Raman response of collective modes in multicomponent superconductors

TL;DR

This work develops a gauge-invariant, microscopic theory for the Raman response of collective modes in multicomponent superconductors by starting from a Bogoliubov–de Gennes Hamiltonian with separable pairing and including long-range Coulomb interactions via a scalar field. It provides a group-theoretical selection rule and a detailed classification of Raman-active modes across all crystal point groups, linking symmetry channels to possible Leggett-, Bardasis–Schrieffer–, and clapping-type fluctuations. The theory is applied to a multicomponent, fully gapped odd-parity state in UTe$_2$, revealing sharp in-gap Raman resonances arising from intraband relative modes between order-parameter components, with their visibility governed by the Raman vertex. This framework enables direct computation of Raman spectra from BdG data and offers a scalable route to identify Raman-active collective modes in candidate multiband superconductors, including potential extensions to coupling with phonons or other bosons. The results underscore how symmetry dictates possible couplings while microscopic details shape spectral weight, providing practical fingerprints for experimental Raman probes.

Abstract

We formulate a microscopic theory of the Raman response of superconducting collective modes in multicomponent superconductors. Starting from a general Bogoliubov--de Gennes (BdG) Hamiltonian with a separable pairing interaction, we derive a gauge-invariant expression for the Raman susceptibility, including a long-range Coulomb interaction. The resulting Raman susceptibility is directly computable for an arbitrary BdG Hamiltonian, which contains single- and multiband systems, spin-singlet and triplet order parameters, and time-reversal-symmetric and time-reversal-symmetry-breaking superconducting states. Based on the microscopic coupling between a Raman source field and collective modes, we derive a symmetry selection rule for Raman-active collective modes and show a group-theoretical classification for all crystalline point groups. This classification provides a unified framework based on the ``higher-order Lifshitz-invariant'' to identify Raman-active collective modes such as Leggett mode, Bardasis-Schrieffer (BS) mode, and clapping mode. As an application, we focus on an effective model of the heavy-fermion superconductor UTe$_2$ with a fully gapped multicomponent odd-parity pairing state. We find sharp in-gap Raman resonances below the quasiparticle continuum, which do not correspond to a conventional Leggett mode but arise from the {\it intraband} relative modes between different pairing components.

Figures (5)

• Figure 1: Feynman diagram of (a,b) the Raman response in superconductors, and (c) the effective interaction $U_{\text{eff}}(\omega)$ within the random phase approximation. The diagram (a) corresponds to the quasiparticle excitation. The diagram (b) corresponds to the collective excitations, including the charge density fluctuation, and we replace $\bm{\delta}\Delta \equiv (\delta\Delta, \delta ^{*}\Delta)$ and $U_{\text{eff}}$ to $\bm{\delta}\Delta_{\phi} \equiv (\bm{\delta}\Delta,\phi)$ and $U_{\text{eff},\phi}$. $U$ is the bare interaction and $\Pi$ is the polarization bubble.
• Figure 2: (a) Crystal structure (point group $D_{2h}$) of UTe$_2$, where a unit cell (red) includes two U atoms: U1 and U2. (b) Cylindrical Fermi surface of UTe$_2$.
• Figure 3: Energy spectrum of the BdG Hamiltonian in Eq. (\ref{['eq-main: BdG-Hamiltonian-multi']}) for the UTe$_2$, plotted as a function of $k_z$. For each $k_z$, all eigenvalues $E_n(\bm{k})$ obtained by scanning $(k_x,k_y)$ over the Brillouin zone are shown, so that the vertical spread reflects the residual $(k_x,k_y)$ dispersion. The energy gap is given by $E_{\text{gap}} \approx 0.074$.
• Figure 4: (a) Frequency dependence of the eigenvalues of the inverse fluctuation matrix $U^{-1}_{\mathrm{eff},\phi}(\bm{q}=\bm{0},\omega)$ in Eq. (\ref{['eq: general-Anderson-Higgs-main']}) with the tight-binding model of UTe$_2$. The vertical dotted line indicates $\omega\simeq E_{\text{gap}}$. (b) Raman susceptibility for $\bm{e}^{\text{in}}_x$ and $\bm{e}^{\text{out}}_x$ at $\bm{q}=0$. The blue line shows the full $\chi_{RR}(\bm{0},\omega)$ of Eq. (\ref{['eq: raman-numerical']}), which includes the scalar field and Coulomb interaction, while the red line shows the bare response $\pi_{RR}(\bm{0},\omega)$ [Eq. (\ref{['eq: Raman_without_scalar-app']})] without the scalar field. The two curves are indistinguishable, numerically confirming that long-range Coulomb screening does not affect the Raman response: $\chi_{RR}(\bm{0},\omega)=\pi_{RR}(\bm{0},\omega)$. At $\omega \approx 0.063 \ (\textrm{ii}), 0.071 \ (\textrm{iii})$, the peak structures appear corresponding to the relative collective modes in (a).
• Figure 5: (a) Energy spectrum of the BdG Hamiltonian in Eq. (\ref{['eq-main: BdG-Hamiltonian-multi']}) plotted as a function of $k_z$, (b) fequency dependence of the eigenvalues of the inverse fluctuation matrix $U^{-1}_{\mathrm{eff},\phi}(\bm{q}=\bm{0},\omega)$ in Eq. (\ref{['eq: general-Anderson-Higgs-main']}), (c) Raman susceptibility $\chi_{RR}(\bm{0},\omega)$ of Eq. (\ref{['eq: raman-numerical']}) for $\bm{e}^{\text{in}}_x$ and $\bm{e}^{\text{out}}_x$ at $\bm{q}=0$.