Trigonal warping enables linear optical spectroscopy in single-valley superconductors

TL;DR

The paper shows that in single-Fermi-surface superconductors with trigonal warping, fluctuations in competing pairing channels (BaSch and clapping modes) can become optically active in linear response, lifting previously dark modes without requiring an external supercurrent. The mechanism relies on an inversion-even velocity component $ ilde{oldsymbol{v}}_{oldsymbol{k}}$ from trigonal warping, which couples light to order-parameter fluctuations and yields observable features in the longitudinal and Hall optical conductivities, as well as in the inductive response. The analysis predicts a clapping-mode resonance near $ ext{Ω}_c oughly ext{=} oot2 hinspace ext{2} riangle_{ ext{mf}}$ with degeneracy lifted by channel competition, and a measurable optical signature including delta-function peaks at $ ext{Ω}_c$ and $2 riangle_{ ext{mf}}$ as well as a finite Hall response. Rhombohedral graphene multilayers ($ ext{R}n ext{G}$) with a single, trigonal-warped Fermi surface emerge as a promising platform to observe these collective modes via microwave impedance spectroscopy and related linear optical probes, enabling new insights into exotic superconducting order parameters.

Abstract

In superconductors with multiple pairing channels, Bardasis-Schrieffer modes and clapping modes arise as fluctuations in channels whose angular momenta differ from that of the pair condensate. Crystal symmetries often impose selection rules which keep these modes optically dark. We show that if pairing occurs around a single Fermi surface, trigonal warping renders both of these modes, as well as the quasiparticle excitation gap, visible in the longitudinal and Hall optical responses. Our results suggest that rhombohedral graphene multilayers, which are believed to host the required ingredients, might offer an ideal setting for the study of exotic superconducting collective modes.

Figures (2)

• Figure 1: Trigonal warping of a single Fermi surface (a) corresponds to an inversion-even contribution $\tilde{\vb{v}}$ to the electron velocity (b). Zero-momentum Cooper pairs thus acquire a non-zero velocity, enabling linear optical response in the superconducting state.
• Figure 2: Linear optical response of a single-valley $p+ip$ superconductor with trigonal warping. (a) Real part of longitudinal conductivity. (b) Hall conductivity. (c) Longitudinal superfluid stiffness. (d) Longitudinal inductance of a square sample. $g_{\ell}$ is the pairing strength in channel $\ell=1,2$. All panels use parameters inspired by rhombohedral graphene supp: $\mu = 1$ meV, $\nu = 5$ eV$^{-1}$nm$^{-2}$, $\eta = 20$ meV nm$^3$, $\Delta_{\text{mf}} = 15$$\mu$eV, and $\Lambda = 150$$\mu$eV. We give the frequency $\Omega / 2 \pi$ a small imaginary part, $1.5 \times 10^{-4}$ GHz.