particle electron-positron Green’s function, describing the
probability amplitude for an electron and a positron propagating between two
different space-time points. The DFT can be generalized to the problem at hand
by including the positron density in the form of a 2-component DFT Boroński
and Nieminen (1986); Puska and Nieminen (1994). In the present calculations
the electron-positron correlations are taken into account by a multiplicative
(enhancement) factor, which results from the inclusion of the electron-
positron interaction in the form of an effective one-particle potential as
formulated by Boroński and Nieminen Boroński and Nieminen (1986) or
alternatively by Drummond Drummond et al. (2011). Many-body DMFT corrections
are taken into account only for the electronic subsystem and no many-body
correlation is included for the positron. The Green’s function of the
electron-positron pair in momentum representation can be expressed using the
multiple scattering Green’s function and the eigenfunctions of the momentum
operator $\phi_{{\bf p}\sigma^{\prime}}$ as:

$\displaystyle G^{\mathrm{X}}_{\sigma\sigma^{\prime}}({\bf p}_{e},{\bf
p}_{p},E_{e},E_{p})=\frac{1}{N\Omega}\int d^{3}{\bf r}\,\int d^{3}{\bf
r^{\prime}}\phi_{{\bf p}_{e}\sigma}^{e\dagger}({\bf r})\imaginary
G^{\vphantom{\dagger}}_{e\sigma}({\bf r},{\bf r^{\prime}}\,E_{e})\phi_{{\bf
p}_{e}\sigma}^{e}({\bf r^{\prime}})\,\phi_{{\bf
p}_{p}\sigma^{\prime}}^{p\dagger}({\bf r})\imaginary
G^{\vphantom{\dagger}}_{p\sigma^{\prime}}({\bf r},{\bf
r^{\prime}},E_{p})\phi_{{\bf p}_{p}\sigma^{\prime}}^{p}({\bf r^{\prime}})\,,$
(1)

where ${\mathrm{X}=\text{LSDA(+DMFT)}}$ and $({\bf p}_{e}\sigma)$ and $({\bf
p}_{p}\sigma^{\prime})$ are the momenta and spin of electron and positron,
respectively. Taking into account the positron thermalization (which in metals
occurs very quickly compared to the lifetime of the positron), one may
consider the positron to be in a state with ${{\bf p}_{p}=0}$ with $s$-type
symmetry and the energy corresponding to the bottom of the positronic band.
The positron will almost always annihilate via two photons from an anti-
parallel positron-electron spin configuration. As consequence, the electron-
positron pair has zero total spin angular momentum and
${\sigma=-\sigma^{\prime}}$ holds. The two-photon momentum density resulted
from the annihilation process can be expressed using the electron-positron
Green’s function as

$\rho^{2\gamma,\mathrm{X}}_{\sigma}({\bf
p})=-\frac{1}{\pi}\\!\int_{-\infty}^{\infty}\\!\\!\\!\\!\\!\mathrm{d}E_{e}\\!\\!\int_{-\infty}^{\infty}\\!\\!\\!\\!\\!\mathrm{d}E_{p}\,G^{\mathrm{X}}_{\sigma\sigma^{\prime}}({\bf
p}_{e},{\bf p}_{p},E_{e},E_{p}),$ (2)

whereas for the electron’s momentum density (EMD) the electronic Green’s
function in the momentum representation is employed,

$\rho^{\text{EMD},\mathrm{X}}_{\sigma}({\bf
p})=-\frac{1}{\pi}\int_{-\infty}^{\infty}\mathrm{d}E_{e}\,\imaginary
G^{\mathrm{X}}_{\sigma\sigma^{\prime}}({\bf p}_{e},E_{e})\,.$ (3)

The energy integrations in Eqs. (2) and (3) are performed along a contour in
the complex plane. For the two-photon momentum densities, in Eq. (2),
integration over $E_{p}$ is not required as it is taken as a parameter
corresponding to the bottom of the positronic band. Only the filled states
contribute to the $2D$-ACAR spectra, as the momentum distributions are
obtained through energy integrals up to the Fermi level. The computed
$2D$-ACAR/EMD spectrum $\tilde{\rho}(p_{x},p_{y})$ is a $2D$ projection of the
$3D$ momentum-density distribution
${\rho^{2\gamma/\text{EMD},\mathrm{X}}_{\sigma}({\bf p})}$ along a chosen
$p_{z}$,

$\tilde{\rho}^{2\gamma/\text{EMD},\mathrm{X}}_{\sigma}(p_{x},p_{y})=\int\mathrm{d}{p_{z}}\,\rho^{2\gamma/\text{EMD},\mathrm{X}}_{\sigma}({\bf
p})\,.$ (4)

In the following, the spin index $\sigma$ can be dropped, as the calculations
are performed for the non-magnetic state.

In order to expose details of the momentum densities the anisotropy spectrum
$\rho_{an}({\bf p}_{\perp})$ is obtained by subtracting an angular averaged
contribution $\bar{\rho}(p)$. The same procedure is usually used for the
analysis of the experimental measurements, which helps to emphasise the
anisotropic features in the momentum density which arise from wavefunction
anisotropy or, in a metal, from the presence of the Fermi surface. All EMD and
ACAR calculations were performed with cut off $|\mathbf{p}|_{\max}=16$ a.u.
and step size $0.01$ a.u.

We analyse the two-dimensional projections of the electronic momentum
densities along the $[100]$ and $[111]$ directions (integration directions).
In the left plot in Fig. 5 we show the radial anisotropy of EMD (upper panels)
and $2D$-ACAR spectra (lower panels) obtained with the LDA exchange-
correlation (left column) and including dynamical correlations (right column),
in the extended momentum space. Given that the effects of electronic
correlations amount to slight difference between the spectral function and the
one-particle band structure, see Fig. 1, very similar EMD spectra are
expected. Indeed we note that correlation effects does not change
significantly the LDA EMD and $2D$-ACAR spectra. However, a direct comparison
of the EMD and $2D$-ACAR spectra allows certain differences to be identified
stemming form the presence of the positron in the computation. The most
notable differences happen along the direction $(\Gamma,X)-W-(U,K)-(X,W)$ and
in the second Brillouin zone around e.g., ${\mathbf{p}_{\perp}\approx(\pm
1.5,\pm 0.25)}$ a.u. points. Another visible effect happens in the region
around the $W$-point in which the $2D$-ACAR spectra are depleted with respect
to the EMD. In the region between $W$ and $(U,K)$ the $2D$-ACAR has a stronger
signal, while further out following the same direction the spectra is loosing
weight. The “jungle-gym” structure in the ${X-W-X}$ direction remains visible
in the EMD, also after the projection along $[001]$ which is because no
intense momentum density exists around the $X$-point. In contrast, in the same
region the electronic momentum density is enhanced by the presence of
positrons as seen in the $2D$-ACAR spectra and the “jungle-gym” structure is
covered by the response from the $X$-hole pockets. Strong positron enhancement
effects are identified in the second Brillouin zone around e.g.:
${\mathbf{p}_{\perp}\approx(\pm 1.5,\pm 0.25)}$ a.u. points. These might be
used to asses the effects of electron-positron enhancement factor (in our case
modeled by Drummond parametrization) if the experimental results would be
available.

To compute the momentum densities in the corresponding $2D$-ACAR Brillouin-
zones we employ the LCW back-folding procedure Lock et al. (1973),

$\tilde{\rho}^{2\gamma/\text{EMD},\mathrm{X}}_{\mathrm{LCW}}(\vec{k}_{\perp})=\\!\\!\sum_{\\{{\bf
K}_{\perp}\\}}\\!\\!\tilde{\rho}^{2\gamma/\text{EMD},\mathrm{X}}({\bf
p}_{\perp})\Big{|}_{{\bf p}_{\perp}={\bf k}_{\perp}+{\bf K}_{\perp}}\\!\\!,$
(5)

where the sum runs over a set of translation vectors in the projection plane,
${\\{{\bf K}_{\perp}\\}}$. We analyzed the LCW back-folded radial anisotropy
of EMD and $2D$-ACAR instead of commonly computed LCW back-folded EMD and
$2D$-ACAR. In this case the FS features are better visible and electron- and
hole-like FS sheets better recognisable (cf. Figs. 5-8 with Figs. 9-11).
Unlike the “high-pass” like filters, where the Gaussian convoluted data is
subtracted from the unconvoluted one, the order of computing the radial
anisotropy and LCW back-folding is not interchangeable: first the radial
anisotropy should be computed and obtained result,
$\tilde{\rho}^{2\gamma/\text{EMD},\mathrm{X}}_{\mathrm{an}}({\bf p}_{\perp})$,
back-folded with the LCW procedure (5). The radial anisotropy offers some
advantage to high-pass filters, because the experimental $2D$-ACAR
measurements contain both the core-electron and filled-band contributions
which in general rather isotropic, and can be eliminated by this procedure.
Moreover, positron annihilation experiments are sensitive to the presence of
open-volume defects, and it has been shown that performing the back-folding
procedure on the radial anisotropy can suppress their deleterious effect in
the LCW densities Dugdale and Laverock (2014).

Figure 5: Anisotropy (left) and LCW back-folded anisotropy (right) plots of
$2D$ projection of EMD (upper rows) and $2D$-ACAR spectra (bottom rows) from
LDA (left column) and LDA+DMFT (${U=1.0}$ eV, ${J=0.3}$ eV) (right column)
calculations. The $2D$ projection of EMD and $2D$-ACAR spectra are normalized
by the number of valence electrons. The false-color map is normalized by
maximal amplitude of anisotropy taken for all four cases. Projections of the
high-symmetry points and directions (black crosses and lines), as well as
extended BZ (white lines) are also shown.

The back-folded spectra of the radial-anisotropy is shown in the left plot in
Fig. 5. All Fermi surface sheets can be identified in the back-folded radial
anisotropy plots (also cf. Fig. 4): large electron sheet around the
$\Gamma$-point can be identified as red squares with centers at
$\Gamma$-point; In the EMD, the corners of the projected $\Gamma$-centered
electron Fermi surface sheets are“chopped” (values are negative), while in the
$2D$-ACAR the corners become positively weighted again. one sees the small
circle shaped deeps in the center of this red squares corresponding to two out
of six face centered $X$-hole pockets (square faces of fcc BZ); remaining four
that overlap could be identified at $(X,W)$-points as dark-blue squares; the
“jungle-gym” (tubes together with $X$-hole fins) becomes also clearly visible
in both EMD and $2D$-ACAR back-folded spectra (plus shapes with center at
either $(\Gamma,X)$-points or $(X,W)$-points); ellipsoid shaped $L$-hole
pockets are well recognized in $2D$-ACAR in both LDA and LDA+DMFT cases as
well es in EMD for LDA+DMFT. Note that, redistribution of electronic momenta
around the $L$-point due to the presence of electronic correlation or
positrons does not alter the signal from the $L$-hole pockets and it should be
visible in a high resolution $2D$-ACAR experiment e.g., using a spectrometer
based on high-density avalanche chambers Dugdale et al. (2013). In Fig. 6 we
show broadened $2D$-ACAR LDA+DMFT results for the relativistic ${c_{0}/c=1}$
and non-relativistic ${c_{0}/c=0}$ cases. Comparing results for relativistic
calculations, where we expect $L$-hole pockets in FS, with non-relativistic
ones, which does not exhibit this feature, one can distinguish the $L$-hole
pocket features with a momentum resolution FWHM of $=0.04$ a.u., which is
about the width of the $L$-hole pocket (see Table 1). Therefore we conclude
that $2D$-ACAR measurements with a comparable or a better resolution will be
able to resolve this FS feature directly.

Figure 6: LCW back-folded anisotropy plots of $2D$-ACAR spectra with
broadening of FWHM=$0.04$ a.u. (left column) and FWHM=$0.06$ a.u. (right
column) from nonrelativistic (upper row) and relativistic (lower row) LDA+DMFT
(${U=1.0}$ eV, ${J=0.3}$ eV) (right column) calculations. The $2D$-ACAR
spectra are normalized by the number of valence electrons. The false-color map
is normalized by maximal amplitude of anisotropy taken for all four cases.
Projections of the high-symmetry points and directions (black crosses and
lines), as well as extended BZ (white lines) are also shown.

###  IV.1 Momentum densities and electron-positron correlation functionals

The theory of the annihilation probability of a positron in a homogeneous
electron gas has a long history in many-body physics Carbotte and Kahana
(1965); Carbotte (1967); Arponen and Pajanne (1979); Rubaszek and Stachowiak
(1988); Boroński et al. (1981). The electron-positron attraction leads to an
increase of the electron density near the positron. It manifests itself in the
annihilation characteristics and leads to a strongly increased total
annihilation rate which is the integral of $\rho({\bf p})$ over the momentum
p. This effect is qualitatively well understood and is called “enhancement”.

Since many decades, a central topic of theoretical positron physics is the
inclusion of the enhancement effect into the theory of annihilating electron-
positron pairs. Whereas this effect is — at least in principle — well
understood for positrons annihilating in a homogeneous electron gas Kahana
(1963); Boroński et al. (1981); Carbotte (1967), the evaluation of Kahana’s
equation:

$\displaystyle\begin{split}&G_{{\rm ep}}(x,x^{\prime};y,y^{\prime})=G_{{\rm
e}}(x,y)G_{{\rm p}}(x^{\prime},y^{\prime})\\\
&\\!+\\!\\!\frac{i}{\hbar}\\!\\!\int\\!\\!{\mathrm{d}}z{\mathrm{d}}z^{\prime}\\!G_{{\rm
e}}(x,z)G_{{\rm p}}(x^{\prime},z^{\prime})V^{{\rm ep}}\\!(z,z^{\prime})G_{{\rm
ep}}(z,z^{\prime};y,y^{\prime})\\!\end{split}$ (6)

in spatially inhomogenous electron gases is still an extremely laborious task.
In Eq. (6) we employ composite coordinates ${x\equiv({\bf x},t_{x})}$ etc.,
where $G_{{\rm e}}$ and $G_{{\rm p}}$ are the single-particle Green’s
functions for an electron and a positron, respectively, and $V^{{\rm ep}}$
describes the effective potential between the two fermions (electron and
positron). The simplest approximation for the electron-positron Green’s
function is given by the first term of Eq. (6): ${G_{{\rm
ep}}(x,x^{\prime};y,y^{\prime})\approx G_{{\rm e}}(x,y)\,G_{{\rm
p}}(x^{\prime},y^{\prime})}$ and it ignores all terms which include the
electron-positron potential $V^{ep}$. This means that all direct correlation
effects between the annihilating particles (leading to the so-called
enhancement of the electron-positron annihilation rate) are neglected. It
represents the so-called independent particle model (IPM). However, this does
not mean that electron and positron interactions with their electronic
environment are completely “switched off” in this approximation. These
interactions can be, at least, partly included into the theory by self-energy
insertions into the one-particle propagators $G_{e}$ and $G_{p}$.

Using the product of the electron and positron Green’s functions, the
electron-positron momentum density ($\rho^{2\gamma}$) can be written in terms
of the Fourier transform of the overlap of Bloch functions of the annihilating
electron and positron. A frequently used non-ab initio approach for a
description of $\rho^{2\gamma}$ including direct electron-positron
correlations is based on the insertion of a local enhancement function, into
the overlap of the electron and annihilating positron wave functions. Such a
procedure was first discussed by Carbotte Carbotte (1967) and Fujiwara et al.
Fujiwara et al. (1972), and has been further developed by Sormann Sormann
(1996) under the name Bloch-modified ladder (BML) theory. The BML theory is
quite successful in giving — at least qualitatively — a correct description of
the electron-positron momentum density in metals of very different electronic
structures Sormann and Šob (2001); Sormann and Kontrym-Sznajd (2006), but on
the other hand, this approach is numerically expensive and has some
convergence problems with respect to the unoccupied electron bands.

For the electron momentum density in the IPM, there are no dynamical electron-
electron or electron-positron correlations. Only in the Fourier integral
correlations between the annihilating particles are approximately taken into
account by the non-dynamical (static) enhancement function. During the last
decades, several proposals for this enhancement function have been made. Here
we shall mention briefly only the two most representative ones, namely: i)
formulas for a momentum-independent enhancement function, based on Lantto’s
work Lantto (1987) about spatially homogeneous electron-positron plasmas and
presented by Boronski and Nieminen (see Ref. Boroński and Nieminen (1986)) and
by Puska et al. Puska et al. (1995), and ii) a state-dependent enhancement
factor published by Alatalo et al. Alatalo et al. (1996) and by Barbiellini et
al. Barbiellini et al. (1997a). As has been already mentioned in connection
with the electron-positron correlation potential, this enhancement factor can
also be used either on an LDA level Boroński and Nieminen (1986); Puska et al.
(1995) or on a GGA level Barbiellini et al. (1995, 1996). This approach was
generalized to an energy dependent form Mijnarends and Singru (1979) and was
later extended to include orbital dependence Singh et al. (1986). Since this
was formulated within DFT Boroński and Nieminen (1986); Rubaszek and
Stachowiak (1988); Barbiellini et al. (1995, 1996); Barbiellini et al.
(1997b); Sormann (1996); Sormann and Kontrym-Sznajd (2006); Kontrym-Sznajd et
al. (2012); Kontrym-Sznajd and Sormann (2014a, b); Boroński (2014) the results
maintain their static mean-field character, but now they also reflect the
inhomogeneity of the electron gas of real materials. Sormann and Kontrym-
Sznajd investigated the effectiveness of various enhancement models applied to
different metals like Mg, Cd, Cu, and Y. They compared the theoretical MDAP
distributions with experimental data and with the ab-initio approach BML (see
Ref. Sormann et al. (2002)). The main result of this work was that all
investigated non-ab initio enhancement models have their advantages and
disadvantages, i.e. they succeed for one metal and fail for another.
Nevertheless, it should be emphasized here that all theoretical approaches
mentioned above (including BML) describe a static enhancement without any
dynamical effects.

Figure 7: Closed path ${X-(\Gamma,X)-L-X}$ (triangle) across the LCW back-
folded data. Upper row corresponds to the results presented in Fig. 5

To model the enhancement we considered the formulation proposed by Drummond et
al. Drummond et al. (2011). In Fig. 7 we show sections of the back-folded
momentum densities through the $L$-hole and $X$-hole pockets. The general
shape of the $2D$-ACAR and the EMD is similar, and the structure seen in these
curves can be related to the Fermi-surface topology and the nature of the wave
functions. One clearly sees deeps at $(\Gamma,X)$\- and $(W,X)$-points where
the projected $X$-hole pockets are expected and one can also read out the
linear size of the ellipsoid of revolution. At about a half of the
$(W,X)-(\Gamma,X)$ distance, where one still expects the contributions from
the $\Gamma$-centered electron surface, the calculated weight dives below $0$
in the EMD, whereas in the $2D$-ACAR spectra one can recognize the edge of
this large electron Fermi surface. We note that the $L$-hole pocket has a
similar width in the EMD and $2D$-ACAR spectra, while it is deeper in the case
of the $2D$-ACAR spectra in which enhancement of momentum density is visible.
Further on one sees that the relativistic effects are the determining factor
in the appearance of the $L$-hope pocket. In the lower panel of Fig. 7 where
we show the comparison of the relativistic versus the non-relativistic
($c/c_{0}=0$) $2D$-ACAR spectra one sees that the non-relativistic spectra are
flattened where the $L$-hole pocket is expected in comparison with the
relativistic case. Whereas the linear sizes of the $X$-hole pocket remain
almost similar in the non-relativistic ($c/c_{0}=0$) case (see the lower panel
of Fig. 7).

Figure 8: Anisotropy (left) and LCW back-folded anisotropy (right) plots of
$2D$-ACAR spectra along $[111]$ high-symmetry direction of experimental
measurements (upper rows) and LDA+DMFT calculations (${U=1.0}$ eV, ${J=0.3}$
eV) (lower rows). The EMD (lower-left quadrants). The Drummond parametrization
of the enhancement factor is used for the $2D$-ACAR spectra (bottom-right
quadrants). The false-color map is normalized by maximal amplitude of
anisotropy in each cases.

###  IV.2 Comparison to the experimental measurement

As already discussed in the Sec. III, the $2D$-ACAR spectra along $[110]$
direction will be beneficial in observing the $L$-hole pockets, since in this
direction other Fermi surface sheets/pockets do not overlap with the $L$-hole
pocket in the projected spectra. Nonetheless in absence of recent accurate
measurements, on high quality bulk single crystal of Pd along the $[110]$
direction, we analyze the accuracy of our calculation and draw conclusions
about the ability of simulations to reproduce/predict the observed structure.
This projection however overlays contributions from around the $L$-point to
those from the central $\Gamma$-point and the larger size $X$-hole pockets,
and therefore does not provide unambiguous information about $L$-pockets.

A $2D$-ACAR experiment was performed on a bulk single-crystal of Pd using a
spectrometer at the University of Bristol, the design of which is described in
Ref. West et al. (1981). The measurement was made at a temperature of 30K. The
angular resolution full widths at half-maximum (FWHM) was $\approx 0.09$ a.u.
and $\approx 0.14$ a.u. along the $[1\bar{1}0]$ and $[11\bar{2}]$ directions,
respectively.

In order to expose details of the momentum densities the anisotropy spectrum
$\rho_{an}({\bf p}_{\perp})$ is obtained by subtracting an angular averaged
contribution $\overline{\rho}(p)$. This procedure eliminates radially
symmetric contributions from core electrons and in metals can be used to
highlight the Fermi surface features.

In Fig. 8 we present a direct comparison of the experimental measurement and
theoretical calculations (including electronic correlations) for the
anisotropy spectra in the extended momentum scheme (left plot) and the
corresponding LCW back-folded spectra (right plot). We observe the very good
agreement between the experimental and the computed $2D$-ACAR spectra which
includes the Drummond’s Drummond et al. (2011) enhancement scheme. We have
also included the EMD spectra. The theoretical distributions were convoluted
with a Gaussian with $0.097$ a.u. and $0.137$ a.u. FWHM presenting the
experimental resolution in $[110]$ and $[112]$ directions, respectively.
Characteristic to the experimental spectra for this particular projection is
the (six fold symmetry) hexagonal pattern of bright and dark areas which is
very well captured by the $2D$-ACAR calculation. Although in several momentum
regions the departure with respect to the experiment is clearly seen, overall,
the inclusion of electron-positron enhancement effects increases the intensity
of the anisotropy signal for small momenta, and improves the comparison with
the experiment.

On the right hand side of Fig. 8 we show the LCW back-folded spectra. Around
the $\Gamma$-point electronic momentum of considerable weight is found within
a hexagonal shaped region. The $2D$-ACAR computation reproduces very well the
size and the shape of the hexagonal region. For the EMD spectra we see that
the hexagon’s boundary (sides) are somewhat reduced in size and are slightly
concave. Outside the central region inside the triangles, produce by
overlapping of the projections of neighboring BZs, small high positive
intensity regions are seen following the hexagonal pattern. There are also
ellipse-shaped negative intensity features due to $X$\- and $L$-hole pockets
($X$-hole pocket projection completely covering the $L$-hole pocket
projection) organized in the hexagonal pattern around the central dominant
high intensity convex hexagon with its center at $\Gamma$. All these features
are also well reproduced by the theoretical $2D$-ACAR calculations including
positron wave function and the electron-positron interaction enhancement
factor parametrized by Drummond Drummond et al. (2011).

In a recent paper Ketels et al. (2021) the differences between the momentum
distributions observed by positron annihilation and Compton scattering were
investigated. The focus of that study was a new approach for tomographical
reconstruction of $2D$ electron momentum distributions from the $1D$ Compton
profiles rather than observing the impact of electron correlation and
relativistic effects on electronic structure of Pd and its Fermi surface. The
experimental resolution of $2D$-ACAR spectra presented in the current paper
($0.097$ a.u. and $0.137$ a.u.), however, is significantly better than the
values reported in Ref. Ketels et al. (2021) ($0.17$ and $0.21$ a.u.). The
inferior experiment resolution reported in Ref. Ketels et al. (2021) is due to
the spectrometer’s relatively short detector-detector distance.

##  V Summary

The primary purpose of the present computations was to investigate to what
extent electronic correlations may affect the Fermi surface of Pd. The main
result is that band theory including relativistic effects accounts relatively
well for the experimentally observed features of the Fermi surface, in
particular the existence of small $L$-hole pockets. Within the current
advances of realistic many-body computations we have performed LDA+DMFT
electronic structure calculation and discussed the modifications in the band
structure and Fermi surface subject to the change in the strength of
relativistic effects.

The Fermi surface of Pd has been explored by a variety of different
methodologies such as angle-resolved photoemission (ARPES), de Haas-van Alphen
(dHvA) and ultrasonic attenuation, which finally established the existence of
$L$-hole pockets not in a direct way, though. Complementary to these methods
we discuss here the two-dimensional angular correlation of annihilation
radiation ($2D$-ACAR) positron spectroscopy which directly probes the
electronic structure by mapping out the Fermi surface and enables the
observation of fine structures such as the $L$-hole pockets in discussion.
Since the positron tends to occupy open volumes as it tries to avoid
positively charged ionic cores, the positron spatial distribution in a crystal
is in general quite non-uniform. In the specific case of Pd annihilation
events happen between the positrons and Bloch state electrons of the compact
fcc crystal structure thus, one also expects that the enhancement (electron-
positron interaction effects) might be essential. A comparison of the
experimental and theoretical $2D$-ACAR spectra (momentum densities in the
extended and back-folded schemes) has been done for the [111] projection. The
comparison shows that there is a reasonable agreement to the overall shape as
well as to the fine structure in the spectra. The importance of enhancement
effects becomes visible by contrasting the $2D$-ACAR with the electron
momentum densities. By comparing different functionals (results not shown) for
the electron positron interaction we found the Drummond’s parametrization
captures the experimental features the best. Finally we note that, in terms of
electronic correlations, for the strength of the Coulomb and Hund parameters
($U$,$J$) that properly describe the ground state properties in Pd the changes
in the $2D$-ACAR spectra are mild.

Because of the observed good agreement between the calculated and measured
$2D$-ACAR spectra it is natural to expect that $2D$-ACAR measurements along
$[110]$ direction with a resolution higher than or about $0.04$ a.u. will be
able to reveal the $L$-hole pockets structure in Pd Fermi surface topology.

###### Acknowledgements.

Financial support by the Deutsche Forschungsgemeinschaft through TRR80
(project E2) Project number 107745057 is gratefully acknowledged. M.S. was
partially supported by the Shota Rustaveli Georgian National Science
Foundation through the grant N FR-19-11872. We would like to thank J. Minàr
and H. Ebert for a fruitful collaboration. *

##  Appendix A LCW back-folded data

For the sake of completeness in Fig. 9 and Fig. 11 we show the LCW back-folded
EMD and $2D$-ACAR results corresponding to the radial anisotropies shown in
the right plots in Fig. 5 and Fig. 8, respectively. The former is typically
reported in the publications, but we find that LCW back-folded radial
anisotropy better highlights electron and hole like features of the Fermi
surface sheets. As a counter part of the data shown in Fig. 7, in Fig. 10 we
show the sections of the LCW back-folded EMD and $2D$-ACAR (sections of the
data shown in Fig. 9) (upper panel) together with non-relativistic results.
One can also see the $L$-pocket structure in the relativistic calculation.
Note it’s absence in the non-relativistic case, as similarly can be seen in
the cuts of LCW back-folded anisotropy (see Fig. 7).

Figure 9: LCW back-folded $2D$ projection of EMD (upper row) and $2D$-ACAR
spectra (bottom row) from LDA (left column) and LDA+DMFT (${U=1.0}$ eV,
${J=0.3}$ eV) (right column) calculations. The false-color map is normalized
for each case by the maximum value of
${\max[\rho_{\mathrm{LCW}}(\mathbf{k}_{\perp})]}$. Projections of the high-
symmetry points (cyan crosses), as well as extended BZ (white lines) are also
shown. The corresponding LCW back-folded radial anisotropy is shown in the
right plot in Fig. 5.  Figure 10: Closed path ${X-(\Gamma,X)-L-X}$ (triangle)
across the LCW back-folded data, corresponding to the calculations presented
on Fig. 9. The corresponding radial anisotropy is shown in Fig. 7.  Figure 11:
LCW back-folded $2D$-ACAR spectra along $[111]$ high-symmetry direction of
experimental measurements (upper row) and LDA+DMFT calculations (${U=1.0}$ eV,
${J=0.3}$ eV) (lower row). The false-color map is normalized for each case by
the maximum value of
${\max[\tilde{\rho}_{\mathrm{LCW}}(\mathbf{k}_{\perp})]}$. Projections of the
high-symmetry points (cyan crosses), as well as extended BZ (white lines) are
also shown. The corresponding LCW back-folded radial anisotropy is shown in
Fig. 8.

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