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

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.

💡 Research Summary

This paper investigates how the geometry of the Fermi surface and the presence of Van Hove singularities (VHS) affect the optical Hall response and polar Kerr effect in the unconventional superconductor Sr₂RuO₄. Using a three‑orbital (t₂g) tight‑binding model that includes the quasi‑two‑dimensional d_xy band and the quasi‑one‑dimensional d_xz/d_yz bands, the authors explore the impact of two key tuning parameters: the chemical potential μ (which mimics biaxial strain or carrier doping) and the interlayer hopping g′ that controls the hybridization between the d_xy and d_xz/d_yz orbitals along the c‑axis.

The normal‑state Hamiltonian H_N = H₀ + H_SOC is constructed with realistic hopping amplitudes (t_xx, t_xy, etc.) and a modest atomic spin‑orbit coupling λ. Because the overlap between d_xy and the quasi‑1D orbitals is maximal at k_z = π, the authors set λ → 0 for most of the analysis, focusing on the band reconstruction driven by μ and g′. Increasing μ pushes the γ‑sheet (predominantly d_xy) toward the Brillouin‑zone edge, eventually crossing a Van Hove saddle point and inducing a Lifshitz transition. Simultaneously, raising g′ transfers electrons from d_xy to the d_xz/d_yz orbitals, reduces the density of states (DOS) at the Fermi level, and brings the β and γ sheets into near‑degeneracy along the diagonal direction. This near‑degeneracy dramatically enhances the Berry curvature and, consequently, the intrinsic Hall conductivity.

Superconductivity is treated within a self‑consistent Bogoliubov–de Gennes framework, assuming only intra‑orbital pairing. The pairing interaction is separable, with form factors f_ν(k) that encode the symmetry of each orbital channel. The authors examine several irreducible representations of the D₄h point group: extended s‑wave (s′), dₓ²₋ᵧ², d_xy, g‑wave, and p‑wave. For the quasi‑2D d_xy orbital, the leading candidates are real dₓ²₋ᵧ² and a complex combination dₓ²₋ᵧ² + i g (denoted d + ig). For the quasi‑1D orbitals, a chiral p‑wave (pₓ + i p_y) belonging to the Eu irrep is imposed, consistent with previous proposals that time‑reversal symmetry breaking (TRSB) resides primarily on these bands.

Critical temperature Tc is computed as a function of μ and g′ for each pairing symmetry. When μ approaches the critical value μ_c ≈ 0.148 eV, the γ‑sheet undergoes a Lifshitz transition; the DOS exhibits a sharp peak at the VHS. In this regime, the d‑wave based pairings (dₓ²₋ᵧ² and d + ig) show a pronounced Tc enhancement—more than a factor of two—because their gap functions remain finite at the saddle points. By contrast, p‑wave and mixed s′ + id states have nodes exactly at the VHS points, so their Tc is largely insensitive to the Lifshitz transition. Increasing g′ also raises Tc for the d‑wave channels by bringing β and γ bands closer together, but it suppresses the DOS of the d_xy band, which can offset the gain.

The optical Hall conductivity σ_xy(ω) is obtained from the Kubo formula using the self‑consistent gap functions. The calculation reveals that the dominant contribution comes from quasiparticle excitations away from the Fermi level (energies of order the superconducting gap), while low‑energy states near the Fermi surface contribute negligibly. As g′ grows, the β‑γ near‑degeneracy allows both bands to contribute comparably, amplifying σ_xy and the resulting Kerr angle. Including spin‑orbit coupling splits the β‑γ degeneracy and reduces the Hall response, confirming the delicate balance between orbital mixing and Berry curvature.

For the polar Kerr effect, the authors adopt the multigap formalism of Ref.