- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review B

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Raman response of collective modes in multicomponent superconductors

Yuki Yamazaki and Takahiro Morimoto

PDF

Phys. Rev. B 114, 034502 – Published 1 July, 2026

DOI: https://doi.org/10.1103/w431-62pc

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 mode, and clapping mode. As an application, we focus on an effective model of the heavy-fermion superconductor UTe2 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 intraband relative modes between different pairing components.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (85)

  1. P. W. Anderson, Random-phase approximation in the theory of superconductivity, Phys. Rev. 112, 1900 (1958).
  2. A. Schmid, The approach to equilibrium in a pure superconductor the relaxation of the Cooper pair density, Phys. Kondens. Mater. 8, 129 (1968).
  3. P. B. Littlewood and C. M. Varma, Gauge-invariant theory of the dynamical interaction of charge density waves and superconductivity, Phys. Rev. Lett. 47, 811 (1981).
  4. P. B. Littlewood and C. M. Varma, Amplitude collective modes in superconductors and their coupling to charge-density waves, Phys. Rev. B 26, 4883 (1982).
  5. D. Pekker and C. Varma, Amplitude/Higgs modes in condensed matter physics, Annu. Rev. Condens. Matter Phys. 6, 269 (2015).
  6. R. Shimano and N. Tsuji, Higgs mode in superconductors, Annu. Rev. Condens. Matter Phys. 11, 103 (2020).
  7. N. Tsuji, I. Danshita, and S. Tsuchiya, Higgs and Nambu–Goldstone modes in condensed matter physics, in Encyclopedia of Condensed Matter Physics, 2nd ed. (Academic Press, Oxford, 2024).
  8. Y. Nambu and G. Jona-Lasinio, Dynamical model of elementary particles based on an analogy with superconductivity. I, Phys. Rev. 122, 345 (1961).
  9. J. Goldstone, Field theories with “superconductor” solutions, Nuovo Cimento 19, 154 (1961).
  10. P. W. Anderson, Plasmons, gauge invariance, and mass, Phys. Rev. 130, 439 (1963).
  11. P. W. Higgs, Broken symmetries and the masses of gauge bosons, Phys. Rev. Lett. 13, 508 (1964).
  12. A. J. Leggett, Number-phase fluctuations in two-band superconductors, Prog. Theor. Phys. 36, 901 (1966).
  13. A. Bardasis and J. R. Schrieffer, Excitons and plasmons in superconductors, Phys. Rev. 121, 1050 (1961).
  14. P. Wölfle, Order-parameter collective modes in He3-A, Phys. Rev. Lett. 37, 1279 (1976).
  15. L. Tewordt, Collective order parameter modes and spin fluctuations for spin-triplet superconducting state in Sr2RuO4, Phys. Rev. Lett. 83, 1007 (1999).
  16. A. V. Balatsky, P. Kumar, and J. R. Schrieffer, Collective mode in a superconductor with mixed-symmetry order parameter components, Phys. Rev. Lett. 84, 4445 (2000).
  17. N. R. Poniatowski, J. B. Curtis, A. Yacoby, and P. Narang, Spectroscopic signatures of time-reversal symmetry breaking superconductivity, Commun. Phys. 5, 44 (2022).
  18. G. Blumberg, A. Mialitsin, B. S. Dennis, M. V. Klein, N. D. Zhigadlo, and J. Karpinski, Observation of Leggett's collective mode in a multiband MgB2 superconductor, Phys. Rev. Lett. 99, 227002 (2007).
  19. F. Kretzschmar, B. Muschler, T. Böhm, A. Baum, R. Hackl, H.-H. Wen, V. Tsurkan, J. Deisenhofer, and A. Loidl, Raman-scattering detection of nearly degenerate s-wave and d-wave pairing channels in iron-based Ba0.6K0.4Fe2As2 and Rb0.8Fe1.6Se2 superconductors, Phys. Rev. Lett. 110, 187002 (2013).
  20. T. Böhm, A. F. Kemper, B. Moritz, F. Kretzschmar, B. Muschler, H.-M. Eiter, R. Hackl, T. P. Devereaux, D. J. Scalapino, and H.-H. Wen, Balancing act: Evidence for a strong subdominant d-wave pairing channel in Ba0.6K0.4Fe2As2, Phys. Rev. X 4, 041046 (2014).
  21. D. Jost, J.-R. Scholz, U. Zweck, W. R. Meier, A. E. Böhmer, P. C. Canfield, N. Lazarević, and R. Hackl, Indication of subdominant d-wave interaction in superconducting CaKFe4As4, Phys. Rev. B 98, 020504(R) (2018).
  22. G. He, D. Li, D. Jost, A. Baum, P. P. Shen, X. L. Dong, Z. X. Zhao, and R. Hackl, Raman Study of Cooper Pairing Instabilities in (Li1−xFex)OHFeSe, Phys. Rev. Lett. 125, 217002 (2020).
  23. H. Matsumoto, S. Neri, T. Kobayashi, A. Maeda, D. Manske, and R. Shimano, A new collective mode in an iron-based superconductor with electronic nematicity, arXiv:2507.14466.
  24. C. M. Varma, Higgs boson in superconductors, J. Low Temp. Phys. 126, 901 (2002).
  25. N. Tsuji and H. Aoki, Theory of Anderson pseudospin resonance with Higgs mode in superconductors, Phys. Rev. B 92, 064508 (2015).
  26. A. F. Kemper, M. A. Sentef, B. Moritz, J. K. Freericks, and T. P. Devereaux, Direct observation of Higgs mode oscillations in the pump-probe photoemission spectra of electron-phonon mediated superconductors, Phys. Rev. B 92, 224517 (2015).
  27. T. Cea, C. Castellani, and L. Benfatto, Nonlinear optical effects and third-harmonic generation in superconductors: Cooper pairs versus Higgs mode contribution, Phys. Rev. B 93, 180507(R) (2016).
  28. N. Tsuji, Y. Murakami, and H. Aoki, Nonlinear light–Higgs coupling in superconductors beyond BCS: Effects of the retarded phonon-mediated interaction, Phys. Rev. B 94, 224519 (2016).
  29. T. Jujo, Quasiclassical theory on third-harmonic generation in conventional superconductors with paramagnetic impurities, J. Phys. Soc. Jpn. 87, 024704 (2018).
  30. M. Silaev, Nonlinear electromagnetic response and Higgs-mode excitation in BCS superconductors with impurities, Phys. Rev. B 99, 224511 (2019).
  31. L. Schwarz, B. Fauseweh, N. Tsuji, N. Cheng, N. Bittner, H. Krull, M. Berciu, G. S. Uhrig, A. P. Schnyder, S. Kaiser, and D. Manske, Classification and characterization of nonequilibrium Higgs modes in unconventional superconductors, Nat. Commun. 11, 287 (2020).
  32. N. Tsuji and Y. Nomura, Higgs-mode resonance in third harmonic generation in NbN superconductors: Multiband electron-phonon coupling, impurity scattering, and polarization-angle dependence, Phys. Rev. Res. 2, 043029 (2020).
  33. G. Seibold, M. Udina, C. Castellani, and L. Benfatto, Third harmonic generation from collective modes in disordered superconductors, Phys. Rev. B 103, 014512 (2021).
  34. R. Haenel, P. Froese, D. Manske, and L. Schwarz, Time-resolved optical conductivity and Higgs oscillations in two-band dirty superconductors, Phys. Rev. B 104, 134504 (2021).
  35. M. Udina, J. Fiore, T. Cea, C. Castellani, G. Seibold, and L. Benfatto, THz non-linear optical response in cuprates: Predominance of the BCS response over the Higgs mode, Farad. Discuss. 237, 168 (2022).
  36. C.-g. Oh, H. Watanabe, and N. Tsuji, Role of quantum geometry in the competition between Higgs mode and quasiparticles in third-harmonic generation of superconductors, arXiv:2512.01200.
  37. R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang, and R. Shimano, Higgs amplitude mode in the BCS superconductors Nb1−xTixN induced by terahertz pulse excitation, Phys. Rev. Lett. 111, 057002 (2013).
  38. R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka, K. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki, and R. Shimano, Light-induced collective pseudospin precession resonating with Higgs mode in a superconductor, Science 345, 1145 (2014).
  39. R. Matsunaga, N. Tsuji, K. Makise, H. Terai, H. Aoki, and R. Shimano, Polarization-resolved terahertz third-harmonic generation in a single-crystal superconductor NbN: Dominance of the Higgs mode beyond the BCS approximation, Phys. Rev. B 96, 020505(R) (2017).
  40. K. Katsumi, N. Tsuji, Y. I. Hamada, R. Matsunaga, J. Schneeloch, R. D. Zhong, G. D. Gu, H. Aoki, Y. Gallais, and R. Shimano, Higgs mode in the d-wave superconductor Bi2Sr2CaCu2O8+x driven by an intense terahertz pulse, Phys. Rev. Lett. 120, 117001 (2018).
  41. H. Chu, M. Kim, K. Katsumi, et al., Phase-resolved Higgs response in superconducting cuprates, Nat. Commun. 11, 1793 (2020).
  42. T. Kamatani, S. Kitamura, N. Tsuji, R. Shimano, and T. Morimoto, Optical response of the Leggett mode in multiband superconductors in the linear response regime, Phys. Rev. B 105, 094520 (2022).
  43. R. Nagashima, S. Tian, R. Haenel, N. Tsuji, and D. Manske, Classification of Lifshitz invariant in multiband superconductors: An application to Leggett modes in the linear response regime in Kagome lattice models, Phys. Rev. Res. 6, 013120 (2024).
  44. C. Lee and S. B. Chung, Linear optical response from the odd-parity Bardasis-Schrieffer mode in locally non-centrosymmetric superconductors, Commun. Phys. 6, 307 (2023).
  45. B. A. Levitan, Y. Oreg, E. Berg, M. S. Rudner, and I. Iorsh, Linear spectroscopy of collective modes and the gap structure in two-dimensional superconductors, Phys. Rev. Res. 6, 043170 (2024).
  46. B. A. Levitan and É. Lantagne-Hurtubise, Trigonal warping enables linear optical spectroscopy in single-valley superconductors, arXiv:2508.09387.
  47. T. Matsushita, J. Ieda, Y. Araki, T. Morimoto, I. Vekhter, and Y. Yanase, Microwave Kerr/Faraday resonance in two-dimensional chiral superconductors, arXiv:2601.10151.
  48. L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon press, Oxford, 1969).
  49. M. V. Klein and S. B. Dierker, Theory of Raman scattering in superconductors, Phys. Rev. B 29, 4976 (1984).
  50. T. P. Devereaux and R. Hackl, Inelastic light scattering from correlated electrons, Rev. Mod. Phys. 79, 175 (2007).
  51. R. Sooryakumar and M. V. Klein, Raman scattering by superconducting-gap excitations and their coupling to charge-density waves, Phys. Rev. Lett. 45, 660 (1980).
  52. M.-A. Méasson, Y. Gallais, M. Cazayous, B. Clair, P. Rodière, L. Cario, and A. Sacuto, Amplitude Higgs mode in the 2H-NbSe2 superconductor, Phys. Rev. B 89, 060503(R) (2014).
  53. R. Grasset, T. Cea, Y. Gallais, M. Cazayous, A. Sacuto, L. Cario, L. Benfatto, and M.-A. Méasson, Higgs-mode radiance and charge-density-wave order in 2H-NbSe2, Phys. Rev. B 97, 094502 (2018).
  54. R. Grasset, Y. Gallais, A. Sacuto, M. Cazayous, S. Mañas Valero, E. Coronado, and M.-A. Méasson, Pressure-induced collapse of the charge density wave and Higgs mode visibility in 2H-TaS2, Phys. Rev. Lett. 122, 127001 (2019).
  55. A. Majumdar, D. VanGennep, J. Brisbois, D. Chareev, A. V. Sadakov, A. S. Usoltsev, M. Mito, A. V. Silhanek, T. Sarkar, A. Hassan, O. Karis, R. Ahuja, and M. Abdel-Hafiez, Interplay of charge density wave and multiband superconductivity in layered quasi-two-dimensional materials: The case of 2H-NbS2 and 2H-NbSe2, Phys. Rev. Mater. 4, 084005 (2020).
  56. H. C. Lee and H. Y. Choi, Electronic Raman scattering of two-band superconductors: A time-dependent Landau–Ginzburg theory approach, J. Phys.: Condens. Matter 21, 445701 (2009).
  57. M. V. Klein, Theory of Raman scattering from Leggett's collective mode in a multiband superconductor: Application to MgB2, Phys. Rev. B 82, 014507 (2010).
  58. T. Cea and L. Benfatto, Signature of the Leggett mode in the A1g Raman response: From MgB2 to iron-based superconductors, Phys. Rev. B 94, 064512 (2016).
  59. S. Maiti, T. A. Maier, T. Böhm, R. Hackl, and P. J. Hirschfeld, Probing the pairing interaction and multiple Bardasis-Schrieffer modes using Raman spectroscopy, Phys. Rev. Lett. 117, 257001 (2016).
  60. D. J. Scalapino and T. P. Devereaux, Collective d-wave exciton modes in the calculated Raman spectrum of Fe-based superconductors, Phys. Rev. B 80, 140512(R) (2009).
  61. M. Khodas, A. V. Chubukov, and G. Blumberg, Collective modes in multiband superconductors: Raman scattering in iron selenides, Phys. Rev. B 89, 245134 (2014).
  62. S. Maiti, A. V. Chubukov, and P. J. Hirschfeld, Conservation laws, vertex corrections, and screening in Raman spectroscopy, Phys. Rev. B 96, 014503 (2017).
  63. I. Benek-Lins and S. Maiti, Many-body physics-induced selection rules: Application to Raman spectroscopy, Phys. Rev. B 109, 104505 (2024).
  64. S. Sarkar and S. Maiti, Electronic Raman response of a superconductor across a time reversal symmetry breaking phase transition, Phys. Rev. B 109, 094515 (2024).
  65. K. Takasan and N. Tsuji, Superconducting nonlinear Hall effect induced by geometric phases, arXiv:2503.14589.
  66. S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz, S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and N. P. Butch, Nearly ferromagnetic spin-triplet superconductivity, Science 365, 684 (2019).
  67. D. Aoki, J.-P. Brison, J. Flouquet, K. Ishida, G. Knebel, Y. Tokunaga, and Y. Yanase, Unconventional superconductivity in UTe2, J. Phys.: Condens. Matter 34, 243002 (2022).
  68. H. Matsumura, H. Fujibayashi, K. Kinjo, S. Kitagawa, K. Ishida, Y. Tokunaga, H. Sakai, S. Kambe, A. Nakamura, Y. Shimizu, Y. Homma, D. Li, F. Honda, and D. Aoki, Large reduction in the a-axis Knight shift on UTe2 with Tc = 2.1 K, J. Phys. Soc. Jpn. 92, 063701 (2023).
  69. F. Theuss, A. Shragai, G. Grissonnanche, I. M. Hayes, S. R. Saha, Y. S. Eo, A. Suarez, T. Shishidou, N. P. Butch, J. Paglione, and B. J. Ramshaw, Single-component superconductivity in UTe2 at ambient pressure, Nat. Phys. 20, 1124 (2024).
  70. S. Suetsugu, M. Shimomura, M. Kamimura, T. Asaba, H. Asaeda, Y. Kosuge, Y. Sekino, S. Ikemori, Y. Kasahara, Y. Kohsaka, M. Lee, Y. Yanase, H. Sakai, P. Opletal, Y. Tokiwa, Y. Haga, and Y. Matsuda, Fully gapped pairing state in spin-triplet superconductor UTe2, Sci. Adv. 10, eadk3772 (2024).
  71. Z. Li, C. M. Moir, N. J. McKee, E. Lee-Wong, R. E. Baumbach, M. B. Maple, and Y. Liu, Observation of odd-parity superconductivity in UTe2, Proc. Natl. Acad. Sci. USA 122, e2419734122 (2025).
  72. I. M. Hayes, T. E. Metz, C. E. Frank, S. R. Saha, N. P. Butch, V. Mishra, P. J. Hirschfeld, and J. Paglione, Robust nodal behavior in the thermal conductivity of superconducting UTe2, Phys. Rev. X 15, 021029 (2025).
  73. Q. Gu, S. Wang, J. P. Carroll, K. Zhussupbekov, C. Broyles, S. Ran, N. P. Butch, J. A. Horn, S. Saha, J. Paglione, X. Liu, J. C. S. Davis, and D.-H. Lee, Pair wave function symmetry in UTe2 from zero-energy surface state visualization, Science 388, 938 (2025).
  74. S. Wang, K. Zhussupbekov, and J. E. A. Carroll, Odd-parity quasiparticle interference in the superconductive surface state of UTe2, Nat. Phys. 21, 1555 (2025).
  75. W. C. Wu and J. P. Carbotte, c-axis Raman spectra of a normal plane-chain bilayer cuprate and the pseudogap, Phys. Rev. B 56, 6327 (1997).
  76. B. Valenzuela, M. J. Calderón, G. León, and E. Bascones, Optical conductivity and Raman scattering of iron superconductors, Phys. Rev. B 87, 075136 (2013).
  77. T. P. Devereaux, A. Virosztek, and A. Zawadowski, Multiband electronic Raman scattering in bilayer superconductors, Phys. Rev. B 54, 12523 (1996).
  78. G. R. Boyd, T. P. Devereaux, P. J. Hirschfeld, V. Mishra, and D. J. Scalapino, Probing the pairing symmetry of the iron pnictides with electronic Raman scattering, Phys. Rev. B 79, 174521 (2009).
  79. C. Sauer and G. Blumberg, Screening of the Raman response in multiband superconductors: Application to iron pnictides, Phys. Rev. B 82, 014525 (2010).
  80. Higher-dimensional IRs are regarded as 1D IRs of the subgroup of G.
  81. While our explicit calculations focus on fluctuations of the condensed components included in the mean-field BdG Hamiltonian, the selection rule and the classification tables are more general. In particular, by taking one of (Δα,Δβ) to be condensed and the other to represent a fluctuation in a subdominant (noncondensed) pairing channel, the same classification also contains Raman-active noncondensed pairing modes (with the corresponding subdominant coupling kept as an additional input in the kernel).
  82. T. Shishidou, H. G. Suh, P. M. R. Brydon, M. Weinert, and D. F. Agterberg, Topological band and superconductivity in UTe2, Phys. Rev. B 103, 104504 (2021).
  83. D. Aoki, H. Sakai, P. Opletal, Y. Tokiwa, J. Ishizuka, Y. Yanase, H. Harima, A. Nakamura, D. Li, Y. Homma, Y. Shimizu, G. Knebe, J. Flouquet, and Y. Haga, First observation of the de Haas–van Alphen effect and Fermi surfaces in the unconventional superconductor UTe2, J. Phys. Soc. Jpn. 91, 083704 (2022).
  84. T. I. Weinberger, Z. Wu, D. E. Graf, Y. Skourski, A. Cabala, J. Pospíšil, J. Prokleška, T. Haidamak, G. Bastien, V. Sechovský, G. G. Lonzarich, M. Vališka, F. M. Grosche, and A. G. Eaton, Quantum interference between quasi-2D Fermi surface sheets in UTe2, Phys. Rev. Lett. 132, 266503 (2024).
  85. D. Aoki, I. Sheikin, N. Marquardt, G. Lapertot, J. Flouquet, and G. Knebel, High field superconducting phases of ultra clean single crystal UTe2, J. Phys. Soc. Jpn. 93, 123702 (2024).

Outline

Information

Phys. Rev. B 114, 034502– Published 1 July, 2026
  • Received 9 February 2026
  • Revised 21 May 2026
  • Accepted 10 June 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation