Revealing THz optical signatures of Shiba-state-induced gapped and gapless superconductivity

We report a fully self-consistent calculation of the complex renormalization by exchange interactions and hence the complete phase diagram of conventional $s$-wave superconductors with magnetic impurities as well as the related physical properties including the optical response. We show that a small amount of magnetic disorder can drive the system into a gapless superconducting state, where the single-particle excitation gap vanishes whereas the superconducting order parameter $Δ_0$ remains finite. In this phase, the linear optical conductivity exhibits a finite absorption over the low-frequency regime, particularly for photon energies below the conventional threshold $2|Δ_0|$, even at low temperatures, in sharp contrast to the gapped state. The nonlinear response, however, remains coherent and is dominated by the Higgs-mode dynamics rather than gapless quasiparticle background. These findings reveal a fundamental distinction between dissipative single-particle excitations and coherent collective dynamics of the condensate, a feature likely general to other gapless superconductors, and introduces a fundamentally different detection scheme, using THz spectroscopy to probe the signatures of Shiba states.

💡 Research Summary

The authors present a fully self‑consistent theoretical study of conventional s‑wave superconductors doped with magnetic impurities, focusing on how exchange‑induced Yu‑Shiba‑Rusinov (YSR) bound states evolve into impurity bands that can close the single‑particle excitation gap while leaving the superconducting order parameter Δ₀ finite. Starting from a microscopic Hamiltonian that includes an s‑d exchange term J S·σ, they formulate the problem in Nambu‑spin space and derive the Dyson equation for the Green function. Within the random‑phase approximation for randomly positioned and oriented impurity spins, the self‑energy Σ(ω) is obtained, leading to Shiba’s renormalization equations for the complex frequency ˜ω and renormalized gap ˜Δ₀. Two key dimensionless parameters appear: γ_s, the quasiparticle relaxation rate proportional to impurity concentration n_i, and η, which encodes the energy splitting of the YSR doublet.

Because the renormalization equations contain non‑trivial branch cuts, the authors transform the complex relation into a sixth‑order polynomial in the ratio y = ˜ω/˜Δ₀. Solving this polynomial yields six mathematical roots; physical selection is performed in two steps: (i) retain only roots with Im y ≥ 0 (retarded Green function condition) and (ii) choose the root whose real part is closest to the impurity‑free solution y = x (with x = (ω + i0⁺)/Δ₀). This procedure guarantees continuity as impurity strength is turned on and eliminates unphysical branches.

With the unique solution for (˜ω, ˜Δ₀) in hand, the anomalous Green function F_k(ω) is constructed, and the self‑consistent gap equation is solved numerically for Δ₀ at arbitrary temperature T and impurity concentration n_i. The algorithm converges rapidly across the entire phase diagram, enabling the authors to map out regions of conventional gapped superconductivity, a crossover regime, and a true gapless superconducting (GSC) phase where the density of states ρ(ω) is finite down to zero energy.

In the GSC phase, linear THz conductivity σ₁(ω) exhibits a finite low‑frequency absorption even for photon energies ℏω < 2|Δ₀|, in stark contrast to the ideal BCS case where σ₁(ω) vanishes below the pair‑breaking threshold. This absorption directly reflects the impurity‑induced in‑gap states and scales with γ_s. Remarkably, the nonlinear THz response—probed via third‑harmonic generation or pump‑probe spectroscopy—remains dominated by the Higgs amplitude mode at ℏω_H ≈ 2|Δ₀|. The Higgs resonance retains its sharpness and coherence despite the presence of abundant low‑energy quasiparticles, indicating that collective condensate dynamics are largely immune to the gapless quasiparticle background.

The paper therefore establishes a clear dichotomy: magnetic impurities dramatically enhance dissipative single‑particle channels (linear response) while leaving the coherent collective channel (nonlinear Higgs dynamics) essentially intact. This distinction provides a powerful experimental diagnostic: THz time‑domain spectroscopy can detect the emergence of a gapless phase through the appearance of sub‑gap linear absorption, whereas the persistence of a well‑defined Higgs peak confirms that superconducting order remains robust.

Beyond the immediate context, the authors argue that similar behavior should appear in any superconductor where impurity bands or other mechanisms close the quasiparticle gap (e.g., strong spin‑orbit scattering, proximity‑induced states). Moreover, because YSR bands can acquire non‑trivial topology when magnetic atoms are arranged in specific patterns, the methodology presented here can be extended to probe topological superconductivity and Majorana bound states via bulk THz probes, complementing local STM measurements.

In summary, the work delivers a comprehensive, numerically exact phase diagram for magnetic‑impurity‑doped s‑wave superconductors, elucidates the optical signatures of the gapless regime, and proposes THz spectroscopy as a bulk, contact‑free tool to distinguish between dissipative quasiparticle excitations and coherent Higgs dynamics—offering new avenues for both fundamental studies and potential quantum‑device applications.