Influence of Fermi Surface Geometry and Van Hove Singularities on the Optical Response of Sr$_2$RuO$_4$

TL;DR

The paper addresses how Fermi surface geometry, particularly Lifshitz transitions and Van Hove singularities, shapes the optical Hall response and polar Kerr effect in Sr$_2$RuO$_4$. It uses a two-dimensional three-orbital tight-binding model with self-consistent Bogoliubov–de Gennes theory, tuning the chemical potential $\mu$ and interlayer hopping $g'$ to map pairing symmetries and transport. The main findings are that the leading gap structures on the $d_{xy}$ orbital are $d_{x^2-y^2}$ and $d_{x^2-y^2}+ig$, and that the Kerr response can be enhanced by band proximity between the $\beta$ and $\gamma$ sheets; the Hall response for these pairings is essentially identical, indicating TRSB in the $d_{xy}$ gap is not essential. The Kerr angle is further modulated by interorbital transfer and SOC, with a pronounced peak near $g'\approx 6$ meV linked to near-degeneracy of low-energy states. Overall, the work provides a framework for interpreting Kerr-effect experiments in multi-orbital superconductors and clarifies how Fermi-surface topology controls optical transport in Sr$_2$RuO$_4$.

Abstract

Motivated by the sensitivity of Sr$_2$RuO$_4$ to Fermi surface reconstructions under strain, we investigate how Fermi surface geometry and Van Hove singularities influence the optical Hall response and polar Kerr effect. Within a three-orbital model, we explore the impact of chemical potential and interlayer hopping on superconducting pairing and response functions. We find that $d_{x^2-y^2}$ and $d_{x^2-y^2}+ig$ symmetries are the leading candidates for the quasi-2D orbital, while a chiral $p$-wave state in the quasi-1D orbitals is essential for generating an accessible Kerr angle. The Lifshitz transition is shown to affect coherence factors and density-of-states peaks, producing sharp signatures in $T_c$ and optical transport. Inter-orbital charge transfer further enhances these effects by modifying the balance between quasi-1D and quasi-2D contributions. These results provide a framework for interpreting Kerr effect experiments in multi-orbital superconductors.

Figures (5)

• Figure 1: Illustration of the effect of the parameter $g'$ on the Fermi surface: (a) $g' = 1\text{meV}$, and (b) $g' = 6\text{meV}$, highlighting the touching of the two bands, $\beta$ and $\gamma$, in the diagonal region, which predominantly contributes to the Berry curvature and Hall transport. (c) and (d) depict the variations in orbital weight on the $\gamma$-sheet along a quarter of the Brillouin zone for the respective values of $g'$, where $0<\theta=\arctan (k_y/k_x)<\pi/2$. (e) Temperature dependence of the superconducting gap for different order parameters.
• Figure 2: The critical temperature versus (a) the chemical potential $\mu$ and (b) the $z$-direction hopping $g^\prime$ for the variety of pairings. The increase in $\mu$ grows up the filling of the band $\gamma$ leading to the charge transfer from the orbitals $d_{xz/yz}$ to the orbital $d_{xy}$ while the coupling $g^\prime$ decreases the filling of the band $\gamma$. The dash-dotted vertical line shows the critical chemical potential. These plots can be criterion for choosing the favored pairing.
• Figure 3: The comparison between the dynamical Hall conductivity, $\sigma^H(\omega)$, for the pairing channels $d+ig$ and $d_{x^2-y^2}$ that indicates the role of the breaking of the time reversal symmetry on the gap function of the orbital $d_{xy}$ with (a) $g^\prime=1$meV, and (b) $g^\prime=6$meV.
• Figure 4: The DOS in the superconducting state is shown for the $p+ip$ wave pairing on the quasi-1D orbitals and the $d+ig$ wave pairing on the quasi-2D orbital at $T=0$: (a) for different $g^\prime$ values with $\mu = 142~\mathrm{meV}$, and (b) for different $\mu$ values with $g^\prime = 1~\mathrm{meV}$. The insets show the same data with a zoomed-in view for the low-energy regimes.
• Figure 5: (a) Kerr rotation angle (in unit of nanoradian) versus photon energy. The non-solid curves represent the influence of the interband electron-electron scattering on the Kerr angle in the hydrodynamic regime, $\varrho=1/\tau\gg1/\tau_\iota\sim1/\tau_\kappa$. The blue solid curve indicates the Kerr signal in the high frequencies limit (HFL), where the longitudinal conductivity is described by the Drude formula. The Kerr polar angle is shown: (b) as a function of the chemical potential $\mu$ with $g^\prime = 1$meV and $\Delta = 0.14$meV, (c) as a function of the interorbital coupling or $z$-direction hopping $g^\prime$ with $\mu = 142$meV and $\Delta = 0.14$meV, we plotted it in the possible range from $0$ to $8$meV Oda2019Paramagnetic, and (d) as a function of the gap function magnitude $\Delta$ with $g^\prime = 1$meV and $\mu = 142$meV. We set $\omega =0.8$eV Xia2006High, although the general behavior remains consistent across other frequency ranges. The maximum signal corresponds to the formation of near-Fermi-energy degenerate states between the $\beta$ and $\gamma$ bands. The region near the maximum in plot (c) corresponds to specific $z$-direction hopping values that induce a concave curvature in the Fermi surface of the $\gamma$ sheet. The effect of SOC on the polar Kerr angle is shown for the upper pseudospin sector in panel (e), and for both pseudospin sectors (up and down) in panel (f). The presence of SOC causes a decrease in the Kerr angle and the disappearance of the observed peak at higher SOCs