Superconductivity’s Hidden Vibrations Unlocked by New Raman Response Theory

Researchers are increasingly focused on understanding collective excitations within multicomponent superconductors, and a new study by Yuki Yamazaki and Takahiro Morimoto, both from the Department of Applied Physics at The University of Tokyo, details a microscopic theory of Raman responses to these modes. Their work establishes a gauge-invariant expression for Raman susceptibility, applicable to a broad range of superconducting systems including those with multiband structures and unconventional pairing symmetries. Significantly, this research provides a unified framework, based on higher-order Lifshitz invariants, to classify Raman-active collective modes and predict novel in-gap resonances, demonstrated through application to the heavy-fermion material UTe, offering crucial insights into the behaviour of complex superconducting states.

Microscopic theory predicts Raman spectra and classifies collective excitations in multicomponent superconductors

Scientists have developed a comprehensive microscopic theory detailing how light interacts with collective excitations in multicomponent superconductors. This work establishes a directly computable Raman susceptibility, applicable to a broad range of superconducting systems including those with single or multiple energy bands, spin-singlet or triplet order parameters, and both time-reversal-symmetric and symmetry-breaking states.
The resulting framework allows for the prediction of Raman spectra given a specific Bogoliubov, de Gennes (BdG) Hamiltonian, effectively linking material properties to observable spectroscopic signatures. A key achievement is the derivation of a symmetry selection rule, classifying Raman-active collective modes across all crystalline point groups using a “higher-order Lifshitz-invariant” approach.

This classification unifies the identification of crucial modes such as the Leggett mode, Bardasis-Schrieffer (BS) mode, and clapping mode, providing a systematic way to understand their behaviour. Researchers applied this theory to the heavy-fermion superconductor UTe2, possessing a fully gapped multicomponent odd-parity pairing state, revealing unexpected Raman resonances.

These resonances, appearing below the quasiparticle continuum, originate from intraband relative modes between different pairing components rather than a conventional Leggett mode. The study formulates a gauge-invariant expression for Raman susceptibility, incorporating long-range Coulomb interactions and enabling accurate calculations for complex superconducting Hamiltonians.

By considering Gaussian fluctuations and a nonresonant Raman vertex, the research provides a robust foundation for predicting and interpreting Raman scattering experiments in a diverse range of superconducting materials. This advancement promises to deepen understanding of the internal structure and symmetry of superconductors, potentially guiding the development of novel materials with tailored properties. The ability to predict in-gap resonances, as demonstrated with UTe2, offers a new avenue for probing the intricacies of unconventional superconductivity and identifying exotic pairing states.

Raman susceptibility calculation via Bogoliubov de Gennes formalism and symmetry analysis

A gauge-invariant expression for the Raman susceptibility was derived beginning with a general Bogoliubov, de Gennes (BdG) Hamiltonian incorporating a separable pairing interaction. This formulation directly computes the Raman susceptibility for any BdG Hamiltonian, accommodating single- and multiband systems, alongside both spin-singlet and triplet order parameters, and extending to states with and without time-reversal symmetry.

The research establishes a symmetry selection rule governing Raman-active collective modes, utilising a group-theoretical classification based on a “higher-order Lifshitz-invariant” to identify modes including the Leggett mode, Bardasis-Schrieffer (BS) mode, and clapping mode. To achieve this, the study assumes separable pairing interactions, Gaussian fluctuations around the mean-field solution, a nonresonant Raman vertex, and neglects couplings to other bosonic modes and impurity-induced damping, operating within the clean limit.

Once a BdG Hamiltonian is defined, the Raman spectrum is calculated directly from its eigenvalues and eigenvectors via explicit kernel expressions, representing an extension of prior work on spin-singlet superconductors to encompass spin-triplet pairings. This methodology remains applicable regardless of basis choice, time-reversal symmetry, or the specific superconducting pairing structure.

Furthermore, a group-theoretical classification was developed to analyse the linear coupling between collective modes and Raman source fields, analogous to the symmetry analysis of Lifshitz invariants for linear optical responses. This approach identifies combinations of order-parameter components that can couple to a given Raman vertex field across all crystalline point groups.

Applying this theory to UTe2, a candidate time-reversal-symmetric spin-triplet superconductor, revealed sharp in-gap Raman resonances originating from intraband relative modes between different pairing components, rather than a conventional Leggett mode. These resonances manifest as peak structures within the Raman spectrum.

Raman susceptibility and collective mode identification in multiband superconductors

Researchers derived a gauge-invariant expression for the Raman susceptibility, directly computable for any Bogoliubov, de Gennes (BdG) Hamiltonian encompassing single- and multiband systems, alongside both spin-singlet and triplet order parameters. This formulation accommodates time-reversal-symmetric and time-reversal-symmetry-breaking superconducting states, establishing a unified framework for analysing Raman responses.

The resulting susceptibility facilitates the identification of Raman-active collective modes, including the Leggett mode, Bardasis-Schrieffer (BS) mode, and clapping mode, through a classification based on higher-order Lifshitz invariants. Applying this to a model of the heavy-fermion UTe, the study revealed sharp in-gap Raman resonances below the quasiparticle continuum.

These resonances do not correspond to a conventional Leggett mode but originate from intraband relative modes between different pairing components. The effective action for bosonic fields was obtained after decoupling the attractive interaction using complex bosonic Hubbard, Stratonovich fields and integrating out the fermions, resulting in an action defined by terms involving pairing field fluctuations and a scalar field.

The matrix U−1eff,φ(Qp), representing the inverse propagator for coupled fluctuations, determines the dispersion of collective modes, including an Anderson, Higgs mass that vanishes at q = 0. Raman susceptibility, χRR(Qp), was calculated as 1/4ΦRR(Qp) − 1/8QT R,φ(Qp) Ueff,φ(Qp) QR,φ(−Qp), providing a central microscopic result applicable to any BdG Hamiltonian.

This allows for the evaluation of kernels and the determination of the gauge-invariant Raman response, incorporating all superconducting collective modes and coupling to the scalar field. The Raman intensity, I(ω), is proportional to the imaginary part of χRR(q →0, ω).

Raman susceptibility and collective mode classification in multicomponent superconductivity

A comprehensive theoretical framework for understanding Raman spectroscopy of superconducting collective modes in multicomponent superconductors has been established. The research derives a gauge-invariant expression for Raman susceptibility, applicable to diverse superconducting systems including those with single or multiple bands, spin-singlet or triplet order parameters, and both time-reversal symmetric and broken states.

This formulation allows direct computation of the Raman response by integrating out fermions and considering fluctuations in the order parameter alongside Coulomb interactions, clearly separating contributions from quasiparticles and collective modes. Furthermore, symmetry selection rules and a group-theoretical classification of Raman-active collective modes have been developed, utilising a “second-order Lifshitz-invariant” to identify modes such as the Leggett mode, Bardasis-Schrieffer mode, and clapping mode within a unified framework.

Application of this formalism to the heavy-fermion material UTe2, possessing a fully gapped multicomponent odd-parity pairing state, revealed in-gap Raman resonances originating from relative modes between different pairing components. The authors acknowledge that the intensity of Raman signals is influenced by microscopic details like spin-orbit coupling and the structure of the Raman vertex, in addition to symmetry considerations.

Future work could extend this formulation to include additional particle-hole interactions and bosonic degrees of freedom, such as phonons or magnons, to explore hybridization effects with superconducting collective modes. These findings demonstrate the potential for identifying distinct polarization-dependent fingerprints of fully gapped multicomponent superconductivity through Raman spectroscopy, offering a pathway to experimentally probe complex superconducting systems.