Spin-Orbital Entanglement and the Breakdown of Singlets and Triplets in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ Revealed by Spin- and Angle-Resolved Photoemission Spectroscopy

Spin-orbit coupling has been conjectured to play a key role in the low-energy electronic structure of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. By using circularly polarized light combined with spin- and angle-resolved photoemission spectroscopy, we directly measure the value of the effective spin-orbit coupling to be $130\pm 30\mathrm{meV}$. This is even larger than theoretically predicted and comparable to the energy splitting of the $d_{xy}$ and $d_{xz,yz}$ orbitals around the Fermi surface, resulting in a strongly momentum-dependent entanglement of spin and orbital character in the electronic wavefunction. As demonstrated by the spin expectation value $⟨\overrightarrow{s_k}·\overrightarrow{s_{-k}}⟩$ calculated for a pair of electrons with zero total momentum, the classification of the Cooper pairs in terms of pure singlets or triplets fundamentally breaks down, necessitating a description of the unconventional superconducting state of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ in terms of these newly found spin-orbital entangled eigenstates.

Figure 1

(a) Electronic band structure from along the high-symmetry directions and (b) $k_z=0$ Fermi surface calculated without (thin black) and with (thick, color-coded to show $⟨\mathrm{l⃗}·\mathrm{s⃗}⟩$) the inclusion of SO coupling; at the $\mathrm{\Gamma }$ point, the latter gives rise to a ${\zeta }_{\text{eff}}\sim 90\mathrm{meV}$ splitting [note that $Z\equiv (0,0,\pi /c)$, $\mathrm{\Gamma }\equiv (0,0,0)$, $M\equiv (\pi /a,0,0)$, $X\equiv (\pi /a,\pi /a,0)$]. (c,d) Three-dimensional Fermi surface sheets color coded to show (c) orbital character and (d) the expectation value $⟨\mathrm{l⃗}·\mathrm{s⃗}⟩$, in the first Brillouin zone [25]. The energy and momentum location of the spin-resolved ARPES spectra presented in Fig. 2 is marked in yellow in panels (a) and (b).

Figure 2

(a) Spin-integrated ARPES data measured with 24 eV photons at $\mathrm{\Gamma }$, as highlighted in Fig. 1. (a,b) Measured polarization asymmetry of the photoemitted electrons, and (d-f) corresponding spin-resolved ARPES intensities for $x$, $y$, and $z$ crystal axes, obtained with right ($\oplus$) or left ($\ominus$) circular polarization [see inset of (a) for experiment schematics]. (c) Intensity from each underlying state for the $z$ direction, corrected for light incident at 45° with respect to the spin–orbit quantization axis, as detailed in Supplemental Material [24]. Vertical error bars represent statistical uncertainty based on number of counts in the Mott polarimeters, plotted at 95% confidence together with locally-weighted scatter plot smoothing fits [24].

Figure 3

Momentum-dependent Ru-$d$ orbital projection of the wave function for the $\beta$, $\gamma$, and $\alpha$ bands at selected momentum locations along the three-dimensional Fermi surface. The surface color represents the momentum-dependent $s_z$ expectation value along the direction defined by the spherical ($\theta$, $\varphi$) angles, $⟨s_z{⟩}_{(\theta ,\varphi )}$ [24]; as indicated by the color scale at upper left, blue/red correspond to spin $\uparrow /\downarrow$ for one state of the Kramers-degenerate pair (with the opposite spin state not shown [30]). The strongly mixed colors on some of the orbital projection surfaces for the $\beta$ and $\gamma$ bands indicate strong, momentum-dependent spin–orbital entanglement.

Figure 4

Calculated two-particle spin expectation value $⟨\overrightarrow{s_k}·\overrightarrow{s_{-k}}⟩$ for states with zero total momentum along the $k_z=0$ Fermi surface sheets for (a) $\alpha$, (b) $\beta$, and (c) $\gamma$ bands. The $k_x$–$k_y$ plane location is defined by the angle $\mathrm{\Theta }$ for each band, as illustrated in (d). The complete set of results for the full $k_z$ range is shown in Fig. 5S of the Supplemental Material [24].