Current-Enabled Optical Conductivity of Collective Modes in Unconventional Superconductors

TL;DR

The paper develops a general path-integral–based framework to compute current-enabled linear optical conductivity for superconductors with multiple pairing channels and bands, capturing both quasiparticle and collective-mode contributions in a gauge-invariant form. It applies the formalism to one- and two-band models with competing $s$- and $d$-wave pairings, showing that Bardasis-Schrieffer (BS) modes and Leggett modes become optically active only when a supercurrent is injected, and that the MSBS mode persists below the gap across the transition. It then extends to Rashba spin-orbit–coupled interband-pairing systems, where Lifshitz invariants render BS modes optically active even without current, and where a chiral $p+ip$ ground state yields multiple BS and relative-phase modes with both longitudinal and transverse optical signatures, including ac Hall responses. Overall, the work provides a general, experimentally relevant framework for diagnosing sub-dominant pairing channels and multi-component order parameters in unconventional superconductors, with practical implications for multiband and SOC-enabled materials.

Abstract

We theoretically investigate the current-enabled linear optical conductivity of collective modes in superconductors with unconventional pairing symmetries. After deriving general formulas for the optical conductivity of a superconductor featuring multiple pairing channels and bands using the path integral formalism, we apply these formulas to several models. Using a model of competing s- and d-wave pairing interactions, we find that several known collective modes generate peaks in the optical conductivity upon injection of a supercurrent. This includes single- and multiband versions of Bardasis-Schrieffer modes, mixed-symmetry Bardasis-Schrieffer modes, and Leggett modes. Using a model for interband p-wave superconductivity with Rashba spin-orbit coupling, we find that in such a system Bardasis-Schrieffer modes are optically active even without introducing a supercurrent. In a p+ip chiral ground state, these modes turn out to produce peaks in the longitudinal and transverse optical conductivity. Other collective modes belonging to the chiral p+ip order parameter turn out to be unaffected by the spin-orbit coupling but contribute to the optical response when a supercurrent is introduced. These results promise new avenues for the observation of collective modes in a variety of superconducting systems, including multiband superconductors and superconductors that feature multiple pairing channels or multi-component order parameters, such as chiral p- or d-wave superconductors.

Figures (27)

• Figure 1: Schematic representation of current-enabled optical response of collective modes in unconventional superconductors. One can, for example, excite a mixed-symmetry Bardasis-Schrieffer (MSBS) mode linearly by light in an $s+id$-wave superconductor in the presence of externally injected supercurrent $\bm{J}$.
• Figure 2: Feynman diagrams for each term in the effective action $S_\text{eff}^\text{FL}[\Delta^{\nu\mu};\bm{A}]$ (see Eq. (\ref{['effectiveActionFluctuations']})). The dashed (wavy) lines represent the fluctuations $\Delta^\nu_\mu$ (the vector potential $\bm{A}$) with the corresponding vertex $\varphi^\mu\tau^\nu$ ($v^i$).
• Figure 3: Diagrammatic representation for the effective coupling $V_\text{eff}$ [Eq. (\ref{['eq: V_eff']})].
• Figure 4: Diagrammatic representation for the components of the effective action [Eq. (\ref{['eq: S_eff^EM']})] that contribute to the real part of the optical conductivity.
• Figure 5: Schematics of the two tight-binding Hamiltonians considered for the competing $s$- and $d$-wave interactions: (a) The single layer model with one band and (b) the bilayer model with two bands. The dotted lines in blue and red illustrate attractive interactions in the $s$- and $d$-wave channels, while the solid lines in black and green represent intralayer and interlayer single-particle hoppings, respectively.