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The samples of Sr2RuO4 were grown using the floating-zone technique, following a previously published procedure43. Single crystals were postcleaved in an ultrahigh vacuum at a base pressure of 1 × 10−10 mbar and a temperature of 20 K (and 77 K). The temperature was kept constant throughout the measurements. The experiment was performed at the NFFA–APE Low Energy beamline laboratory at the Elettra synchrotron radiation facility and designed with an APPLE-II aperiodic source for polarized extreme UV radiation and a vectorial twin-VLEED spin-polarization detector downstream of a DA30 Scienta ARPES analyser44. The photon energy used for our measurements was 40 eV, which was found to maximize the spectral intensity, as shown previously45. The energy and momentum resolutions were better than 12 meV and 0.018 Å−1, respectively. Importantly, as already mentioned, to eliminate the geometrical contribution to the circular polarization, the crystals were aligned as in Fig. 1c,d. For completeness, seminal works on ARPES and dichroism that might aid the understanding of our measurements can be found in refs. 39,41,46,47,48.

In the following sections, we report additional measurements that help to corroborate the message and conclusions given in the main text.

Sample alignment and experimental geometry

When using circularly polarized light, the disentanglement between geometrical and intrinsic matrix elements is crucial but problematic. 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. 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 carefully. In the present case, the symmetric character of the material’s Fermi surface45,49,50 allows us to carefully align the sample with the incoming beam of photons lying in a mirror plane. The alignment of the sample was carried out by monitoring the experimental Fermi surface and by making sure that the analyser slit direction was perpendicular to the mirror plane. As shown in Extended Data Figs. 1 and 2, we estimated our alignment to be better than 0.9° from the ideal configuration, a value within the uncertainty considering our angular azimuthal precision (about 1°). Furthermore, different samples gave us the same results, corroborating the robustness of the measurement outputs within this azimuth uncertainty.

In the NFFA–APE Low Energy beamline laboratory, our sample was placed in the manipulator in normal emission conditions, with the synchrotron light impinging on the sample surface at an angle of 45°. This means that standard linear polarizations, such as linear vertical and linear horizontal (Extended Data Fig. 1), would act differently on the matrix elements’ selection rules. In particular, linear vertical light would be fully within the sample plane, whereas linear horizontal light would have one component within the plane and one out of plane (with 50% intensity each). Now, when using circularly polarized light, to distinguish between real and geometrical matrix element effects, the incoming light needed to be aligned within the experimental error, within one of the mirror planes of the sample.

To estimate the azimuthal value we fitted the k-loci of the Fermi surface contours (red markers in Extended Data Fig. 2a,b) and we then aligned the horizontal and vertical axes (see ‘Details of the fitting’). In our configuration, there is negligible misalignment between the states at positive and negative values of k (Extended Data Fig. 2c,d). In Extended Data Fig. 2, we show that by extracting momentum distribution curves (coloured horizontal lines in Extended Data Fig. 2c), the peak positions are symmetric within the resolution of the instrument (12 meV for energy and 0.018 Å−1). We can therefore confidently perform the measurements shown in the main text.

Details of the fitting

The k-loci of the Fermi surfaces shown in Extended Data Fig. 2a,b and the positions of the peaks in Extended Data Fig. 1d have been extracted by fitting the ARPES data. The fitting procedure used is standard and consists of fitting both energy distribution curves (EDCs) and momentum distribution curves by using Lorentzian curves convoluted by a Gaussian contribution that accounts for the experimental resolutions. Then, as part of the fit results, we extracted the k positions of the peaks, which are shown as red markers in Extended Data Fig. 2 and the values in Extended Data Fig. 2d.

Spin-ARPES data

To obtain the values reported, the spin data shown have also been 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 normalized to their background, so they matched in both cases. In the present study, the background normalization was done on the high-energy tails of the EDCs far from the region where the spin polarization was observed. After normalization, to extract the spin intensity, we used the following relations:

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

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

where P is the polarization of the system, ITRUE is the intensity value (for either spin-up or -down species) obtained after inclusion of the Sherman (see below) function of the spin detector, and  and ITOT = Ibg.norm(k, ↑) + Ibg.norm(k, ↓) is simply the sum of the intensity for EDCs with spin-up and spin-down after normalization to the background. For the polarization P, the Sherman function from the instrument was included and defined as η = 0.3 (ref. 44). The Sherman function was calibrated from measurements on a single gold crystal. Therefore, P is described by:

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

This procedure was done for all light polarizations. We also characterized the spin channels by using different polarization-vector directions, as shown in Extended Data Fig. 3.

Dichroism and spin-dichroism amplitudes

A way to visualize the breaking of the time-reversal symmetry is to analyse the dichroic signal shown in Fig. 2c but resolved in the two different spin channels, up and down, which gives rise to different amplitude values when measured at ±k (expected for time-reversal symmetry breaking but not expected otherwise). We show this here at selected momentum values. The amplitude values have been extracted from the data shown in Fig. 3a and Extended Data Fig. 3, after including the Sherman function normalization.

To corroborate the claim in the main text, that is, the observation of a signal compatible with the existence of chiral currents, Extended Data Fig. 4 shows the relative amplitudes of the dichroic versus spin-dichroic signal. First, let us consider the spin-integrated dichroism shown in Extended Data Fig. 4a. 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 to be as large as 10%, which is close to a previously reported value39 of 8%. As we will clarify from a theoretical point of view, a small degree of asymmetry in the spin-integrated dichroism can still be expected, although the amplitudes of the dichroism selected in their spin channels are supposed to be 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 (Extended Data Fig. 4d–f) and at positive k (Extended Data Fig. 4g–i). By also considering their residuals, we can compare them to the amplitude of the spin-integrated signal. We reported this comparison in Extended Data Fig. 5. The spin-down channel shows an amplitude as high as 30% and the spin-up one is 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 summing the positive and negative momentum is also counteracting any possible effects caused by small sample misalignment.

Data and temperature

For completeness, we also performed C+(+k, ↑) and C(−k, ↓) on the sample after cleaving it, also at high temperature (70 K), which is above the magnetic transition of Sr2RuO4. We report the results in Extended Data Fig. 6. In particular, in Extended Data Fig. 6a–c, the top panels with blue lines show the difference between C+(+k, ↑) and C(−k, ↓), normalized by their sum, at three values of k and at low temperature, but the bottom line is the same for the data collected at 70 K. If in the low-temperature configuration we observe a varying finite signal, at high temperature we did not see such a variation. It is important to mention that even with our resolution, we do not see any finite signal, although there might be some differences that could be observed above the magnetic transition, because it is likely that not all magnetic excitations are turned off immediately, although a reduction should be still observed. Furthermore, the high-temperature data are more noisy. Even if we cleaved the samples at high temperature, and the ARPES shown in Extended Data Fig. 6d,e confirms their presence, they are much weaker than at low temperature and are broadened thermally. Such a thermal broadening is not surprising to see in ARPES. Nevertheless, even with reduced intensity, the surface states are still clearly visible.

Calibrating the VLEED

Within the uncertainty of the instrument (1° integration region), the VLEED has been calibrated by acquiring spin EDCs at various angles, both positive and negative, 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). Afterwards, by summing both circular polarizations and both spin species, we can reconstruct the ARPES spectra (Extended Data Fig. 7). This procedure was done by using only the spin detector to directly access the probed states and be sure that, when selecting the angular values on the deflectors, we effectively probe the selected state.

Uncertainties and additional calibration

To evaluate the uncertainty we used a controlled and known sample with no asymmetries in the dichroic signal, as in our previous work39. We used a kagome lattice because at the Γ point there is a well defined energy gap, opened by the action of spin–orbit coupling. Furthermore, at this point the bands are spin-degenerate; the system is also not 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 that obtained by standard ARPES in our set-up: 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%.

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