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arxiv: 2604.15054 · v1 · submitted 2026-04-16 · ❄️ cond-mat.supr-con

Type II Lifshitz invariant and optically active Higgs mode in time-reversal symmetry broken superconductors

Raigo Nagashima , Chihiro Mamiya , Naoto Tsuji This is my paper

Pith reviewed 2026-05-10 09:45 UTC · model grok-4.3

classification ❄️ cond-mat.supr-con
keywords Lifshitz invariantHiggs modetime-reversal symmetry breakingoptical conductivitysuperconductorsGinzburg-Landau theorymagnetic point groupscollective modes
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The pith

A particle-hole odd Lifshitz invariant exists only in time-reversal symmetry broken superconductors and couples their Higgs mode to light.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a type II Lifshitz invariant, a term in the Ginzburg-Landau free energy that is odd under particle-hole transformation and involves one spatial derivative of the order parameters. This term appears exclusively when time-reversal symmetry is broken and renders the Higgs mode detectable in the optical conductivity spectrum. The authors classify every pair of irreducible corepresentations of the order parameters in magnetic point groups that permits such an invariant. Numerical calculations on multiband models of time-reversal symmetry broken superconductors confirm that the optical response matches the symmetry classification. A sympathetic reader would care because the result identifies a broad, symmetry-defined class of superconductors in which the Higgs mode, normally invisible to light, can be observed directly through optics.

Core claim

We show that the type II Lifshitz invariant appears only in superconductors that break time-reversal symmetry and allows the Higgs mode to be visible in the optical conductivity spectrum. We provide a classification of all pairs of irreducible corepresentations of order parameters in the magnetic point groups that admit a type II Lifshitz invariant. We also numerically calculate the optical conductivity for various models of time-reversal symmetry broken multiband superconductors, finding agreement with the group-theoretical analysis.

What carries the argument

The type II Lifshitz invariant, a particle-hole odd term in the Ginzburg-Landau free energy that couples the superconducting order parameters to electromagnetic fields via a single spatial derivative.

If this is right

  • The Higgs mode contributes a distinct feature to the optical conductivity spectrum in any time-reversal symmetry broken superconductor that hosts the type II Lifshitz invariant.
  • A complete group-theoretical list now exists of all allowed pairs of irreducible corepresentations in magnetic point groups for which the effect is symmetry-permitted.
  • Numerical calculations on multiband models reproduce the predicted optical activity and match the symmetry analysis.
  • The results define a universal class of time-reversal symmetry broken superconductors that host an optically active Higgs mode.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Experimental searches for the optical Higgs signature can now be guided by the symmetry classification rather than by trial and error in candidate materials.
  • The same symmetry logic may apply to other collective modes in ordered phases that break time-reversal symmetry, opening routes to optical detection of modes that are usually hidden.
  • If the type II invariant dominates, then time-reversal symmetry breaking becomes a necessary condition for optical visibility of the Higgs mode in superconductors.

Load-bearing premise

The particle-hole odd type II Lifshitz invariant is the dominant mechanism that couples the Higgs mode to light, and the chosen numerical models represent the general symmetry-allowed cases without hidden cancellations from other terms.

What would settle it

Optical conductivity measurements on a time-reversal symmetry broken superconductor whose order-parameter corepresentations are classified as admitting the type II invariant but that show no Higgs peak, or the opposite observation in a symmetry class not allowed by the classification.

Figures

Figures reproduced from arXiv: 2604.15054 by Chihiro Mamiya, Naoto Tsuji, Raigo Nagashima.

Figure 2
Figure 2. Figure 2: Several lattice models of time-reversal symme [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optical conductivity spectrum for each lattice [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
read the original abstract

Lifshitz invariant is a symmetry-allowed term in the Ginzburg-Landau free energy of an ordered phase, involving the order parameters and a single spatial derivative, which serves as a source of unusual optical responses. Here we introduce a ``type II" Lifshitz invariant for superconductors, which changes its sign under the particle-hole transformation and can be distinguished from the ordinary particle-hole even ``type I" Lifshitz invariant. We show that the type II Lifshitz invariant appears only in superconductors that break time-reversal symmetry and allows the Higgs mode to be visible in the optical conductivity spectrum. We provide a classification of all pairs of irreducible corepresentations of order parameters in the magnetic point groups that admit a type II Lifshitz invariant. We also numerically calculate the optical conductivity for various models of time-reversal symmetry broken multiband superconductors, finding agreement with the group-theoretical analysis. Our results establish a universal class of time-reversal symmetry broken superconductors hosting an optically active Higgs mode.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a 'type II' Lifshitz invariant in the Ginzburg-Landau free energy of superconductors, defined to be odd under particle-hole transformation (in contrast to the even 'type I' form). It shows that this term is permitted exclusively in time-reversal symmetry (TRS) broken superconductors, where it couples the Higgs mode to the electromagnetic field and renders the mode visible in the optical conductivity. The central result is a complete classification of all pairs of irreducible corepresentations of the order parameter that allow a type II Lifshitz invariant within the magnetic point groups, together with numerical calculations on multiband TRS-broken models that reproduce the expected optical response in agreement with the symmetry analysis.

Significance. If the classification is exhaustive and the numerical support holds, the work supplies a symmetry-based criterion for identifying a universal class of TRS-broken superconductors in which the Higgs mode acquires optical activity. This could guide both theory and experiment on pairing symmetries and collective modes. The group-theoretical part rests on standard representation theory of magnetic groups and is internally consistent; the numerical checks provide an independent, falsifiable test. Upon reading the full manuscript the stress-test concern about model details does not land: the models are specified with explicit band structures, interaction parameters, and conductivity formulas, and no post-hoc exclusions or unaccounted cancellations appear.

minor comments (3)
  1. [Theory section] The definition of the type II Lifshitz invariant (particle-hole odd term linear in a spatial derivative) would benefit from an explicit comparison, in the text or an appendix, to the conventional type I form to highlight the sign change under particle-hole conjugation.
  2. [Classification section] A compact summary table listing the magnetic point groups together with the allowed corepresentation pairs that admit the type II invariant would make the classification results easier to navigate.
  3. [Numerical results] In the numerical optical-conductivity plots, the frequency axis and the location of the Higgs peak should be labeled with the same reduced units used in the analytic expressions to facilitate direct comparison with the group-theory predictions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The referee's summary accurately reflects the central results on the type II Lifshitz invariant, its restriction to time-reversal symmetry broken superconductors, the exhaustive classification of magnetic point group corepresentations, and the supporting numerical calculations of optical conductivity. We are pleased that the group-theoretical analysis and model checks were found to be internally consistent and falsifiable.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external group theory and independent numerics

full rationale

The central classification of type II Lifshitz invariants relies on standard representation theory of magnetic point groups applied to irreducible corepresentations, which is an external mathematical tool independent of the paper's claims. Numerical optical conductivity calculations for specific multiband models serve as an independent check that reproduces the symmetry-allowed behavior without any fitted parameters being renamed as predictions or self-referential definitions. No load-bearing step reduces to a self-citation chain, ansatz smuggling, or input-output equivalence by construction. The derivation chain remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard magnetic point group representation theory plus the assumption that the type II term dominates the optical response in the chosen models.

axioms (1)
  • standard math Standard representation theory of magnetic point groups governs the allowed invariants for superconducting order parameters.
    Invoked for the classification of irreducible corepresentations that admit the type II Lifshitz invariant.
invented entities (1)
  • Type II Lifshitz invariant no independent evidence
    purpose: A particle-hole odd term in the Ginzburg-Landau free energy that enables optical activity of the Higgs mode.
    Newly defined in the paper and distinguished from the conventional type I term.

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Reference graph

Works this paper leans on

123 extracted references · 123 canonical work pages

  1. [1]

    2(a), where we fixt ij =t= 1.0,ϕ= 0.01, µ=−0.3, andU=−3.0

    Ladder model The first example is the triangular ladder model shown in Fig. 2(a), where we fixt ij =t= 1.0,ϕ= 0.01, µ=−0.3, andU=−3.0. In Fig. 3(a), we show the linear optical conductivity with blue and red curves cor- responding to the response of quasiparticles (σ QP) and collective modes (σ CM), respectively. One can see that the collective-mode respon...

    work page

  2. [2]

    2(b) and (c))

    Square lattice model The second and third examples are the anisotropic and isotropic square lattice models (Fig. 2(b) and (c)). In the anisotropic model, we set the hopping amplitudest 1 = t= 1.0 (solid lines) andt 2 = 0.1 (dashed lines), respec- tively. In the isotropic model, we fixt 1 =t 2 =t= 1.0. In both cases, we setµ=−0.3. The results of the opti- ...

    work page

  3. [3]

    Kagome lattice model with three sites in the unit cell The fourth example is the isotropic Kagome lattice model with three sites in the unit cell (Fig. 2(d)). We sett ij =t= 1.0 andµ= 0. The calculated linear optical conductivity is shown in Fig. 3(d). Both the quasiparticle and collective-mode responses have a peak atω/2∆ = 1 with the same height but wit...

    work page

  4. [4]

    2(e) and (f))

    Kagome lattice model with12sites in the unit cell The fifth and last examples are the anisotropic and isotropic Kagome lattice models with 12 lattice sites in the unit cell (Fig. 2(e) and (f)). In the anisotropic model, we set the hopping amplitudest 1 = 0.5 (solid lines) and t2 = 0.05 (dashed lines), respectively. In the isotropic model, we putt 1 =t 2 =...

    work page

  5. [5]

    Product tables for the crystallographic point groups are widely available

    Complex corepresentation The product tables for the complex corepresentations of the point groupM=G+AGcan be obtained from the product tables for the representations ofG. Product tables for the crystallographic point groups are widely available. See, for example, Ref. [104]. We remark that some of the literature doesn’t distinguish 1Eand 2E, the physicall...

    work page

  6. [6]

    The real representations ofHare, in turn, tab- ulated from its complex representation [105]

    Real corepresentation The product tables of the real corepresentations can be derived by utilizing the identity between the real corepresentations ofMand the real representations of H. The real representations ofHare, in turn, tab- ulated from its complex representation [105]. For the magnetic point groups that do not contain parityless corepresentations,...

    work page

  7. [7]

    Bardeen, L

    J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev.108, 1175 (1957)

    work page 1957

  8. [8]

    P. W. Anderson, Random-Phase Approximation in the Theory of Superconductivity, Phys. Rev.112(1958). 28

    work page 1958

  9. [9]

    Schmid, The approach to equilibrium in a pure su- perconductor the relaxation of the Cooper pair density, Phys

    A. Schmid, The approach to equilibrium in a pure su- perconductor the relaxation of the Cooper pair density, Phys. Kondens. Mater.8(1968)

    work page 1968

  10. [10]

    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)

    work page 1981

  11. [11]

    P. B. Littlewood and C. M. Varma, Amplitude collective modes in superconductors and their coupling to charge- density waves, Phys. Rev. B26, 4883 (1982)

    work page 1982

  12. [12]

    Pekker and C

    D. Pekker and C. Varma, Amplitude/Higgs Modes in Condensed Matter Physics , Annu. Rev. Condens. Mat- ter Phys.6(2015)

    work page 2015

  13. [13]

    Shimano and N

    R. Shimano and N. Tsuji, Higgs mode in superconduc- tors, Annu. Rev. Condens. Matter Phys.11, 103 (2020)

    work page 2020

  14. [14]

    Tsuji, I

    N. Tsuji, I. Danshita, and S. Tsuchiya, Higgs and Nambu–Goldstone modes in condensed matter physics, inEncyclopedia of Condensed Matter Physics (Second Edition)(Academic Press, Oxford, 2024) 2nd ed., pp. 174–186

    work page 2024

  15. [15]

    Superconductor

    J. Goldstone, Field Theories with “Superconductor” So- lutions, Nuovo Cim.19, 154 (1961)

    work page 1961

  16. [16]

    Nambu and G

    Y. Nambu and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Super- conductivity. I, Phys. Rev.122, 345 (1961)

    work page 1961

  17. [17]

    P. W. Anderson, Plasmons, Gauge Invariance, and Mass, Phys. Rev.130, 439 (1963)

    work page 1963

  18. [18]

    Englert and R

    F. Englert and R. Brout, Broken Symmetry and the Mass of Gauge Vector Mesons, Phys. Rev. Lett.13, 321 (1964)

    work page 1964

  19. [19]

    P. W. Higgs, Broken Symmetries and the Masses of Gauge Bosons, Phys. Rev. Lett.13, 508 (1964)

    work page 1964

  20. [20]

    G. S. Guralnik, C. R. Hagen, and T. W. B. Kibble, Global Conservation Laws and Massless Particles, Phys. Rev. Lett.13, 585 (1964)

    work page 1964

  21. [21]

    Tsuji and H

    N. Tsuji and H. Aoki, Theory of Anderson pseudospin resonance with Higgs mode in superconductors, Phys. Rev. B92, 064508 (2015)

    work page 2015

  22. [22]

    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. B92, 224517 (2015)

    work page 2015

  23. [23]

    T. Cea, C. Castellani, and L. Benfatto, Nonlinear opti- cal effects and third-harmonic generation in supercon- ductors: Cooper pairs versus Higgs mode contribution, Phys. Rev. B93, 180507 (2016)

    work page 2016

  24. [24]

    Tsuji, Y

    N. Tsuji, Y. Murakami, and H. Aoki, Nonlinear light– Higgs coupling in superconductors beyond BCS: Effects of the retarded phonon-mediated interaction, Phys. Rev. B94, 224519 (2016)

    work page 2016

  25. [25]

    Jujo, Quasiclassical Theory on Third-Harmonic Gen- eration in Conventional Superconductors with Param- agnetic Impurities, J

    T. Jujo, Quasiclassical Theory on Third-Harmonic Gen- eration in Conventional Superconductors with Param- agnetic Impurities, J. Phys. Soc. Jpn.87, 024704 (2018)

    work page 2018

  26. [26]

    Silaev, Nonlinear electromagnetic response and Higgs-mode excitation in BCS superconductors with im- purities, Phys

    M. Silaev, Nonlinear electromagnetic response and Higgs-mode excitation in BCS superconductors with im- purities, Phys. Rev. B99, 224511 (2019)

    work page 2019

  27. [27]

    Schwarz, B

    L. Schwarz, B. Fauseweh, N. Tsuji, N. Cheng, N. Bit- tner, H. Krull, M. Berciu, G. S. Uhrig, A. P. Schnyder, S. Kaiser, and D. Manske, Classification and characteri- zation of nonequilibrium Higgs modes in unconventional superconductors , Nat. Commun.11(2020)

    work page 2020

  28. [28]

    Tsuji and Y

    N. Tsuji and Y. Nomura, Higgs-mode resonance in third harmonic generation in nbn superconductors: Multi- band electron-phonon coupling, impurity scattering, and polarization-angle dependence, Phys. Rev. Res.2, 043029 (2020)

    work page 2020

  29. [29]

    Haenel, P

    R. Haenel, P. Froese, D. Manske, and L. Schwarz, Time- resolved optical conductivity and Higgs oscillations in two-band dirty superconductors, Phys. Rev. B104, 134504 (2021)

    work page 2021

  30. [30]

    Udina, J

    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 , Faraday Discuss.237(2022)

    work page 2022

  31. [31]

    Derendorf, A

    P. Derendorf, A. F. Volkov, and I. M. Eremin, Nonlinear response of diffusive superconductors to ac electromag- netic fields, Phys. Rev. B109, 024510 (2024)

    work page 2024

  32. [32]

    Sooryakumar and M

    R. Sooryakumar and M. V. Klein, Raman Scattering by Superconducting-Gap Excitations and Their Cou- pling to Charge-Density Waves, Phys. Rev. Lett.45, 660 (1980)

    work page 1980

  33. [33]

    Sooryakumar and M

    R. Sooryakumar and M. V. Klein, Raman scattering from superconducting gap excitations in the presence of a magnetic field, Phys. Rev. B23, 3213 (1981)

    work page 1981

  34. [34]

    M´ easson, Y

    M.-A. M´ easson, Y. Gallais, M. Cazayous, B. Clair, P. Rodi` ere, L. Cario, and A. Sacuto, Amplitude Higgs mode in the 2H-NbSe 2 superconductor, Phys. Rev. B 89, 060503 (2014)

    work page 2014

  35. [35]

    Grasset, T

    R. Grasset, T. Cea, Y. Gallais, M. Cazayous, A. Sacuto, L. Cario, L. Benfatto, and M.-A. M´ easson, Higgs-mode radiance and charge-density-wave order in 2H-NbSe 2, Phys. Rev. B97, 094502 (2018)

    work page 2018

  36. [36]

    Majumdar, D

    A. Majumdar, D. VanGennep, J. Brisbois, D. Cha- reev, 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−NbS 2 and 2H−NbSe 2, Phys. Rev. Mater.4, 084005 (2020)

    work page 2020

  37. [37]

    Matsunaga, Y

    R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang, and R. Shimano, Higgs Amplitude Mode in the BCS Superconductors Nb1−xTixNInduced by Terahertz Pulse Excitation, Phys. Rev. Lett.111, 057002 (2013)

    work page 2013

  38. [38]

    Matsunaga, N

    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 preces- sion resonating with Higgs mode in a superconductor , Science345(2014)

    work page 2014

  39. [39]

    Matsunaga, N

    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. B96, 020505 (2017)

    work page 2017

  40. [40]

    Katsumi, N

    K. Katsumi, N. Tsuji, Y. I. Hamada, R. Matsunaga, J. Schneeloch, R. D. Zhong, G. D. Gu, H. Aoki, Y. Gal- lais, and R. Shimano, Higgs Mode in thed-Wave Su- perconductor Bi 2Sr2CaCu2O8+x Driven by an Intense Terahertz Pulse, Phys. Rev. Lett.120, 117001 (2018)

    work page 2018

  41. [41]

    Chu, M.-J

    H. Chu, M.-J. Kim, K. Katsumi, S. Kovalev, R. D. Daw- son, L. Schwarz, N. Yoshikawa, G. Kim, D. Putzky, Z. Z. Li, H. Raffy, S. Germanskiy, J.-C. Deinert, N. Awari, I. Ilyakov, B. Green, M. Chen, M. Bawatna, G. Cris- tiani, G. Logvenov, Y. Gallais, A. V. Boris, B. Keimer, A. P. Schnyder, D. Manske, M. Gensch, Z. Wang, R. Shimano, and S. Kaiser, Phase-resol...

    work page 2020

  42. [42]

    A. Moor, A. F. Volkov, and K. B. Efetov, Amplitude higgs mode and admittance in superconductors with a moving condensate, Phys. Rev. Lett.118, 047001 (2017)

    work page 2017

  43. [43]

    Nakamura, Y

    S. Nakamura, Y. Iida, Y. Murotani, R. Matsunaga, H. Terai, and R. Shimano, Infrared activation of the higgs mode by supercurrent injection in superconduct- ing NbN, Phys. Rev. Lett.122, 257001 (2019)

    work page 2019

  44. [44]

    Y. Lu, S. Ili´ c, R. Ojaj¨ arvi, T. T. Heikkil¨ a, and F. S. Bergeret, Reducing the frequency of the higgs mode in a helical superconductor coupled to an lc circuit, Phys. Rev. B108, 224517 (2023)

    work page 2023

  45. [45]

    Nagashima, T

    R. Nagashima, T. Mouilleron, and N. Tsuji, Opti- cally active higgs and leggett modes in multiband pair- density-wave superconductors with lifshitz invariant, Phys. Rev. B112, 024503 (2025)

    work page 2025

  46. [46]

    G. R. Stewart, Superconductivity in iron compounds, Rev. Mod. Phys.83, 1589 (2011)

    work page 2011

  47. [47]

    Xu, A review and prospects for Nb 3Sn superconduc- tor development, Superconductor Science and Technol- ogy30(2017)

    X. Xu, A review and prospects for Nb 3Sn superconduc- tor development, Superconductor Science and Technol- ogy30(2017)

    work page 2017

  48. [48]

    Buzea and T

    C. Buzea and T. Yamashita, Review of the supercon- ducting properties of MgB 2, Superconductor Science and Technology14, R115 (2001)

    work page 2001

  49. [49]

    Jiang, T

    K. Jiang, T. Wu, J.-X. Yin, Z. Wang, M. Z. Hasan, S. D. Wilson, X. Chen, and J. Hu, Kagome superconductors AV3Sb5 (A = K,Rb,Cs), National Science Review10 (2023)

    work page 2023

  50. [50]

    A. J. Leggett, Number-Phase Fluctuations in Two-Band Superconductors , Prog. Theor. Phys.36(1966)

    work page 1966

  51. [51]

    A. V. Balatsky, P. Kumar, and J. R. Schrieffer, Collec- tive Mode in a Superconductor with Mixed-Symmetry Order Parameter Components, Phys. Rev. Lett.84, 4445 (2000)

    work page 2000

  52. [52]

    F. J. Burnell, J. Hu, M. M. Parish, and B. A. Bernevig, Leggett mode in a strong-coupling model of iron arsenide superconductors, Phys. Rev. B82, 144506 (2010)

    work page 2010

  53. [53]

    Y. Ota, M. Machida, T. Koyama, and H. Aoki, Col- lective modes in multiband superfluids and supercon- ductors: Multiple dynamical classes, Phys. Rev. B83, 060507 (2011)

    work page 2011

  54. [54]

    Lin and X

    S.-Z. Lin and X. Hu, Massless leggett mode in three- band superconductors with time-reversal-symmetry breaking, Phys. Rev. Lett.108, 177005 (2012)

    work page 2012

  55. [55]

    Marciani, L

    M. Marciani, L. Fanfarillo, C. Castellani, and L. Ben- fatto, Leggett modes in iron-based superconductors as a probe of time-reversal symmetry breaking, Phys. Rev. B88, 214508 (2013)

    work page 2013

  56. [56]

    Bittner, D

    N. Bittner, D. Einzel, L. Klam, and D. Manske, Leggett Modes and the Anderson-Higgs Mechanism in Super- conductors without Inversion Symmetry, Phys. Rev. Lett.115, 227002 (2015)

    work page 2015

  57. [57]

    Krull, N

    H. Krull, N. Bittner, G. S. Uhrig, D. Manske, and A. P. Schnyder, Coupling of Higgs and Leggett modes in non- equilibrium superconductors , Nat. Commun.7(2016)

    work page 2016

  58. [58]

    Murotani, N

    Y. Murotani, N. Tsuji, and H. Aoki, Theory of light- induced resonances with collective Higgs and Leggett modes in multiband superconductors, Phys. Rev. B95, 104503 (2017)

    work page 2017

  59. [59]

    Murotani and R

    Y. Murotani and R. Shimano, Nonlinear optical re- sponse of collective modes in multiband superconduc- tors assisted by nonmagnetic impurities, Phys. Rev. B 99, 224510 (2019)

    work page 2019

  60. [60]

    Giorgianni, T

    F. Giorgianni, T. Cea, C. Vicario, C. P. Hauri, W. K. Withanage, X. Xi, and L. Benfatto, Leggett mode controlled by light pulses, Nat. Phys.15, 341 (2019)

    work page 2019

  61. [61]

    Fiore, M

    J. Fiore, M. Udina, M. Marciani, G. Seibold, and L. Benfatto, Contribution of collective excitations to third harmonic generation in two-band superconduc- tors: The case of mgb 2, Phys. Rev. B106, 094515 (2022)

    work page 2022

  62. [62]

    Seibold, M

    G. Seibold, M. Udina, C. Castellani, and L. Benfatto, Third harmonic generation from collective modes in dis- ordered superconductors, Phys. Rev. B103, 014512 (2021)

    work page 2021

  63. [63]

    Blumberg, A

    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 MgB 2 Super- conductor, Phys. Rev. Lett.99, 227002 (2007)

    work page 2007

  64. [64]

    Kamatani, S

    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. B105, 094520 (2022)

    work page 2022

  65. [65]

    Nagashima, S

    R. Nagashima, S. Tian, R. Haenel, N. Tsuji, and D. Manske, Classification of lifshitz invariant in multi- band superconductors: An application to leggett modes in the linear response regime in kagome lattice models, Phys. Rev. Res.6, 013120 (2024)

    work page 2024

  66. [66]

    L. D. Landau and E. M. Lifshitz,Statistical Physics (Pergamon Press, Oxford, 1969)

    work page 1969

  67. [67]

    V. P. Mineev and K. V. Samokhin, Helical phases in superconductors , JETP105(1994)

    work page 1994

  68. [68]

    V. P. Mineev and K. V. Samokhin, Nonuniform states in noncentrosymmetric superconductors: Derivation of Lifshitz invariants from microscopic theory, Phys. Rev. B78, 144503 (2008)

    work page 2008

  69. [69]

    K. V. Samokhin, Gradient energy of superconductors without inversion symmetry, Physica C: Superconduc- tivity489, 19 (2013)

    work page 2013

  70. [70]

    Fuchs, D

    L. Fuchs, D. Kochan, J. Schmidt, N. H¨ uttner, C. Baum- gartner, S. Reinhardt, S. Gronin, G. C. Gardner, T. Lin- demann, M. J. Manfra, C. Strunk, and N. Paradiso, Anisotropic vortex squeezing in synthetic rashba super- conductors: A manifestation of lifshitz invariants, Phys. Rev. X12, 041020 (2022)

    work page 2022

  71. [71]

    Kochan, A

    D. Kochan, A. Costa, I. Zhumagulov, and I. ˇZuti´ c, Phe- nomenological theory of the supercurrent diode effect: The lifshitz invariant, arXiv (2023)

    work page 2023

  72. [72]

    Kanasugi and Y

    S. Kanasugi and Y. Yanase, Anapole superconductiv- ity fromPT-symmetric mixed-parity interband pairing, Commun. Phys.5, 39 (2022)

    work page 2022

  73. [73]

    Kitamura, S

    T. Kitamura, S. Kanasugi, M. Chazono, and Y. Yanase, Quantum geometry induced anapole superconductivity, Phys. Rev. B107, 214513 (2023)

    work page 2023

  74. [74]

    Kopsky and D

    V. Kopsky and D. G. Sannikov, Gradient invariants and incommensurate phase transitions , J. Phys. C: Solid State Phys.10(1977)

    work page 1977

  75. [75]

    Ishibashi and V

    Y. Ishibashi and V. Dvoˇ r´ ak, Incommensurate Phase Transitions under the Existence of the Lifshitz Invariant , J. Phys. Soc. Jpn.44, 32 (1978)

    work page 1978

  76. [76]

    Sparavigna, Role of Lifshitz Invariants in Liquid Crystals , Materials2(2009)

    A. Sparavigna, Role of Lifshitz Invariants in Liquid Crystals , Materials2(2009)

    work page 2009

  77. [77]

    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). 30

    work page 2024

  78. [78]

    B. A. Levitan and ´Etienne Lantagne-Hurtubise, Trigo- nal warping enables linear optical spectroscopy in single- valley superconductors, arXiv (2025)

    work page 2025

  79. [79]

    Takasan and N

    K. Takasan and N. Tsuji, Superconducting nonlinear hall effect induced by geometric phases, arXiv (2025)

    work page 2025

  80. [80]

    Yamazaki and T

    Y. Yamazaki and T. Morimoto, Raman response of collective modes in multicomponent superconductors, arXiv (2026)

    work page 2026

Showing first 80 references.