# Signatures of a surface spin-orbital chiral metal

Federico Mazzola<EMAIL_ADDRESS>Department of Molecular Sciences
and Nanosystems, Ca’ Foscari University of Venice, Venice IT-30172, Italy
Istituto Officina dei Materiali, Consiglio Nazionale delle Ricerche, Trieste
I-34149, Italy  Wojciech Brzezicki<EMAIL_ADDRESS>Institute of
Theoretical Physics, Jagiellonian University, ulic, S. Łojasiewicza 11,
PL-30348 Kraków, Poland  International Research Centre MagTop, Institute of
Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw,
Poland  Maria Teresa Mercaldo  Dipartimento di Fisica “E. R. Caianiello”,
Università di Salerno, IT-84084 Fisciano (SA), Italy  Anita Guarino  CNR-SPIN,
c/o Universitá di Salerno, IT-84084 Fisciano (SA), Italy  Chiara Bigi
Synchrotron SOLEIL, F-91190 Saint-Aubin, France  Jill A. Miwa  Department of
Physics and Astronomy, Interdisciplinary Nanoscience Center, Aarhus
University, 8000 Aarhus C, Denmark  Domenico De Fazio  Department of Molecular
Sciences and Nanosystems, Ca’ Foscari University of Venice, Venice IT-30172,
Italy  Alberto Crepaldi  Dipartimento di Fisica, Politecnico di Milano, Piazza
Leonardo Da Vinci 32, Milano IT-20133, Italy  Jun Fujii  Istituto Officina dei
Materiali, Consiglio Nazionale delle Ricerche, Trieste IT-34149, Italy
Giorgio Rossi  Dipartimento di Fisica, Università degli studi di Milano,
IT-20133 Milano, Italy  Istituto Officina dei Materiali, Consiglio Nazionale
delle Ricerche, Trieste IT-34149, Italy  Pasquale Orgiani  Istituto Officina
dei Materiali, Consiglio Nazionale delle Ricerche, Trieste IT-34149, Italy
Sandeep Kumar Chaluvadi  Istituto Officina dei Materiali, Consiglio Nazionale
delle Ricerche, Trieste IT-34149, Italy  Shyni Punathum Chalil  Istituto
Officina dei Materiali, Consiglio Nazionale delle Ricerche, Trieste IT-34149,
Italy  Giancarlo Panaccione  Istituto Officina dei Materiali, Consiglio
Nazionale delle Ricerche, Trieste IT-34149, Italy  Anupam Jana  Istituto
Officina dei Materiali, Consiglio Nazionale delle Ricerche, Trieste IT-34149,
Italy  Vincent Polewczyk  Istituto Officina dei Materiali, Consiglio Nazionale
delle Ricerche, Trieste IT-34149, Italy  Ivana Vobornik  Istituto Officina dei
Materiali, Consiglio Nazionale delle Ricerche, Trieste IT-34149, Italy
Changyoung Kim  Department of Physics and Astronomy, Seoul National
University, Seoul, 08826, Korea  Fabio Miletto Granozio  CNR-SPIN, c/o
Complesso di Monte S. Angelo, IT-80126 Napoli, Italy  Rosalba Fittipaldi  CNR-
SPIN, c/o Universitá di Salerno, IT-84084 Fisciano (SA), Italy  Carmine Ortix
Dipartimento di Fisica “E. R. Caianiello”, Università di Salerno, IT-84084
Fisciano (SA), Italy  Mario Cuoco<EMAIL_ADDRESS>CNR-SPIN, c/o
Universitá di Salerno, IT-84084 Fisciano (SA), Italy  Antonio Vecchione
<EMAIL_ADDRESS>CNR-SPIN, c/o Universitá di Salerno, IT-84084
Fisciano (SA), Italy

###### Abstract

The  relation between crystal symmetries, electron correlations, and
electronic structure steers the formation of a large array of unconventional
phases of matter,  including magneto-electric loop currents and chiral
magnetism [1, 2, 3, 4, 5, 6].  Detection of such hidden orders is a major goal
in condensed matter physics. However, to date, nonstandard forms of magnetism
with chiral electronic ordering have been experimentally elusive [7].  Here,
we develop a theory for symmetry-broken chiral ground states and propose a
methodology based on circularly polarized spin-selective angular-resolved
photoelectron spectroscopy to probe them. We exploit the archetypal  quantum
material Sr2RuO4 and  reveal spectroscopic signatures which, even though
subtle, may be reconciled with the formation of spin-orbital  chiral currents
at the material surface [8, 9, 10]. As we shed light on these chiral regimes,
our findings pave the way for a deeper understanding of ordering phenomena and
unconventional magnetism.

Main
One central problem in condensed matter physics is the existence of
unconventional magnetism beyond the traditional forms, arising from the long-
range order of magnetic dipole moments $\mu=(2S+L)$, with $S$ and $L$ being
the electron spin and orbital angular momentum, respectively [11, 12, 13, 14,
15]. Most often, such magnetic dipole moments do not arrange spatially with
ordered patterns in the crystal but other forms of magnetic phases may still
originate from an electronic ordering made of charge currents at the atomic
scale. Such phases are odd in time (time-reversal symmetry is broken),
inherently subtle and difficult to observe, and often associated with a hidden
magnetic order [1, 16, 2].

The transport properties of all metallic quantum materials are determined by
the spin and orbital degrees of freedom of the Fermi surface. Broken symmetry
states (mirror and/or time) with charge currents can have an internal
structure with a combination of spin and orbital angular momentum. Since the
spin and orbital angular momentum are pseudovectors with magnetic dipolar
nature, one can refer to their product as a spin-orbital quadrupole or orbital
quadrupole. Spin and orbital angular momentum  are odd in time and the current
is an odd function of crystal wave vector. Therefore,  currents carrying spin-
orbital or orbital quadrupoles, which are even in time, break time-reversal
symmetry (Fig.1a) [8, 17, 18, 19].

A hallmark of both orbital and spin-orbital quadrupole current is the
appearance of additional symmetry breaking related to mirror, inversion, or
roto-inversion transformations, more generically known as chirality. Hence, a
chiral electronic ordering may be realized uniquely as a consequence of the
intrinsic spin and orbital structure of the charge currents [20, 21, 22].
Chirality is known to set out several unconventional forms of transport and
magnetism [2, 23, 24, 25, 26, 27, 28].  However, chiral effects have been
proven challenging to be detected, because their electronic signature is very
weak. So far, their measurements are limited to only a few material-specific
cases [1, 29, 30, 31].

In a metallic state, chiral orders imprint themselves on the spin and orbital
textures of the electronic states close to the Fermi level [32, 33, 34]. The
action of mirror- and time-reversal symmetries connects the amplitude of spin-
and orbital-angular momentum of the electron states at symmetry-related
momenta. For instance, for time-reversal symmetric electronic states (Fig.1b),
the associated spin-angular momentum must have opposite orientations, i.e.
$(-\bm{k},\uparrow)$ transforms into $(+\bm{k},\downarrow)$ [35]. The same
behaviour applies to the orbital-angular momentum $L$. On the other hand, for
mirror-symmetric electronic states, due to the axial nature of $S$ and $L$,
upon mirror transformation the components lying within the mirror plane change
sign, while the ones perpendicular to the plane remain unchanged. Apart from
the dipolar one, the interaction between spin and orbital degrees of freedom
can set out physical observables with tensorial character. Here, the time-
reversal and mirror symmetries are also expected to affect the behavior of the
spin-orbital ($L_{i}S_{k}$) and orbital ($L_{i}L_{k}$) quadrupole components,
when probed at symmetry-related momenta. This means that the spin-orbital
dipolar and quadrupolar structures are the relevant observables of the onset
of a symmetry breaking and can be exploited to assess the nature of the
realized electronic ordering. For a ground state hosting chiral currents,
there is an antisymmetric combination of $L$ and $S$. In this situation, the
spin-orbital texture of the electronic states at the Fermi level exhibit a
distinctive behavior:  Chiral currents carrying spin-orbital quadrupoles give
rise to orbital moments with the same parity as for mirror-symmetric systems,
while spin-orbital quadrupoles themselves show neither a time nor mirror-
symmetric profile. It is this physical case, that we focus on in this work.

To exemplify the concept of a chiral metal, one can employ the analogy with
chiral crystals and their symmetry properties.  One can then generally
identify a chiral metal with an electronic state that has a well-defined
handedness, due to the lack of inversion, mirror or other roto-inversion
symmetries [36, 37]. In this study, we start from this description to
introduce the concept of surface spin-orbital chiral metal to  indicate a
conducting electronic state of matter that has a well-defined handedness due
to an interaction driven by magnetochiral order that lacks mirror symmetries,
resulting from the internal spin-orbital structure,  but possesses the same
translational symmetry as the hosting crystal.

Here, we  show the relationship between the spin-orbital textures of the
electronic states, for both dipolar and quadrupolar channels, and the
consequential occurrence of a chiral electronic ordering. By supporting the
theory with circularly polarized spin-selective angular-resolved photoelectron
spectroscopy, we  introduce a methodology to probe, otherwise elusive,
symmetry-broken chiral electronic states.  To this aim, we exploit the
archetypal quantum material Sr2RuO4 (see Methods for growth and measurements
details) and reveal  signatures of a broken symmetry phase [8], compatible
with the formation of spin-orbital quadrupole currents at the material
surface.

Figure 1: Currents and symmetries in an  electronic system. a Legend of
possible charge, spin and orbital currents that can be created in a material.
As the charge can give rise through its motion to a conventional current, i.e.
charge current, spin and orbital dipoles or quadrupoles can also generate more
complex types of currents. Indeed, the spin and orbital angular momentum are
pseudo-vectors that change sign upon time-reversal symmetry, and then the spin
and orbital currents carrying spin or orbital dipoles are time-reversal
conserving. Instead, the currents carrying orbital or spin-orbital quadrupoles
break time-reversal symmetry and yield non-vanishing amplitudes for the dipole
and quadrupole observables at given momentum. b Examples of (top) mirror
preserving and (bottom) mirror broken configurations. In a system which
preserves time-reversal, a charge with its spin at a certain positive
momentum, under the action of such symmetries, goes into a charge with
opposite spin (directed still along the same direction but opposite in sign)
at negative (symmetry related) momentum. For mirror-symmetric configurations
the sign changes occurs when the spin lies in the mirror plane as depicted in
the sketch. The situation is different when a spin-orbital quadrupolar
current, which flows along a given direction is included. Such current couples
to the orbital ($L$) and spin ($S$) angular momentum, creating a strong
asymmetry in their product $LS$ (quadrupole) between symmetry-related momenta
in the Brillouin zone. This is why the spin is not longer reversed and picks
up an angle. c Experimental configuration adopted to measure the asymmetry of
$LS$ caused by the chiral current-driven breaking of mirror symmetry. The
Fermi surface (shown in both sheared and front view for clarity) of Sr2RuO4,
employed here as a test-bed for our theory. d Energy of the electrons as a
function of momentum collected by photoelectron spectroscopy for Sr2RuO4 along
the direction indicated in c by the dashed grey line.

Dichroic and spin-dichroic photoemission effects
As anticipated, assessing whether the profile of the $LS$ quadrupole
components has a mirror-and time-broken character is key for the detection of
electronic phases with electronic chiral currents.  Sr2RuO4 is an ideal
candidate to host symmetry-broken chiral ground states because of the recent
low-energy muon spin spectroscopy [8] and scanning tunnelling microscopy (STM)
measurements [38], which unveil the existence of unconventional magnetism and
electronic ordering forming at the surface.

To date, nevertheless, there are no accepted methodologies to ascertain the
existence of such  symmetry-broken chiral states. From our theory, in order to
detect the action of symmetries on $LS$, one has to probe the out-of-plane
components of the orbital angular-momentum ($L_{z}$) and spin ($S_{z}$).
Experimentally, the $LS$ asymmetry can be tackled by circularly polarized
spin-selective angular-resolved photoelectron spectroscopy (CP-Spin-ARPES)
[35, 39].  This approach requires extreme caution in the alignment and
geometry of the apparatus (Fig.1c). Indeed, photoelectrons from circularly
polarized light host a combination of intrinsic and geometric matrix elements
[40, 41]. Nevertheless, as shown in Ref.[41], these can be disentangled (see
Methods).

Figure 2: Circularly polarized spin-integrated ARPES. a (left) Unpolarized
ARPES spectrum from Sr2RuO4 along the direction orthogonal to the crystal
mirror plane, corresponding to the grey dashed line in Fig.1c. The spectrum
has been obtained by summing both contributions from (middle) right- and
(right) left-circularly polarized light. Here, we will refer to as
$C^{+,-}(\pm k,\uparrow,\downarrow)$ for indicating signals from right- (or
left-) circularly polarized light, collected at momentum $\pm k$, and with
spin-up (or down) component, respectively. b Circular dichroism of ARPES
spectrum obtained by subtracting the contributions from right and left-
circularly polarized light. Remarkably, the signal changes sign from $+k$ to
$-k$, with incoming light within the mirror plane. The asymmetry seen is
discussed in both the main text and Methods. c Energy-dependent circular
dichroism collected with spin-detector (VLEED) at the $k$-points indicated in
b. Figure 3: Circularly polarized spin-ARPES. a EDCs taken at six selected
momenta ($\pm k_{i}$, $i=1,2,3$) with fixed spins and circular polarizations.
In particular, the orange curves are obtained by measuring the EDCs at
positive $k$ values, right-circularly polarized light and spin-up channel
($C^{+}(k,\uparrow)$), while the green curves are obtained with negative $k$
values, left-circularly polarized light and spin-down channel
($C^{+}(-k,\downarrow)$). b ARPES spectra with reversed spin and circularly
polarized light configurations: orange curves refer to $C^{+}(-k,\uparrow)$,
while the green curves are obtained for $C^{-}(k,\downarrow)$. c ARPES image
indicating the $k$ values at which the EDCs have been taken. Note that both a
and b configurations show a difference, which is larger than the experimental
uncertainty. d The amplitudes of the circular dichroism (at $k$-summed up to
see the actual residual) are reported for both spin-integrated and spin-
resolved measurements. The data show that while the spin-integrated signal
(grey curve) shows a finite value, as large as 10$\%$ (which is also very
similar to the experimental uncertainty of 8$\%$ as shown in [39]), the spin-
resolved channels show a significantly larger amplitude, of a factor $\times
2$ and $\times 3$ for up and down channels, respectively. The amplitude values
have been extracted from the data shown in a-b and in Methods Fig.7, after
including the Sherman function and calculating the true spin polarization, as
described in the Methods. The other $k$ points indicated, as well as the
dichroic amplitude in MDC fashion are shown in Methods Figures 9-10 and
corroborate the validity of our result.

In Fig.2a, we show ARPES spectra collected  in the geometry of Fig.1c,
compatible with signals coming from both the bulk and surface states. The
latter appear weaker in spectral intensity but clearly visible with increased
contrast (see Methods). From left to right, in Fig.2a, spectra with
unpolarized, right-, and left-circularly polarized photons are shown. The
former shows an overall symmetric intensity pattern between features at $+k$
and at $-k$. From these spectra, the circular dichroism ($CD$) is extracted
(Fig.2b), and the signal $CD(+k)$ goes to $-CD(-k)$ at opposite momenta,
consistent with previous studies [42]. This behaviour is also seen in the
momentum-distribution curve (MDC) at the Fermi level (see the inset extracted
along the mauve line). Importantly, an attentive look at the spin-integrated
circular dichroism of Fig.2b reveals a small asymmetry in the residual of the
amplitudes (see Methods Figures 9-10), quantified as large as 10$\%$. This
value is slightly larger than the estimated experimental error on the
dichroism for this experimental setup (around 8$\%$). In addition, there might
be  a component of asymmetry related to the character of the chiral electronic
ordering (discussed in Supplementary Information). However, such an asymmetry
remains significantly smaller than the one measured for spin-resolved signals.
For completeness, spin-integrated data are collected with the VLEED spin
detector (Fig.2c at two $\pm k$ points) and show the same behaviour.

Importantly, the spin-integrated data of Fig.2c are not only consistent with
other ARPES works, but also, they do not reveal mirror-symmetry breaking
relatable to an anomalous behavior of $L_{z}$. On the other hand, quantities
such as  spin-orbital chiral currents, which strongly depend on the spin as
much as on the orbital angular-momentum, cannot be imaged by standard
circularly polarized ARPES. Here, we will refer to $C^{+,-}(\pm
k,\uparrow,\downarrow)$ for indicating signals from right- (or left-)
circularly polarized light, collected at momentum $\pm k$, and with a spin-up
(or down) component, respectively. In a perfectly symmetry-preserving
situation, the circularly-polarized spin-ARPES intensity transforms under the
mirror operator, from $C^{+}(+k,\uparrow)$ to $C^{-}(-k,\downarrow)$ (or
equivalently from $C^{+}(-k,\uparrow)$ to $C^{-}(+k,\downarrow)$). This means
that $C^{+}(+k,\uparrow)$ and $C^{-}(-k,\downarrow)$ (or $C^{+}(-k,\uparrow)$
and $C^{-}(+k,\downarrow)$) are expected to be the same under mirror symmetry
in the case that the latter is preserved [35]. However, if loop currents are
present along the surface, the mirror symmetry is broken and these quantities
are not equivalent anymore. This scenario  is what we tested with CP-Spin-
ARPES and reported in Fig.3a-b:  we observe subtle differences at different
$k$ points, as noted in Fig.3c, between $C^{+}(+k,\uparrow)$ and
$C^{-}(-k,\downarrow)$ (or $C^{+}(-k,\uparrow)$ and $C^{-}(+k,\downarrow)$).
These differences, despite being very small, result instead in a sizeable
asymmetry in the amplitudes of spin-up and down dichroism (Fig.3d). Note that
for the latter, positive and negative momenta have been summed, compensating
for possible instrumental asymmetry of the measurements.

Such a mirror-symmetry breaking in the amplitude of the spin-dichroism seems
compatible with the presence of spin-orbital quadrupole currents, as predicted
by the theory. In addition to this, the finite dichroism difference observed
experimentally is not seen for a temperature of 77 K, higher than the magnetic
transition temperature as indicated by the muons spectroscopy [8], however,
this requires further investigation as the thermal broadening becomes
significantly more pronounced (see Methods).  The important finding here, is
that the estimated difference in spin from the dichroic signals is up to a
factor of 3 times larger than the spin-integrated one . In Fig.3d, the
amplitude of the spin-integrated dichroic signal is approximately 10$\%$ (grey
curve), while the spin-resolved is as high as 20$\%$ for spin-up (red curve)
and 30$\%$ for spin-down (blue curve); see Supplementary Information for
possible explanation of the spin-integrated asymmetry observed. We emphasize
that a quantitative analysis is difficult and the signals detected are subtle.
To quantify these effects correctly, future measurements as a function of
photon energy and various geometries will be desirable. Nevertheless, the
presence of a sizeable asymmetry in the amplitude signal from $k$ and $-k$ is
observed and this is consistent with the theoretical predictions.

Figure 4: Orbital and spin-orbital textures in the presence of chiral
currents. a Computed Fermi surface of the Sr2RuO4. b Broken symmetry state
with an electronic pattern marked by either orbital-quadrupole (top panel) or
spin-orbital quadrupole currents (bottom panel), The sketch indicates a
current in real space connecting the Ru sites along the [110] direction. c
Electronic phase with chiral orbital-quadrupole currents: amplitude of the
orbital angular momentum ${L_{z}(n,k)}$ of the bands $|\psi_{n,k}\rangle$ with
$n=1,2$ evaluated along the ${\Gamma-X}$ direction
(${L_{z}(n,k)}=\langle\psi_{n,k}|\hat{L}_{z}|\psi_{n,k}\rangle$). For clarity,
we plot both ${L_{z}(n,k)}$ (blue) and $-{L_{z}(n,-k)}$ (yellow) for any given
momentum ${k}$ to directly compare the amplitudes at opposite momenta. d
Electronic phase with chiral orbital-quadrupole currents: amplitude of the
spin projected orbital angular momentum related to the out-of-plane spin-up
($+$) and down ($-$) components as selected by the projector
$(1\pm\hat{s}_{z})$. The amplitude is given by
${L^{\pm}_{z}(n,k)}=\langle\psi_{n,k}|(1\pm\hat{s}_{z})\hat{L}_{z}|\psi_{n,k}\rangle$.
The amplitudes of ${L_{z}(n,k)}$ and ${L^{\pm}_{z}(n,k)}$, displayed in c and
d, do not show any symmetry and do not match at ${k}$ and $-{k}$. e Electronic
phase with chiral spin-orbital quadrupole currents with antisymmetric $L\,S$
content with respect to the current flow direction $k$ (i.e. ${\bf
k}\cdot(\hat{\bf{L}}\times\hat{{\bf{s}}})$: for this configuration
${L_{z}(n,k)}$ and ${L_{z}(n,-k)}$ coincide. f Electronic phase with chiral
spin-orbital quadrupole currents with antisymmetric $L\,S$ combination: spin
projected orbital moment ${L^{\pm}_{z}(n,k)}$ at opposite momenta are unequal
in amplitude. Similar trends occur for the other bands (see Supplementary
Information).

Chiral currents phase
We now analyze our experimental results from a theoretical point of view. This
can be done with a two-dimensional tight-binding description of the electronic
structure of Sr2RuO4 based on the Ru $d$-orbitals ($d_{xy}$,$d_{xz}$,$d_{yz}$)
in such a way to capture the profile of the experimental Fermi surface
(Fig.4a). In this model, we include the broken symmetry states hosting orbital
and spin-orbital quadrupole currents that are driven by the $d-d$ Coulomb
interactions (see Supplementary Information for details). The internal
structure of the charge currents is provided by either the orbital quadrupole
$\hat{L}_{p}\hat{L}_{q}$ or by the spin-orbital quadrupole
$\hat{L}_{p}\hat{s}_{q}$ tensors (see also Fig.1a for differences between
these currents). For a given direction $k_{l}$ in momentum space, the
amplitude of the charge current propagating through the lattice can in
principle contain various components of the type
$\hat{j}^{l}_{o}=\sin(k_{l})\hat{L}_{p}\hat{L}_{q}$ and
$\hat{j}^{l}_{so}=\sin(k_{l})\hat{L}_{p}\hat{s}_{q}$ for the orbital and spin-
orbital quadrupole, respectively (note that the first does not contain any
spin channel, the second instead does). In the presence of currents flowing
along e.g. $l=x$ that break all mirror symmetries (i.e. $M_{j=x,y,z}$), the
orbital and spin-orbital quadrupoles have to include components with $\hat{s}$
and $\hat{L}$ that are perpendicular to $x$, thus lying in the $yz$ planes.
For instance, a term of the type $\sin(k_{x})\hat{L}_{y}\hat{L}_{z}$ does not
preserve the mirror symmetry for any choice of the $M_{j=x,y,z}$
transformations. The latter is exactly the case of Sr2RuO4, where a uniform
charge current flows with the direction of the momentum aligned along the
$l=[110]$ direction of the Ru lattice (see Fig. 4b). This pattern is
compatible with symmetry-allowed loop currents involving charge current
flowing from ruthenium to oxygen atoms at the octahedra length scale [8].  The
qualitative outcomes of the results are not altered by surface reconstruction
or by having currents flowing along other symmetry directions (see
Supplementary Information for details). We select a spin-orbital chiral state
that breaks the $C_{4}$ rotational symmetry  because it is compatible with the
scanning tunnelling microscopy findings [38]. However, spin-orbital chiral
states with loop currents that are rotational invariant can be also
constructed. They break translational symmetry and do not modify the
qualitative outcomes of the analysis (see Supplementary Information).

Additionally, the charge current along [110] is directly relevant when probing
the electronic states along the corresponding direction of the Brillouin zone
(along $\Gamma$-X as in our experiment). To evaluate the orbital and spin-
orbital textures for the electronic states in the presence of either orbital
or spin-orbital quadrupole currents, we will focus on such direction. For each
band eigenstate $|\psi_{n,k}\rangle$ at a given momentum $k$, we determine the
amplitude of the out-of-plane orbital moment
${L_{z}(n,k)}=\langle\psi_{n,k}|\hat{L}_{z}|\psi_{n,k}\rangle$ and the spin-
projected orbital moment
${L^{\pm}_{z}(n,k)}=\langle\psi_{n,k}|(1\pm\hat{s}_{z})\hat{L}_{z}|\psi_{n,k}\rangle$
(i.e. the out-of-plane spin-up ($+$) and down ($+$) components as singled out
by the projector $(1\pm\hat{s}_{z})$). These observables are related to the
dichroic and spin-dichroic amplitudes probed by ARPES, respectively (see
Supplementary Information for details). For clarity and simplicity, we display
only two representative bands, e.g. $n=1,2$ (the behavior for the other bands
is however qualitatively similar and reported in Supplementary Information).
As shown in Fig.4c, a broken symmetry state with orbital quadrupole currents
exhibits an asymmetry in the orbital angular moment at opposite momenta, i.e.
${L_{z}(n,k)}\neq-{L_{z}(n,-k)}$. Moreover, both the amplitude and the sign of
${L^{\pm}_{z}(n,k)}$ and ${L^{\pm}_{z}(n,-k)}$ are dissimilar (Fig.4d).
Instead, for spin-orbital currents (Fig.4b) with an antisymmetric combination
of $L$ and $S$, i.e. $j^{l}_{so}=\sin(k_{l})(\hat{L}\times\hat{s})_{l}$, we
find that the orbital angular momentum turns out to be  antisymmetric at $k$
and $-k$, namely ${L_{z}(n,k)}=-{L_{z}(n,-k)}$ (Fig.4e), while the spin-
projected orbital moment does not exhibit any symmetry relation among the
states at opposite momentum (Fig.4f). Importantly, the last asymmetry is the
same observed experimentally by the CP-Spin-ARPES. One way to grasp the origin
of this behavior is to inspect the structure of the equations of motion for
the amplitudes of the orbital ${L_{z}(n,k)}$ and spin ${s_{z}(n,k)}$ moments
(see Supplementary Information for details).  The chiral currents lead to
spin-orbital torques that in the case of the spin-orbital quadrupole current
result into a balance such as the amplitude of the orbital moment has a
symmetric behavior at symmetry-related momenta.

Additionally, we have also addressed the role of sublattice chiral currents
due to the surface reconstruction resulting from the octahedral rotations (see
Supplementary Information).  When we consider an inhomogeneous state with a
staggered amplitude modulation of the currents, we find that a small asymmetry
of the orbital moment is obtained. Nevertheless, its amplitude is
substantially smaller than that of the spin-projected orbital moment (see
Supplementary Information). This implies that non-homogeneous chiral currents
are also compatible with the observation of a much larger spin-dichroic
asymmetry as compared to the dichroic one. This behavior holds independently
of the selected band and momentum, differently from that of magnetic states,
e.g. antiferromagnetic, with a pattern in the spin and orbital moments that
break time and mirror symmetries (see Supplemental Information).

Conclusions
In conclusion, we discovered a spin-orbital and angular-momentum sensitive
methodology based on CP-Spin-ARPES, able to probe symmetry breaking compatible
with the existence of spin-orbital chiral currents.  While the study has been
applied to the archetypal Sr2RuO4, this methodology is general to all chiral
surface metals and constitutes a tantalizing experimental avenue to detect
symmetry-broken chiral states.  However, since the effect is subtle one cannot
exclude that the observed symmetry breaking might arise from other real space
chiral orderings that break both time and mirror symmetries. Nevertheless, our
work stimulates the combined use of circular dichroism and spin-selective
photoemission to investigate how all three quantities $L$, $S$, and $LS$
behave and their relationship to the crystal symmetries, which are markers for
hidden ordered phases.

The spin-dichroic signal we have used to detect the putative presence of
quadrupole spin-orbital currents at the surface of Sr2RuO4 can be used without
restrictions in other quantum materials, even when currents appear in the bulk
of a (centrosymmetric) crystal. In this situation, the currents might preserve
the combination of time-reversal with inversion symmetry, with the consequent
absence of dichroism. Nevertheless, the asymmetry of the spin-dichroic signal
could be still visible and therefore represents an efficient diagnostic tool
for spin-orbital chiral metallic phases.

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Methods
The samples of Sr2RuO4 were grown by floating zone technique, following the
procedure of Ref.[1]. Single crystals were post-cleaved in ultrahigh vacuum at
a base pressure of 1$\times 10^{-10}$ mbar and at a temperature of $20$ K (and
$77$ K). The latter was kept constant throughout the measurements. The
experiment was performed at the NFFA-APE Low Energy beamline laboratory at the
Elettra synchrotron radiation facility designed with an APPLE-II aperiodic
source for polarized EUV radiation and a vectorial twin-VLEED spin
polarization detector downstream of a DA30 Scienta ARPES analyser [2]. The
photon energy used for our measurements was $40$ eV, which is found to
maximize the spectral intensity, as also shown in Ref.[3]. The energy and
momentum resolutions were better than $12$ meV and $0.018$ Å-1, respectively.
Importantly, as already mentioned, we stress that in order to eliminate the
geometrical contribution to the circular polarization, the crystals were
aligned as in Fig.1c-d. For completeness, we report seminal works on ARPES and
dichroism which might help the understanding of the measurements performed in
our article, i.e., Ref.[4, 5, 6, 7, 8].

In this part of the Methods, we report additional measurements which
contributes to corroborate the message and conclusions in the main text.

Sample alignment and experimental geometry
When using circularly polarized light, the disentanglement between geometrical
and intrinsic matrix elements is crucial and challenging. A solution is to
have the incoming radiation exactly within one of the mirror planes of the
system studied and to measure in the direction orthogonal to that plane, as we
show in Fig. 1c of the main text. In such a configuration, the differences in
the CP-Spin-ARPES signal can be attributed to intrinsic differences in $LS$
and the geometrical contributions are well-defined. In this regard, it is of
paramount importance to align the sample very carefully. In the present case,
the symmetric character of the material’s Fermi surface [9, 10, 3] allows us
to carefully align the sample with the incoming beam of photons lying within a
mirror plane. The alignment of the sample was carried out by monitoring the
experimental Fermi surface and by making sure that the analyzer slit direction
was perpendicular to the mirror plane. As one can see in Figs. 5 and 6, we
estimated our alignment to be better than 0.9∘ from the ideal configuration, a
value within the uncertainty considering our angular azimuthal precision
($\approx$1∘). Furthermore, different samples mounted gave us the same
results, corroborating the robustness of the measurements outputs within this
azimuth uncertainty.

Figure 5: Photoemission experimental geometry. The sample, represented by the
purple box, is such that the incoming synchrotron radiation (red wavy arrow)
impinges with an angle of 45∘ with respect to its surface. In this
configuration, with linear polarizations, we would have linear vertical (LV,
green double-headed arrow) lying completely on the sample surface. Instead,
linear horizontal (LH, blue double-headed arrow) would have both in- and out-
of-plane components, projected along the $y$\- and $z$-axis, respectively. The
slit of the analyser is along the scattering plane (vertical slit).

In the laboratory APE-LE, our sample is placed in the manipulator in normal
emission conditions, with the synchrotron light impinging on the sample
surface at 45∘ angle. This means that standard linear polarizations, such as
linear vertical and linear horizontal (See Fig. 5), would act quite
differently on the matrix elements selection rules: In particular, linear
vertical light would be fully within the sample plane, while linear horizontal
would have a component within the plane and one out-of-plane (with 50$\%$
intensity each). Now, when using circularly polarized light, in order to
distinguish between real and geometrical matrix element effects, the incoming
light needs to be aligned within the experimental error, within one of the
mirror planes of the sample.

Figure 6: ARPES identification of the surface states and sample alignment. a
Fermi surface collected at 40 eV (sum of the two circularly polarized lights)
showing both bulk bands and surface states. The latter are weaker than the
bulk in intensity but still visible. To better appreciate the precise sample
alignment we fitted the data and extracted the $k$ positions, reported in the
image as red markers. The mirror plane deviates from the (b) ideal condition
by 0.9∘. c) Energy versus momentum dispersion collected in the same
experimental conditions of (Fig.5a)) showing a very symmetric character. To
better appreciate this, we extracted MDCs and plotted them in panel d along
with their extracted $k$ values.

To estimate the azimuthal value we have performed fitting of the $k$-loci of
the Fermi surface contours (red markers in Fig.6 a-b) and we have then aligned
the horizontal and vertical axis (see the next subsection for fitting
details). As one can see, in our configuration, there is negligible
misalignment between the states at positive and negative $k$, see Fig. 6 c-d
for example. The latter shows, that by extracting momentum-distribution curves
(MDCs), coloured horizontal lines in Fig. 6 c) the peak positions are
symmetric within the resolutions of the instrument ( 12 meV for energy and
$0.018$ Å-1). Therefore, we are in a good condition to safely perform the
measurements shown in the main text.

Details of the fitting
The Fermi surfaces $k$-loci shown in Fig. 6 a-b and the positions of the peaks
in Fig.5 d have been extracted by fitting the ARPES data. The fitting
procedure used is standard and consists of fitting both EDCs and momentum-
distribution curves by using Lorentzian curves convoluted by a Gaussian
contribution which accounts for the experimental resolutions. Then, as part of
the fit results, we extracted the $k$ positions of the peaks, which are those
reported as red markers in Fig. 6 and the values in Fig. 6 d.

Spin-ARPES data
For the extraction of the values reported, the spin-data shown in the
manuscript have been also normalized to include the action of the Sherman
function of the instrument. In particular, the data for spin-up and spin-down
channels have been normalised to their background, such that this matched in
both cases. In the present study, the background normalisation was executed on
the high-energy tails of the EDCs far from the region where the actual spin
polarization is observed. After normalisation, in order to extract the spin
intensity, we used the following relationships:

$I^{TRUE}(k,\uparrow)=\frac{I^{TOT}(k)}{2}*(1+P)$

$I^{TRUE}(k,\downarrow)=\frac{I^{TOT}(k)}{2}*(1-P)$

where $P$ is the polarization of the system and
$I^{TOT}=I^{bg.norm}(k,\uparrow)+I^{bg.norm}(k,\downarrow)$ is simply the sum
of the intensity for EDCs with spin-up and spin-down after normalisation to
the background. For the polarisation $P$, the Sherman function from the
instrument was included (i.e. $\eta$=0.3 [2] and calibrated from measurements
on gold single crystal). This is described by:

$P(k)=\frac{1}{\eta}*\frac{I^{bg.norm}(k,\uparrow)-I^{bg.norm}(k,\downarrow)}{I^{bg.norm}(k,\uparrow)+I^{bg.norm}(k,\downarrow)}$

This procedure was done for all light polarizations. Additionally, we also
characterized the spin channels by using different polarization vector
directions, as shown in Fig.7.

Figure 7: Spin-resolved data with unpolarised light. Energy distribution
curves collected at momenta a-d $k_{y}=0.75$ Å-1, $k_{y}=0.70$ Å-1,
$k_{y}=0.65$ Å-1, $k_{y}=0.60$ Å-1. The data are with sum of circular right
and left light and spin-up and spin-down channels have been shown in red and
blue, respectively.

Dichroism and spin-dichroism amplitudes
A way to visualise the breaking of the time-reversal symmetry is to analyse
the dichroic signal already shown in Fig. 2c of the main text but resolved in
the two different spin channels, i.e. up and down, which gives rise to very
different amplitude values when measured at $\pm k$ (expected for time-
reversal symmetry breaking but not expected otherwise).  We show this here at
a selected momentum values. The amplitude values have been extracted from the
data shown in a-b and in Fig. 7, after including the Sherman function
normalization.

Figure 8:  Spin-integrated and spin-resolved dichroism. Spin integrated
circular dichroism collected at a $\bm{k_{y}}=\pm{0.73}{\AA}^{-1}$ ($k_{1}$),
b $\bm{k_{y}}=\pm{0.68}{\AA}^{-1}$ ($k_{2}$), and c
$\bm{k_{y}}=\pm{0.72}{\AA}^{-1}$ ($k_{3}$), as indicated in the main text
Figure 3c. Green curves indicate negative $k$, and orange curve positive $k$.
d-e-f Spin-resolved circular dichroism collected at negative $k$ for the three
momenta indicated. g-i Same but collected at positive momenta.  Figure 9:
Amplitude of the dichroism, EDC and MDC. a-b-c The amplitudes of the dichroism
(at $k$-summed up to see the actual residual) are reported for $k_{1,2,3}$.
These show that, while d the spin-integrated signal (grey curve) shows a
finite value, as large as 10$\%$ (which is also very similar to the
experimental uncertainty as reported in [8] \- purple stripe), the spin-
resolved channels indicate a significantly larger amplitude, of a factor
larger than $\times 2$ and $\times 3$ for up and down channels, respectively.
e The amplitude of the dichroism have been also collected by using MDC at two
binding energies, i.e., at the Fermi level and at 150 meV below it. As one can
see, the grey line, which is the spin-integrated dichroism is nearly flat (in
average is 7$\%$ \- obtained by summing up all the points), while the spin-up
and spin-down channels are varying and well-different. The fact that these are
also varying is quite remarkable and indicates that our signal is intrinsic in
nature.

To better understand the claim in the main text, we here show in Fig.8 the
relative amplitudes of the dichroic versus spin-dichroic signal. First of all,
let us focus on the spin-integrated dichroism shown in Fig.8a. Here, the
orange and green curves represent positive and negative $k$ values,
respectively and their behaviour is overall symmetric with respect to zero.
However, a small asymmetry can still be noticed, estimated as large as 10$\%$,
a value which is very close to previously reported ones [8] ( 8$\%$). As we
will clarify from a theoretical point of view, a small degree of asymmetry in
the spin-integrated dichroism can be still expected, however, the amplitudes
of the dichroism selected in their spin channels are supposed to be quite
larger. To demonstrate this difference, we have shown how the dichroism
curves, resolved in their spin channels (up-red and down-blue), appear at
negative $k$ (Fig.8d-f) and at positive $k$ (Fig.8g-i). By also considering
their residuals, we can compare them to the amplitude of the spin-integrated
signal. We reported this comparison in Fig.9: The spin-down channel shows an
amplitude as high as 30$\%$, the spin-up as high as 20$\%$. These values are
three times and two times bigger, respectively, than the residual extracted
for the spin-integrated signal. Such a large difference corroborates the
validity of our methodology and the claims of our work. Note that by summing
the positive and negative momentum is also counteracting possible effects due
to small sample misalignment.

Data and temperature
For completeness, we have also performed $C^{+}(+k,\uparrow)$ and
$C^{-}(k,\downarrow)$ on the sample after cleaving it also at high temperature
(HT, 70 K), which is above the magnetic transition of Sr2RuO4. We report the
results in Fig.10. In particular, in Fig.10 a-c, the upper panels with blue
lines show the difference between $C^{+}(+k,\uparrow)$ and
$C^{-}(k,\downarrow)$ \- normalized by their sum - at three values of $k$ and
at low temperature (LT), while the lower line is the same for the data
collected at 70 K. If in the LT configuration we do observe a varying finite
signal, at HT we did not see such a variation. It is important to mention that
even with our resolution, we do not see any finite signal, there might be
still some differences which could be observed above the magnetic transition,
because it is likely that not all magnetic excitations are turned off
immediately, however, a reduction should be still observed. In addition, the
HT data are more noisy: even if we cleaved the samples at HT, and as the ARPES
shown in Fig.10 d-e confirms their presence, they are much weaker than in LT
and broadened thermally. Such a thermal broadening is not surprising to see in
ARPES. Nevertheless, even if with reduced intensity, the surface states are
still well visible.

Figure 10: Temperature dependence data. a-c The upper line (blue curves) shows
the difference between $C^{+}(+k,\uparrow)$ and $C^{-}(k,\downarrow)$ \-
normalized by their sum - at three values of $k$ and at low temperature (LT),
while the lower line is the same for the data collected at 70 K (high
temperature, HT), above the magnetic transition. When at LT we do observe a
varying finite signal, it starts from negative and it switches sign into
positive as a function of $k$, at HT we did not see such a variation. d and e
show the Fermi surface maps and energy versus momentum dispersion for both LT
and HT data. The surface states are visible in both cases.

Calibrating the V-LEED
Within the uncertainty of the instrument (1∘ integration region), the V-LEED
has been calibrated by acquiring spin EDCs at various angles, both positive
and negative values, for the sample. This is done for both spin species and
with the used light polarizations. In the present case, for consistency, we
did this with circularly polarized light (both left and right-handed). After,
by summing up both circular polarizations and both spin species, we can
reconstruct as in Fig. 11 the ARPES spectra. This procedure was done by using
only the spin-detector to access directly the states probed and be sure that
when selecting the angular values on the deflectors we probe effectively the
selected state.

Figure 11: Example of some ARPES from the spin-detector a Energy dispersions
as a function of negative $k$-values obtained by using the spin-detector only
in a course alignment scan. b Same as a but interpolated for a thicker angular
grid to see the bands better. c Energy dispersions as a function of positive
$k$-values obtained by using the spin-detector only in a course alignment
scan. d Same as c but interpolated for a thicker angular grid to see the bands
better.

Discussion about uncertainties and additional calibration
For evaluating the uncertainty, we used a controlled and known sample with no
asymmetries in the dichroic signal, as in our previous work (see Ref. [8]). We
used a kagome lattice because at the $\Gamma$ point there is a well-defined
energy gap opened due to the action of spin-orbit coupling. In addition, in
this point, the bands are spin-degenerate. Plus, the system is non-magnetic.
This allowed us to check the asymmetry not only in the circular dichroism
signal, but also in the spin-resolved circular dichroism. We estimated the
uncertainty to be approximately 10$\%$ on the residual of the dichroism. Note
that this is also consistent with what is obtained by standard ARPES in our
setup: at the centre of the Brillouin zone, the difference between circular
right and circular left polarized spectra (each spectrum was normalized by its
own maximum intensity beforehand) is indeed 10$\%$.

Supplementary Information

##  I Model and broken symmetry phases with chiral currents

We consider a two-dimensional system whose electronic states at the Fermi
level are described by $d$-orbitals belonging to the $t_{2g}$ manifold
($d_{xy}$,$d_{xz}$,$d_{yz}$) and can accommodate electronic phases which are
marked by non-trivial loop currents.

The model Hamiltonian for the surface of the Sr2RuO4 is compatible with the
$C_{4v}$ point group symmetry. For the $t_{2g}$ manifold of the $d_{a}$
orbitals we introduce the components of the local orbital angular momentum.
For the spin space, hereafter, we employ the Pauli matrices
$\hat{s}_{j=x,y,z}$. Assuming that the basis of the local creation operator of
electrons for the $d$-orbitals for a given spin orientation is
$\hat{d}^{\dagger}_{\bm{k}}=[c^{\dagger}_{xy,\bm{k}},c^{\dagger}_{yz,\bm{k}},c^{\dagger}_{xz,\bm{k}}]$,
we have that the components of the orbital angular momentum are given by:

$\displaystyle\hat{L}_{x}=\begin{pmatrix}0&-i&0\\\ i&0&0\\\
0&0&0\end{pmatrix}\hat{L}_{y}=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix}\hat{L}_{z}=\begin{pmatrix}0&0&0\\\ 0&0&-i\\\
0&i&0\,.\end{pmatrix}$

with $\hat{L}_{i}$ fulfilling the usual angular momentum algebra.

Then, in the complete spin-orbital basis, given by
$\hat{C}^{\dagger}_{\bm{k}}=[c^{\dagger}_{xy,\uparrow\bm{k}},c^{\dagger}_{yz,\uparrow\bm{k}},c^{\dagger}_{zx,\uparrow\bm{k}},c^{\dagger}_{xy,\downarrow\bm{k}},c^{\dagger}_{yz,\downarrow\bm{k}},c^{\dagger}_{zx,\downarrow\bm{k}}]$,
the Hamiltonian can be generally expressed as

$\hat{\mathcal{H}}=\sum_{\bm{k}}\hat{C}^{\dagger}_{\bm{k}}\hat{H}(\bm{k})\hat{C}_{\bm{k}},$
(2)

with $\hat{H}({\bf k})$ given by

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{\bm{k}}[(\epsilon_{xy}(k_{x},k_{y})-\mu)\mathbb{P}_{z}+(\epsilon_{xz}(k_{x},k_{y})-\mu)\mathbb{P}_{y}+$
$\displaystyle(\epsilon_{yz}(k_{x},k_{y})-\mu)\mathbb{P}_{x}+\epsilon_{xz,yz}(k_{x},k_{y})(\hat{L}_{x}\hat{L}_{y}+\hat{L}_{y}\hat{L}_{x})+$
$\displaystyle\alpha_{OR}(\sin(k_{x})\,\hat{L}_{y}-\sin(k_{y})\hat{L}_{x})]\hat{\sigma}_{0}+\lambda_{\mathrm{SO}}\hat{\bm{L}}\cdot\hat{\bm{\sigma}}\,.$

We recall that $\mathbb{P}_{a}=(\hat{L}^{2}-2\hat{L}_{a}^{2})/2$ is the
projector on the orbital state with a given distribution that is perpendicular
to the $a$ direction (e.g. for $a=z$ the projection is on the $xy$ orbital
configuration). Here, $\mu$ is the chemical potential. The term proportional
to $\epsilon_{xz,yz}(k_{x},k_{y})$ corresponds to the symmetry allowed
hybridization processes of the $(xz,yz)$ orbitals along the $[110]$ and
$[\bar{1}10]$ orientations. The choice of the following electronic parameters
reproduces with good accuracy the experimental profile of the Sr2RuO4 Fermi
lines at the surface as measured by ARPES (as in this paper and in Ref. 11,
see e.g. Fig. 2):

$\displaystyle\epsilon_{xy}(k_{x},k_{y})=$
$\displaystyle-2t_{3}[\cos(k_{x})+\cos(k_{y})]+$
$\displaystyle-4t_{4}\cos(k_{x})\cos(k_{y})+$
$\displaystyle-2t_{5}[\cos(2k_{x})+\cos(2k_{y})]$
$\displaystyle\epsilon_{yz}(k_{x},k_{y})=$
$\displaystyle-2t_{2}\cos(k_{x})-2t_{1}\cos(k_{y})$
$\displaystyle\epsilon_{xz}(k_{x},k_{y})=$
$\displaystyle-2t_{1}\cos(k_{x})-2t_{2}\cos(k_{y})$
$\displaystyle\epsilon_{xz,yz}(k_{x},k_{y})=$
$\displaystyle-4t_{6}\sin(k_{x})\sin(k_{y})$

with $t_{1}=0.145$, $t_{2}=0.016$, $t_{3}=0.08$, $t_{4}=0.039$, $t_{5}=0.005$,
$t_{6}=0.001$, $\mu=0.122$, $\lambda_{\text{SO}}=0.035$, in units of eV [11].
We assume a weak amplitude for the inversion symmetry breaking term
$\alpha_{OR}\sim 0.01$ eV, since the induced splitting is below the
experimental resolution. The term proportional to $\alpha_{OR}$ represents the
orbital Rashba interaction [12, 13, 14, 15] and couples the atomic angular
momentum ${\bf L}$ with the crystal wave-vector ${\bf k}$ due to the breaking
of inversion symmetry at the surface.

Then, in order to take into account the breaking of time-reversal and mirror
symmetry, we consider the spin-orbital chiral current phase along the
$l=[110]$ direction, as given the following term in the Hamiltonian:

$\hat{j}_{\text{so}}=\bm{g}_{\text{so}}\cdot(\hat{\bm{L}}\times\hat{\bm{\sigma}})\sin
k_{l}\,.$ (3)

Here, we take a representative $\bm{g}_{\text{so}}$ vector with amplitude
$|\bm{g}_{\text{so}}|=0.04$ eV, with generic coefficients, so that it does not
point in any high-symmetry direction in the lattice. The outcome of the
orbital and spin-orbital quadrupole moments within the Brillouin zone is not
qualitatively affected by the choice of the coefficients. This current term
can arise by a mean-field decoupling of the Coulomb interaction for nearest-
neighbor Ru centers (see Sect. IV). In Fig. 12 we report the orbital and spin-
orbital textures for all the bands assuming different types of broken symmetry
chiral phases marked by orbital and spin-orbital quadrupole currents.

Figure 12: Orbital and spin-orbital textures in the presence of chiral
currents. a Electronic phase with chiral orbital-quadrupole currents:
amplitude of the orbital angular momentum ${L_{z}(n,k)}$ of all bands
$|\psi_{n,k}\rangle$ evaluated along the ${\Gamma-X}$ direction
(${L_{z}(n,k)}=\langle\psi_{n,k}|\hat{L}_{z}|\psi_{n,k}\rangle$). For clarity
we plot both ${L_{z}(n,k)}$ and $-{L_{z}(n,-k)}$ for any given momentum ${k}$
to directly compare the amplitudes at opposite momenta. b Electronic phase
with chiral orbital-quadrupole currents: amplitude of the spin projected
orbital angular momentum
(${L^{\pm}_{z}(n,k)}=\langle\psi_{n,k}|(1\pm\hat{s}_{z})\hat{L}_{z}|\psi_{n,k}\rangle$).
The amplitudes of ${L_{z}(n,k)}$ and ${L^{\pm}_{z}(n,k)}$, displayed in a and
b, do not show any symmetry and do not match at ${k}$ and $-{k}$. c Electronic
phase with chiral spin-orbital quadrupole currents with antisymmetric $L\,S$
combination: ${L_{z}(n,k)}$ and ${L_{z}(n,-k)}$ coincide. d Electronic phase
with chiral spin-orbital quadrupole currents with antisymmetric $L\,S$
combination: spin projected orbital moment ${L^{\pm}_{z}(n,k)}$ at opposite
momenta are unequal in amplitude.

In order to capture the different behavior of the orbital and spin-orbital
moments in the presence of chiral currents one can consider the equation of
motion associated with the expectation values of the spin and orbital moments
for a given eigenstate at momentum $k$. In general, the time dependent
evolution of the spin and orbital angular momentum is provided by the equation
of motion $\frac{d{\bm{M}}}{dt}=\bm{\tau_{M}}$ with $\bm{M}=\bm{s,L}$ and
${\bm{\tau}}_{M}$ being the corresponding torques. For a given eigenstate
$|\psi_{nk}\rangle$ of the band $n$ at momentum $k$, one can write a torque
equation for the amplitudes of the spin
${\bm{s}}_{nk}=\langle\psi_{nk}|{\hat{\bm{\sigma}}}|\psi_{nk}\rangle$ and
orbital ${\bm{L}_{nk}}=\langle\psi_{nk}|\hat{{\bm{L}}}|\psi_{nk}\rangle$
moments. The equation of motion reads as
$\frac{d{\bm{M}}_{nk}}{dt}=\bm{\tau}^{M}_{nk}$ with $\textbf{M}=\textbf{s,L}$
and
$\bm{\tau}^{M}_{nk}=i\langle\psi_{nk}|[H,{\bm{\hat{M}}}]|\psi_{nk}\rangle$.
Now, since we deal with equilibrium configurations in time-independent
conditions the amplitudes of the spin and orbital moments have to fulfill the
relations $\tau^{M}_{nk}=0$ with $M=s,L$. These relations in turn set out the
amplitude of the spin and orbital moments.

For convenience and clarity, we introduce a Cartesian reference $xyz$ and,
without loss of generality, we assume that the chiral current in the broken
symmetry phase flows parallel to the $x$-direction. The spin-orbital current
in Eq. (3) can be then generally expressed as

$\displaystyle\hat{j}_{\text{so}}$
$\displaystyle=[g^{x}_{so}(\hat{L}_{y}\hat{\sigma}_{z}-\hat{L}_{z}\hat{\sigma}_{y})+g^{y}_{so}(\hat{L}_{z}\hat{\sigma}_{x}-\hat{L}_{x}\hat{\sigma}_{z})$
(4)
$\displaystyle+g^{z}_{so}(\hat{L}_{x}\hat{\sigma}_{y}-\hat{L}_{y}\hat{\sigma}_{x})]\sin
k_{x}$

Furthermore, it is also convenient to write the Hamiltonian in the following
compact form:

$\displaystyle\hat{H}=H_{0}+\hat{H}_{so}+\hat{j}_{\text{so}}$ (5)

with $H_{0}$ containing the kinetic and the orbital Rashba coupling terms and
$\hat{H}_{so}=\lambda_{so}\hat{\bm{L}}\cdot\hat{\bm{\sigma}}$. We notice that
$H_{0}$ preserves the vertical mirrors ($M_{x}$ and $M_{y}$) and the time-
reversal symmetries. Then, we point out that among the components of
$\hat{j}_{\text{so}}$, the term related to $g^{x}_{so}$ is the source of
chiral symmetry breaking, as all the mirror symmetries are broken. This is
because the components of $\hat{\mathbf{L}}$ and $\hat{\bm{\sigma}}$ are
pseudovectors and are perpendicular to the current flow direction. Instead,
the other components, related to $g^{y}_{so}$ and $g^{z}_{so}$, preserve the
$M_{x}$ vertical mirror symmetry.

Let us now consider the spin and orbital amplitudes of the torque by taking
for instance the corresponding orientations along the $x$-direction. For this
configuration, the commutators of the Hamiltonian with the $x$ component of
spin and orbital angular momentum yield

$\displaystyle[\hat{H}_{0},\hat{\sigma}_{x}]$ $\displaystyle=$ $\displaystyle
0$ (6) $\displaystyle\left[\hat{H}_{so},\hat{\sigma}_{x}\right]$
$\displaystyle=$ $\displaystyle
i\lambda_{so}(\hat{L}_{z}\hat{\sigma}_{y}-\hat{L}_{y}\hat{\sigma}_{z})\equiv
i\lambda_{so}\hat{A}$ (7)
$\displaystyle\left[\hat{j}_{so},\hat{\sigma}_{x}\right]$ $\displaystyle=$
$\displaystyle
i\left[h_{x}({k_{x}})\hat{B}_{x}+h_{y}({k_{x}})\hat{B}_{y}+h_{z}({k_{x}})\hat{B}_{z})\right]$
(8)

where $\hat{B}_{x}=\hat{L}_{y}\hat{\sigma}_{y}+\hat{L}_{z}\hat{\sigma}_{z}$,
$\hat{B}_{y}=\hat{L}_{x}\hat{\sigma}_{y}$,
$\hat{B}_{z}=-\hat{L}_{x}\hat{\sigma}_{z}$ and
$h_{i}(k_{x})=g_{so}^{i}\sin(k_{x})$ with $i=x,y,z$. On the basis of these
relations, the equilibrium condition for the spin torque yields:

$\displaystyle\tau^{s_{x}}_{nk}=\tau^{s_{x},so}_{nk}+\sum_{i=x,y,z}\tau^{s_{x},j^{i}_{so}}_{nk}=0\,$
(9)

with

$\displaystyle\tau^{s_{x},so}_{nk}$ $\displaystyle=$
$\displaystyle-\lambda_{so}\langle\psi_{nk}|\hat{A}|\psi_{nk}\rangle$ (10)
$\displaystyle\tau^{s_{x},j^{i}_{so}}_{nk}$ $\displaystyle=$ $\displaystyle-
h_{i}(k_{x})\langle\psi_{nk}|\hat{B}_{i}|\psi_{nk}\rangle\,.$ (11)

We notice that in the equation for $\tau^{s_{x}}_{nk}$ the torque arising from
the $x$ component of the spin-orbital current (i.e.
$\tau^{s_{x},j^{i}_{so}}_{nk}$) breaks the $M_{x}$ mirror symmetry. Instead,
the terms related to $g_{so}^{y,z}$ are mirror-symmetric. We also observe that
in the absence of the spin-orbital current the spin torque due to the spin-
orbit coupling is identically zero. This implies that the spin and orbital
moments must be collinear at any $k$. On the other hand, the presence of the
chiral spin-orbital currents leads to a deviation from the collinearity
between $s$ and $L$.

Let us now consider the analogous equations for the orbital moment. For the
commutators with $\hat{L}_{x}$ we have:

$\displaystyle[\hat{H}_{0},\hat{L}_{x}]$ $\displaystyle=$ $\displaystyle-
if_{y}({\bf k})\hat{L}_{z}+$
$\displaystyle-i\sum_{a=x,y,z}(g_{ay}\\{\hat{L}_{a},\hat{L}_{z}\\}-g_{zy}\\{\hat{L}_{a},\hat{L}_{y}\\}\;$
$\displaystyle\left[\hat{H}_{so},\hat{L}_{x}\right]$ $\displaystyle=$
$\displaystyle-i\lambda_{so}\hat{A}$
$\displaystyle\left[\hat{j}_{so},\hat{L}_{x}\right]$ $\displaystyle=$
$\displaystyle
i(-h_{x}({k_{x}})\hat{B}_{x}+h_{y}({k_{x}})\hat{C}_{y}+h_{z}({k}_{x})\hat{C}_{z})$

with $\hat{C}_{y}=\hat{L}_{y}\hat{\sigma}_{x}$ and
$\hat{C}_{z}=\hat{L}_{z}\hat{\sigma}_{x}$, and
$[\hat{H}_{0},\hat{L}_{x}]=i{\hat{Q}}({\bf k})$. Here, the coefficient
$f_{y}({\bf k})=\alpha_{OR}\sin(k_{x})$ is related to the orbital Rashba
coupling. Hence, the equation for the orbital moment torque becomes:

$\displaystyle\tau^{L_{x}}_{nk}=\tau^{L_{x},0}+\tau^{L_{x},so}_{nk}+\sum_{i=x,y,z}\tau^{L_{x},j^{i}_{so}}_{nk}=0\,$
(12)

with

$\displaystyle\tau^{L_{x},0}_{nk}$ $\displaystyle=$
$\displaystyle-\langle\psi_{nk}|\hat{Q}({\bf k})|\psi_{nk}\rangle$ (13)
$\displaystyle\tau^{L_{x},so}_{nk}$ $\displaystyle=$
$\displaystyle\lambda_{so}\langle\psi_{nk}|\hat{A}|\psi_{nk}\rangle$ (14)
$\displaystyle\tau^{L_{x},j^{x}_{so}}_{nk}$ $\displaystyle=$ $\displaystyle
h_{x}(k_{x})\langle\psi_{nk}|\hat{B}_{x}|\psi_{nk}\rangle$ (15)
$\displaystyle\tau^{L_{x},j^{i=y,z}_{so}}_{nk}$ $\displaystyle=$
$\displaystyle-h_{i}(k_{x})\langle\psi_{nk}|\hat{C}_{i}|\psi_{nk}\rangle\,.$
(16)

Now, we observe that $\tau^{s_{x},so}_{nk}=-\tau^{L_{x},so}_{nk}$ and
$\tau^{s_{x},j^{i}_{so}}_{nk}=-\tau^{L_{x},j^{i}_{so}}_{nk}$. This relation is
a hallmark of the phase with spin-orbital quadrupole currents. Taking into
account these relations and the Eq. (9) for the spin torque, one can replace
$\tau^{s_{x},so}_{nk}$ and $\tau^{s_{x},j^{i}_{so}}_{nk}$ in the Eq. (12) with
the combination of $\tau^{s_{x},j^{a=y,z}_{so}}_{nk}$. Hence, one can deduce
an equation for the orbital torque that contains only terms that are mirror
symmetry compatible. This implies that the expectation value of the orbital
moment has a definite parity as for mirror-symmetric systems.

Figure 13:  Fermi surface in the presence of the staggered octahedral rotation
and average asymmetry of orbital and spin-projected orbital moment. a Fermi
surface. b $k$-integrated and band-averaged orbital moment $\langle\delta
L_{z}\rangle=\frac{1}{12}\sum_{n}\int|L_{z}(n,k)+L_{z}(n,-k)|dk$ and
$\langle\delta
L_{z}^{\sigma}\rangle=\frac{1}{12}\sum_{n}\int|L_{z}^{+}(n,k)+L_{z}^{-}(n,-k)|dk$
as a function of the ratio of the staggered spin-orbital quadrupole current
$j^{stg}_{so}$ with respect to the homogeneous spin-orbital quadrupole current
$j_{so}$ component. Figure 14:  Orbital and spin-orbital moments in the
presence of uniform chiral currents and octahedral rotation pattern. Chiral
orbital-quadrupole currents: a orbital ${L_{z}(n,k)}$ for all bands
$|\psi_{n,k}\rangle$ evaluated along the $k_{y}=0$ direction ($\Gamma-X)$, b
spin-orbital ${L^{\pm}_{z}(n,k)}$. Chiral spin-orbital quadrupole currents: c
orbital ${L_{z}(n,k)}$ of all bands $|\psi_{n,k}\rangle$ evaluated along the
$k_{y}=0$ direction, d spin-orbital ${L^{\pm}_{z}(n,k)}$.

Let us now consider the case of the orbital quadrupole current:

$\displaystyle\hat{j}_{\text{o}}=$
$\displaystyle[g^{x}_{o}(\hat{L}_{y}\hat{L}_{z}+\hat{L}_{z}\hat{L}_{y})+g^{y}_{o}(\hat{L}_{z}\hat{L}_{x}+$
(17)
$\displaystyle\hat{L}_{x}\hat{L}_{z})+g^{z}_{o}(\hat{L}_{x}\hat{L}_{y}+\hat{L}_{y}\hat{L}_{x})]\sin
k_{x}\,.$

As for the case of the spin-orbital quadrupole current, the term related to
$g^{x}_{o}$ break both the horizontal ($M_{z}$) and vertical ($M_{y}$ and
$M_{x}$) mirror symmetries, while for the other current components, we have
that $g^{y}_{o}$ and $g^{z}_{o}$ preserve $M_{x}$. For this configuration one
can deduce that the spin torque is not affected (i.e. $\hat{\sigma}$ commutes
with $\hat{j}_{o}$) and that the orbital part includes all the torques due to
the $g^{a=x,y,z}_{o}$ current components. Then, the resulting orbital moment
has to explicitly manifest a breaking of the mirror perpendicular to the
current flow as demonstrated in Fig. 12.

###  I.1 Surface reconstruction

To take into account the surface reconstruction due to the staggered
octahedral rotation we double the unit cell, in order to include both
sublattices of the square lattice within the unit cell. Then, the term related
to the octahedral rotation can be included in the Hamiltonian as a staggered
potential for the $d$-states at the Ru site of the form:

$\hat{V}_{{\rm
stg}}=\eta_{\text{stg}}\hat{\tau}_{z}\left(\hat{L}_{x}\hat{L}_{y}+\hat{L}_{y}\hat{L}_{x}\right),$
(18)

where $\tau_{z}$ is a sublattice pseudo-spin operator given by the
corresponding Pauli matrix and $\eta_{\text{stg}}$ is a characteristic energy
scale of the rotation pattern. This term leads to a staggered splitting of the
$(xz,yz)$ orbitals at the Ru site that arises due to the local vertical
mirrors symmetry breaking resulting from the rotated octahedra. Here, for the
computation of the electronic structure, we use a representative value for
$\eta_{\text{stg}}=5$ meV. Variation of the strength of $\eta_{stg}$ does not
alter the qualitative outcome of the analysis. We also introduce a staggered
spin-orbital current component as given by the following contribution:

$\hat{j}_{\text{so}}^{\text{stg}}=\hat{\tau}_{z}\left[\bm{g}_{\text{so}}^{\text{stg}}\cdot(\hat{\bm{L}}\times\hat{\bm{\sigma}})\sin
k_{l}\right]\,.$ (19)

Hence, regarding the chiral current broken symmetry phase, we assume an
amplitude modulation of the spin-orbital current along the [110] direction so
that it is given by $j^{so}_{+}=j_{so}+j_{so}^{\text{stg}}$ when connecting Ru
sites in octahedra that are clockwise rotated and
$j^{so}_{-}=j_{so}-j_{so}^{\text{stg}}$ for anticlockwise rotated octahedra.
We have thus considered the possibility of having an amplitude modulation of
the spin-orbital quadrupolar currents that follows the surface reconstruction
due to the rotation of the RuO6 octahedra. Then, it is expected that the
ground state hosts chiral spin-orbital quadrupole currents which are not
spatially homogeneous, being affected by the structural change of the
octahedra, that result into different current amplitudes when linking
ruthenium sites with inequivalent octahedral rotation. We have analyzed this
configuration as a function of the current amplitude unbalance associated with
the two sublattices. For such a current pattern we find that the band resolved
dichroic asymmetry in the reconstructed electronic states (Fig. 13 a) is not
exactly zero (Fig. 13 b) as found for the uniform current configuration. In
order to get an overall estimate of the difference between the orbital and
spin-projected orbital moments as a function of the current sublattice
unbalance, we have followed their integrated values (i.e. summing up all the
band contributions and integrating along the $\Gamma-X$ line). The outcome is
reported in Fig. 13 b where one can clearly see that a non-vanishing orbital
moment is induced by the spatially modulated chiral currents. Nevertheless,
the asymmetry for the orbital moment turns out to be significantly smaller
than that related to the spin projected one. For instance, for a spatial
modulated configuration of the spin-orbital current with a sublattice
unbalance of around 3$\%$ we find that the integrated value of the spin-
projected orbital moment is about five times larger than that of the orbital
moment. The point here is that the spatially homogeneous chiral spin-orbital
current would lead to exact zero asymmetries in the dichroic amplitude due to
the balance among the torques arising from the chiral current and that one
arising from the spin-orbit coupling. Now, since the spin-orbit coupling at
the Ru site is homogeneous, a spatial dependent amplitude of the chiral spin-
orbital current cannot be compensated and a non-vanishing amplitude of the
dichroic signals can occur. Due to the surface reconstruction of the
electronic states in the Sr2RuO4 one might argue that small deviations of the
dichroic signals from zero can be consistent with the theoretical prediction
when considering the orthorhombic configuration due to the staggered
octahedral rotations around the $c$-axis. The main finding is that one can
also account for an amplitude asymmetry of the dichroic signal keeping however
a significantly larger spin-dichroic asymmetry as compared to the dichroic
one.
Finally, we have verified that for a uniform configuration of the spin-orbital
quadrupole currents the asymmetry of the orbital moment at $k$ and $-k$ is
identically zero irrespective of the surface reconstruction (Fig. 14).

###  I.2 C4 rotational invariant spin-orbital chiral phases

We have considered states with spin-orbital chiral loop currents that preserve
the C4 rotational symmetry (see Fig. 15). These configurations lead to a
symmetric dichroic amplitude and an asymmetric spin-dichroic value as related
to the orbital moment ${L_{z}(n,k)}$ and spin projected orbital orbital moment
${L^{\pm}_{z}(n,k)}$ for all bands $|\psi_{n,k}\rangle$ evaluated along the
$k_{y}=0$ direction. In particular, due to the $C_{4}$ symmetries the orbital
moment ${L_{z}(n,k)}$ can be vanishing. This implies that for this type of
configurations we also expect a large difference between the asymmetry in the
dichroic and spin-dichroic signals.

###  I.3 Canted antiferromagnetic order

We consider the case of a magnetic order with a canted antiferromagnetic
configuration. The broken symmetry state is introduced by an effective
magnetic term in the spin channel with uniform/staggered magnetization:

$H_{\text{mag}}=g\mu_{B}\left(\hat{\tau}_{0}\bm{M}_{\text{uni}}+\hat{\tau}_{z}\bm{M}_{\text{stg}}\right)\cdot\hat{\bm{\sigma}}\,,$
(20)

with ${\bm{M}_{\text{uni}}}$ and ${\bm{M}_{\text{stg}}}$ being the uniform and
staggered magnetization components. We have computed, for a representative
antiferromagnetic configuration with canted moments, the orbital and spin-
resolved orbital moment for all the bands crossing the Fermi level (Fig. 16).
As one case see from the inspection of the results in Fig. 16 the orbital and
spin-projected orbital moments exhibit a sizable asymmetry when comparing the
amplitudes at $k$ and $-k$. This is a general feature of all the magnetic
phases with similar symmetry content with respect to mirror and time reversal
symmetry and based on a long-range spatial order of Ru spin moments.

Figure 15: Schematic configuration of a $C_{4}$ rotational invariant spin-
orbital chiral state. a The arrows indicate the current direction on a given
bond associated with the corresponding spin-orbital momentum. b Schematic
representation of a spin-orbital chiral state with $C_{4}$ rotation symmetry.
Figure 16:  Orbital and spin-orbital textures in the presence of canted
antiferromagnetism and staggered octahedral rotation. a orbital moment
${L_{z}(n,k)}$ and b spin projected orbital orbital moment
${L^{\pm}_{z}(n,k)}$ for all bands $|\psi_{n,k}\rangle$ evaluated along the
$k_{y}=0$ direction. The amplitudes of the magnetizations are:
$\bm{M}_{\text{stg}}=\left(0,0,0.1\right)$ and
$\bm{M}_{\text{uni}}=\left(0.02,0,0\right)$ in units of eV$(g\,\mu_{B})^{-1}$
($g$ is the electron $g$-factor and $\mu_{B}$ is the Bohr magneton.

##  II Spin resolved circularly polarized ARPES: matrix elements and orbital
angular momentum

We report here the main steps to deduce the expressions for the dichroic and
spin-dichroic ARPES transition amplitudes. To this aim, we follow the
derivation reported in Ref. [13] and adapt it to the spin-dichroic case too.
The starting point is to consider that the circular dichroic signals probed by
ARPES can be expressed through the normalized amplitude as

$\displaystyle
D(k)=\sum_{\sigma}\frac{\left[I^{R}_{\sigma}-I^{L}_{\sigma}\right]}{I^{R}_{\sigma}+I^{L}_{\sigma}}\,.$
(21)

The sum over the final state spin $\sigma$ indicates the spin-integrated
nature of the detection approach. Thus, one can also introduce the spin-
resolved dichroic amplitude as given by

$\displaystyle
D_{s}(k)=\frac{\left[I^{R}_{s}-I^{L}_{s}\right]}{I^{R}_{s}+I^{L}_{s}}\,,$ (22)

here for convenience we assume that the spin configuration is along the
$z$-axis and are labeled by $s=+(-)$ to indicate the
$\uparrow_{z}(\downarrow_{z})$ spin states, respectively. Similar expressions
can be derived for the all the other spin orientations.
To calculate the amplitude of the dichroic and spin-dichroic signal one has to
evaluate the transition probability $I$ for an optical excitation between the
initial and final states. The interaction with the photon is given by
$H_{int}={\bf A}\cdot{\bf p}$. The vector potentials are ${\bf
A}=\frac{\epsilon_{1}+i\epsilon_{2}}{\sqrt{2}}$ for right circularly polarized
photons, and ${\bf A}^{*}=\frac{\epsilon_{1}-i\epsilon_{2}}{\sqrt{2}}$ for
left circularly polarized photons. Hence, the vector ${\bf A}\times{\bf
A}^{*}=-i{\bf{\hat{k}}}_{ph}$ sets out the incident photon direction. For
convenience and clarity, we report the main steps and expressions of the
derivation to show the link of the ARPES intensity with the orbital and spin-
projected angular momentum. Let us consider as initial state the Bloch
configuration with momentum $k$ that is given by
$|\psi^{I}_{k}\rangle=\frac{1}{\sqrt{N}}\sum_{i,\alpha,\sigma}\exp[ik\cdot
r_{i}]u_{\alpha,\sigma}(k)|i,\alpha,\sigma\rangle$, where
$|u\rangle=\sum_{\alpha,\sigma}u_{\alpha,\sigma}(k)|i,\alpha,\sigma\rangle$ is
the Wannier configuration centered at the site $r_{i}$ associated with the
$t_{2g}$ orbitals labeled by $\alpha=xy,xz,yz$ and spin $\sigma$. A plane-wave
state is assumed for the final state $|F\rangle=\int\exp[ik_{F}r]|r\rangle$
[16, 17] with $k_{F}$ being the Fermi wave vector. We recall that the expected
value of the orbital angular momentum evaluated on the state
$|\psi^{I}_{k}\rangle$ is given by $\langle\hat{L}\rangle=i\sum_{s}u_{s}\times
u^{*}_{s}(k)$, with ${\bf u}_{s}=(u_{yz,s},u_{zx,s},u_{xy,s})$ and the spin
projected orbital components are expressed as
$\langle\hat{L}(1+s\hat{\sigma}_{z})\rangle=iu_{s}(k)\times u^{*}_{s}(k)$,
with $s=\pm$ singling out the spin up and down configuration with respect to
the $z$-direction, respectively. Following the derivation in Ref. [13], one
can show that the spin resolved transition probability can be generally
expressed as

$\displaystyle D_{\sigma}=\frac{({\bf A}\times{\bf A}^{*})\cdot\nabla
g_{\sigma}\times\nabla g^{*}_{\sigma}}{\left[({\bf A}\cdot\nabla
g_{\sigma})({\bf A}^{*}\cdot\nabla g^{*}_{\sigma})+({\bf A}^{*}\cdot\nabla
g_{\sigma})({\bf A}\cdot\nabla g^{*}_{\sigma})\right]}$ (23)

with

$\displaystyle g_{\sigma}(k_{F})={\bf u}_{\sigma}\cdot\nabla_{k_{F}}f(k_{F})$
(24)

where the gradients are respect to the Fermi wave vector $k_{F}$, and $f(k)$
is the Fourier transform of the part of the Wannier function that depends only
on the radial distance $r-r_{i}$ related to the atomic center at $r_{i}$. The
denominator of $D$ is always positive and has a minor role in the dependence
of the matrix elements from the orbital angular momentum. The key quantity for
our purposes is given by the factor $\Gamma_{s}=\nabla g_{s}\times\nabla
g^{*}_{s}$. One can show [13] that the term $\Gamma_{s}$ can be expressed as

$\displaystyle\Gamma_{s}=$
$\displaystyle\frac{1}{2}\varepsilon^{\alpha\beta\gamma}\left({\bf
u}_{s}\times{\bf u}_{s}^{*}\right)_{\alpha}\nabla P_{\beta}f\times\nabla
P_{\gamma}f$ (25) $\displaystyle=$ $\displaystyle
i\frac{1}{2}\varepsilon^{\alpha\beta\gamma}\langle\hat{L}_{\alpha}(1+s{\hat{\sigma}}_{z})\rangle\nabla
P_{\beta}f\times\nabla P_{\gamma}f$ (26)

with the vector ${\bf P}$ having the following components
$(Q_{yz},Q_{zx},Q_{xy})$ with $Q_{ij}=\partial_{i}\partial_{j}$. We notice
that while the structure of the orbital angular momentum is similar to that of
$p$-orbitals because we are using an effective $L=1$ manifold for the $t_{2g}$
sector, the second order differential operator $Q_{ij}$ takes into account the
different structure of the orbital configurations. In a similar way, one can
show that the spin integrated $\Gamma$ amplitude is given by

$\displaystyle\Gamma=i\frac{1}{2}\varepsilon^{\alpha\beta\gamma}\langle\hat{L}_{\alpha}\rangle\nabla
P_{\beta}f\times\nabla P_{\gamma}f\,,$ (27)

with $\varepsilon^{\alpha\beta\gamma}$ the Levi-Civita tensor. Then, one can
see that the amplitude $\Gamma$ and thus the dichroic signal is proportional
to the projected orbital angular momentum components
$\langle\hat{L}_{\alpha}\rangle$ with respect to the incident photon
direction. Additionally, the spin-dichroic signal is proportional to the spin-
projected orbital angular momentum as given
$\langle\hat{L}_{\alpha}(1+s{\hat{\sigma}}_{z})\rangle$ with $s=\pm$ for
projecting spin up and down configurations, respectively. The remaining form
factors depend on the Fermi momentum and on the photon energy. Since our study
aims to have a qualitative understanding of the anomalies in the transition
amplitude probed by Spin CD-ARPES, their contribution does not affect the
conclusions of our findings.
In the employed experimental setup the spin orientation is selected in a
direction that is perpendicular to the surface (i.e. $z$) and the incident
photon direction is primarily selecting the out-of-plane $z$-component of the
orbital angular momentum. Hence, the dichroic and spin-dichroic signal are
proportional to $\langle\hat{L}_{z}\rangle$ and
$\langle\hat{L}_{z}(1\pm{\hat{\sigma}}_{z})\rangle$ as considered in the
manuscript.

##  III Current driven phase by Coulomb interaction

Let us now present how the electronic current phase arises as a broken
symmetry state with a nonvanishing expectation value of the current operator
on the Ru-Ru bond due to the Coulomb interaction. Indeed, in order to
demonstrate this point one needs to introduce the spin-orbital asymmetric
operator

$\displaystyle\phi^{\alpha\beta}_{\sigma,\sigma^{\prime}}(l,m)=i\left(c^{\dagger}_{\alpha,\sigma}(l)c_{\beta,\sigma^{\prime}}(m)-c^{\dagger}_{\beta,\sigma^{\prime}}(m)c_{\alpha,\sigma}(l)\right)$
(28)

for the $l-m$ bond between two Ru atoms with position identified by the
coordinates $R_{l}$ and $R_{m}$. Here,
$c_{\alpha,\sigma}(l)(c^{\dagger}_{\alpha,\sigma}(l))$ are the annihilation
(creation) operators associated with an electronic state with $\alpha$ orbital
and spin $\sigma$ at the atomic site $R_{l}$. Then, the spin and orbital
dependent terms that build up the density-density inter-site Coulomb
interaction $U_{lm}$ for a generic $l-m$ bond can be written in the following
form

$\displaystyle U_{lm}n_{\alpha,\sigma}(l)n_{\beta,\sigma^{\prime}}(m)=$
$\displaystyle-\frac{1}{2}U_{lm}(\phi^{\alpha\beta}_{\sigma,\sigma^{\prime}}(l,m))^{\dagger}\phi^{\alpha\beta}_{\sigma,\sigma^{\prime}}(l,m)$
(29)
$\displaystyle+\frac{1}{2}U_{lm}\left(n_{\alpha,\sigma}(l)+n_{\beta,\sigma^{\prime}}(m)\right)$

where the orbital and spin resolved density operators, $n_{\alpha,\sigma}(l)$,
are defined as
$n_{\alpha,\sigma}(l)=c^{\dagger}_{\alpha,\sigma}(l)c_{\alpha,\sigma}(l)$.
Hence, by decoupling the quartic term, one can introduce an order parameter
associated with the expectation value of
$\phi^{\alpha\beta}_{\sigma,\sigma^{\prime}}(l,m)$ and express the interaction
as

$\displaystyle U_{lm}n_{\alpha,\sigma}(l)n_{\beta,\sigma^{\prime}}(m)\sim$
$\displaystyle-\frac{1}{2}[\langle\phi^{\alpha\beta}_{\sigma,\sigma^{\prime}}(l,m)\rangle(\phi^{\alpha\beta}_{\sigma,\sigma^{\prime}}(l,m))^{\dagger}+h.c.$
(30)
$\displaystyle-|\langle\phi^{\alpha\beta}_{\sigma,\sigma^{\prime}}(l,m)\rangle|^{2}]$

where the average value indicates the summation over all the electronic states
weighted by the Fermi distribution function. Taking into account the spin-
orbital order parameters on the $l-m$ bond, by suitable superposition of the
$\phi$ operators one can construct a bond current order parameter that is
given by the expectation value of the following orbital and spin-orbital
quadrupole current operators:

$\displaystyle
J^{o}_{i,j}(l,m)=i\left(\vec{c}^{\,\,\dagger}(l)\hat{L}_{i}\hat{L}_{j}\vec{c}\,(m)-h.c.\right)$
$\displaystyle
J^{so}_{i,j}(l,m)=i\left(\vec{c}^{\,\,\dagger}(l)\hat{L}_{i}\hat{S}_{j}\vec{c}\,(m)-h.c.\right)$
(31)

with
$\vec{c}^{\,\,\dagger}(l)=\left[c^{\dagger}_{xy,\uparrow}(l),c^{\dagger}_{yz,\uparrow}(l),c^{\dagger}_{zx,\uparrow}(l),c^{\dagger}_{xy,\downarrow}(l),c^{\dagger}_{yz,\downarrow}(l),c^{\dagger}_{zx,\downarrow}(l)\right]$
and $i,j\in\\{x,y,z\\}$.

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Acknowledgements
F.M. greatly acknowledges the SoE action of PNRR, number SOE_0000068. M.C.,
R.F., M.T.M., and A.V. acknowledge support from the EU’s Horizon 2020 research
and innovation program under Grant Agreement No. 964398 (SUPERGATE). M.C. and
G.P. acknowledge financial support from PNRR MUR project PE0000023-NQSTI.
M.C., R.F., A.G., and A.V. acknowledge partial support by the Italian Ministry
of Foreign Affairs and International Cooperation, grant number KR23GR06. W.B.
acknowledges support by Narodowe Centrum Nauki (NCN, National Science Centre,
Poland) Project No. 2019/34/E/ST3/00404 and partial support by the Foundation
for Polish Science through the IRA Programme co-financed by EU within SG OP.
This work was performed in the framework of the Nanoscience Foundry and Fine
Analysis (NFFA-MUR Italy) facility. J.A.M acknowledges financial support from
DanScatt (7129-00011B).