Bardasis–Schrieffer Mode in Superconductors

• Bardasis–Schrieffer mode is a bound collective excitation originating from a subdominant pairing channel in superconductors, observable below the 2Δ pair-breaking threshold.
• It emerges from fluctuations between dominant and competing pairing interactions, with its softening signaling transitions to mixed or unconventional symmetry states.
• The mode is investigated through Raman spectroscopy, pump–probe, and THG methods, and can hybridize with nematic excitations or plasmonic responses in multiband systems.

Searching arXiv for recent Bardasis-Schrieffer mode papers to support the article. The Bardasis–Schrieffer mode is a collective, exciton-like bound state associated with fluctuations in a subdominant attractive pairing channel of a superconductor or, more generally, of a condensate with competing internal channels. In the canonical superconducting setting, a dominant condensate forms in one symmetry channel while a residual attraction survives in another; the latter does not condense, but supports a sharp in-gap excitation below the pair-breaking threshold $2\Delta$ (Müller et al., 2020). In an $s$-wave ground state, a $d$-wave subdominant channel yields the standard Bardasis–Schrieffer mode; in a $d$-wave ground state, an $s$-wave residual channel can generate an analogous excitation, although nodal quasiparticles can strongly damp it unless additional structure, such as nematicity, suppresses that damping (Müller et al., 2020). The mode was originally proposed as an internal fluctuation of the superconducting order parameter and has since become a diagnostic of pairing competition, collective-mode hybridization, and unconventional superconducting symmetry (Costa et al., 2021).

1. Definition and microscopic origin

The defining condition for a Bardasis–Schrieffer mode is the coexistence of a dominant pairing instability and a distinct subdominant attractive interaction. In the formulation used for unconventional superconductors with multiple attractive channels, the condensate selects a ground-state symmetry $l_0$, while fluctuations in a different channel $l \neq l_0$ form a bound-state collective mode below the pair-breaking edge $2\Delta$ (Müller et al., 2019). In this sense the mode is a particle–particle exciton internal to the paired state rather than a quasiparticle continuum feature.

A standard example is an $s$-wave ground state with residual attraction in a $d$-wave channel. The $s$0-wave channel is not strong enough to condense, but it binds a subgap pair fluctuation. In the notation of the non-equilibrium square-lattice model, this is the mode in the subdominant $s$1 sector when the equilibrium state is extended $s$2-wave (Müller et al., 2019). In the nematicity-coupled model, the same logic applies to $s$3- and $s$4-wave competition on a two-dimensional single-band square lattice, where the mode sits at $s$5 and is explicitly associated with the subdominant pairing symmetry (Müller et al., 2020).

The mode softens as the subdominant channel approaches degeneracy with the dominant one. In the pumped unconventional-superconductor study, the Bardasis–Schrieffer mode approaches the Higgs continuum far from the phase boundary, while near the transition between competing ground states it softens toward zero frequency (Müller et al., 2019). The same softening appears in the $s$6- versus $s$7-competition phase diagram of superconductors with competing $s$8- and $s$9-wave interactions, where the one-band $d$0-wave BS mode collapses at the $d$1 boundary and evolves into a mixed-symmetry collective mode inside the $d$2 phase (Maiti et al., 2015).

The basic pole condition is a susceptibility equation in the subdominant channel. In the nematicity-coupled formulation, the single-channel limit recovers the familiar condition $d$3 for an $d$4-wave ground state (Müller et al., 2020). In the current-enabled optical formalism, the collective frequencies are poles of the effective interaction kernel,

$d$5

with the single-band $d$6-phase Bardasis–Schrieffer mode satisfying

$d$7

(Niederhoff et al., 9 Apr 2025). In Raman-based multiband formulations, the same physics appears as zeros of denominators such as $d$8 in the subleading pairing sector (Maiti et al., 2016).

2. Field-theoretic and pseudospin formulations

Several complementary microscopic descriptions are used in the literature. A common approach starts from a BCS or mean-field Hamiltonian with separable pairing interactions in multiple angular-momentum channels, then derives collective modes from Gaussian fluctuations around the saddle point (Müller et al., 2021). In the time-domain studies of unconventional superconductors, the problem is recast in terms of Anderson pseudospins,

$d$9

whose dynamics obey

$d$0

(Müller et al., 2019).

In the nematicity-coupled model, the mean-field Hamiltonian is

$d$1

with

$d$2

and the choice $d$3, $d$4 (Müller et al., 2020). Linearizing the pseudospin equations around equilibrium gives

$d$5

and collective modes satisfy

$d$6

(Müller et al., 2020).

In the Raman-response framework for multicomponent superconductors, one starts from a general BdG Hamiltonian with separable pairing interaction,

$d$7

and derives a gauge-invariant Raman susceptibility in which collective modes appear as poles of an effective fluctuation kernel $d$8 (Yamazaki et al., 5 Feb 2026). In that treatment the Bardasis–Schrieffer mode is one member of a broader class of relative pairing-channel modes, distinct from Leggett and clapping modes by its intraband, subdominant-channel character (Yamazaki et al., 5 Feb 2026).

3. Energetics, softening, and relation to competing phases

A central property of the Bardasis–Schrieffer mode is that it lies below the pair-breaking edge of the dominant condensate. In the pumped square-lattice model this is expressed directly as $d$9, where $s$0 is the Higgs energy for the dominant gap (Müller et al., 2019). In rotationally symmetric $s$1-wave states, the mode is undamped because it sits below the continuum. In nodal $s$2-wave states, however, the analogous subdominant-channel oscillation is strongly damped by nodal quasiparticles (Müller et al., 2019).

The frequency tracks the competition between pairing channels. In the one-band $s$3-wave phase of the $s$4- versus $s$5-competition model, the phase-sector equation for the BS mode is

$s$6

and the solution softens as $s$7, marking the approach to the $s$8 phase boundary (Maiti et al., 2015). In the mixed $s$9 regime the mode no longer remains a pure $l_0$0-channel phase oscillation, but becomes a “mixed-symmetry Bardasis–Schrieffer mode” determined by a coupled equation involving $l_0$1, $l_0$2, and $l_0$3 (Maiti et al., 2015).

The multiband case adds further structure. In the three-pocket model for hole-doped Fe-based superconductors, an $l_0$4-wave ground state supports one well-defined Bardasis–Schrieffer mode below the lowest gap edge and a second damped BS-like mode between the two gap energies (Maiti et al., 2015). These coexist with a damped Leggett mode between the two gap scales (Maiti et al., 2015). This separation of sharp subgap and damped intergap resonances is important experimentally because a subgap peak is not automatically unique to one collective degree of freedom.

A closely related issue is the distinction from other relative modes. In the Raman theory of multicomponent superconductors, the Bardasis–Schrieffer mode is explicitly contrasted with the Leggett mode, which is interband and corresponds to relative-phase oscillation between condensates on different bands, and with clapping modes, which arise in chiral states as fluctuations of Cooper-pair angular momentum (Yamazaki et al., 5 Feb 2026). This distinction becomes blurred in systems with strong symmetry mixing. In FeSe with electronic nematicity, the observed low-energy mode is attributed to a fluctuation between an $l_0$5-wave-like ground state and a subleading pairing channel; it is described as corresponding to the Bardasis–Schrieffer mode but also resembling an intraband Leggett mode (Matsumoto et al., 19 Jul 2025). This suggests that in lowered-symmetry environments the modal taxonomy can become basis dependent even when the underlying subdominant-channel origin remains intact.

4. Hybridization with other collective modes and with symmetry breaking

The Bardasis–Schrieffer mode need not remain isolated. In the nematicity-coupled model, the most important result is that in a rotationally symmetric $l_0$6-wave ground state the $l_0$7-wave Bardasis–Schrieffer fluctuation hybridizes with the nematic collective mode, producing a single in-gap excitation of mixed nematic–BS character (Müller et al., 2020). The relevant $l_0$8 kernel couples the imaginary $l_0$9-wave component $l \neq l_0$0 and the nematic amplitude $l \neq l_0$1,

$l \neq l_0$2

and only one in-gap solution exists for $l \neq l_0$3, with $l \neq l_0$4 (Müller et al., 2020).

This mixing is symmetry controlled. For $l \neq l_0$5 nematicity, where $l \neq l_0$6, the nematic and $l \neq l_0$7-wave pairing sectors hybridize because they transform identically (Müller et al., 2020). If the nematicity instead has $l \neq l_0$8 form factor, $l \neq l_0$9, the off-diagonal couplings vanish and BS and nematic modes remain distinct (Müller et al., 2020). The mixed mode softens both at the nematic critical point $2\Delta$0 and near $2\Delta$1, where $2\Delta$2 degeneracy is approached (Müller et al., 2020).

In bilayer and locally non-centrosymmetric settings, the subdominant-channel fluctuation can take a phase-like form and couple strongly to Coulomb or electromagnetic degrees of freedom. In a superconducting bilayer with strong spin-orbit coupling, the competing odd-parity pairing instability gives rise to a Bardasis–Schrieffer-like phase mode inside the excitation gap when Coulomb interaction is neglected (Hackner et al., 2023). There the mode satisfies

$2\Delta$3

with the pole condition $2\Delta$4 (Hackner et al., 2023). Once long-range Coulomb interaction is included, however, the mode is converted into an antisymmetric plasmon and is generally pushed into the quasiparticle continuum (Hackner et al., 2023). This is unlike conventional single-layer BS modes, which are not primarily longitudinal phase oscillations.

Hybridization also appears in optical and cavity contexts. In cavity proposals, a supercurrent produces a linear light–BS coupling, allowing one photon polarization to hybridize with the BS mode and form upper and lower BS polaritons (Liao et al., 2018). In graphene–superconductor heterostructures, the graphene plasmon hybridizes with the Bardasis–Schrieffer mode of a two-dimensional superconductor, producing clear anticrossings in the near-field reflection coefficient $2\Delta$5 (Costa et al., 2021). These examples show that the BS mode can enter composite bosonic sectors well beyond isolated superconducting fluctuation spectra.

5. Spectroscopic signatures: Raman, pump–probe, THG, and linear optics

Raman spectroscopy remains the canonical probe because the Bardasis–Schrieffer mode is symmetry selective and typically appears below the pair-breaking threshold in the channel corresponding to the subleading pairing symmetry. In the multiband Raman theory, the weights of multiple BS modes are set by overlaps between the Raman vertex and the subleading gap harmonics, allowing several orthogonal subleading eigenfunctions within the same irreducible representation to produce several distinct Bardasis–Schrieffer resonances (Maiti et al., 2016). In centrosymmetric crystals, Raman activity requires

$2\Delta$6

which gives the group-theoretical selection rule for Raman-active collective modes, including BS excitations (Yamazaki et al., 5 Feb 2026).

For iron-based superconductors, the $2\Delta$7 Raman channel is especially important. In hole-doped BaFe$2\Delta$8As$2\Delta$9, a general multiband Raman treatment identifies multiple Bardasis–Schrieffer-type modes and relates their spectral weights to subleading gap structures (Maiti et al., 2016). In systems with nearby $s$0-wave and nematic fluctuations, a single $s$1 in-gap feature can instead be a mixed nematic–BS mode rather than a pure BS excitation, which makes assigning a Raman peak solely to subdominant pairing unreliable except very close to phase boundaries where mixing vanishes (Müller et al., 2020).

Ultrafast pump–probe and THz time-domain methods reveal the mode in non-equilibrium dynamics. In the pumped unconventional-superconductor model, a pump with polarization along the crystal axis induces a finite $s$2 component of the pseudomagnetic field and excites the subdominant channel, while diagonal polarization suppresses the BS excitation because $s$3 (Müller et al., 2019). In an $s$4-wave ground state, $s$5 oscillates at $s$6 whereas $s$7 exhibits Higgs oscillations at $s$8 (Müller et al., 2019). The quasiparticle distribution function $s$9 can carry especially strong BS signatures, suggesting visibility in time-resolved ARPES (Müller et al., 2019).

Third-harmonic generation provides another nonlinear route. In the THG analysis of unconventional superconductors, the Bardasis–Schrieffer mode remains unaffected by long-range Coulomb interaction, while the phase mode is pushed to the plasmon frequency (Müller et al., 2021). The BS mode contributes a sizable signal to third-harmonic currents and has a characteristic polarization dependence. Unlike the Higgs mode, it can generate a current perpendicular to the applied field, with angular structure

$d$0

and a resonance when the effective drive frequency satisfies $d$1 (Müller et al., 2021). This provides a clean way to distinguish BS contributions from charge-density-fluctuation backgrounds near $d$2.

Linear optical access is usually forbidden in centrosymmetric single-band systems, but several mechanisms can lift that restriction. A supercurrent can enable linear optical conductivity from BS modes in superconductors with competing pairing channels (Niederhoff et al., 9 Apr 2025). Rashba spin-orbit coupling can render BS modes optically active even without a supercurrent in interband $d$3-wave systems (Niederhoff et al., 9 Apr 2025). Trigonal warping in single-valley superconductors can also make BS and clapping modes visible in longitudinal and Hall optical response (Levitan et al., 12 Aug 2025). These results suggest that optical darkness is not intrinsic to the mode itself, but rather to a restrictive set of symmetry conditions.

6. Extensions beyond conventional superconductors and current directions

The Bardasis–Schrieffer concept extends beyond conventional superconductors. In excitonic insulators, the same basic mechanism produces subgap bound states in noncondensed internal channels. For an $d$4-wave excitonic insulator, a $d$5-wave Bardasis–Schrieffer mode exists below the gap energy and is optically active at $d$6, because a uniform electric field acts directly on the internal structure of the electron–hole pair (Sun et al., 2020). The mode satisfies a bound-state condition

$d$7

and can hybridize strongly with photons to form Bardasis–Schrieffer polaritons observable in far-field and near-field optics (Sun et al., 2020).

A first-principles extension has also been developed for excitonic insulators using a quasiparticle Bethe–Salpeter equation. In biased WSe$d$8–MoSe$d$9 bilayers, this approach reveals a gapless phase mode, subgap Bardasis–Schrieffer modes, and above-gap scattering states, with a particularly strong optically bright $s$00 BS mode dominating the subgap optical conductivity (Xuan et al., 24 Dec 2025). This suggests that the BS concept is not limited to separable weak-coupling models, but can be carried into ab initio descriptions of symmetry-broken electronic phases.

Recent superconducting experiments continue to sharpen the mode’s materials relevance. In FeSe, terahertz nonlinear spectroscopy detects a collective mode substantially below the superconducting gap energy, distinct from the amplitude Higgs mode, and theory attributes it to a fluctuation between an $s$01-wave-like ground state and a subleading pairing channel activated by electronic nematicity (Matsumoto et al., 19 Jul 2025). In kagome superconductors, scanning tunneling spectroscopy reveals a distinct collective mode below $s$02 whose interpretation remains between a Leggett mode and a Bardasis–Schrieffer mode due to a subleading superconducting component (Hu et al., 2024). These observations indicate that the BS framework is increasingly relevant in multicomponent, symmetry-lowered, and topologically nontrivial superconductors.

A recurring theme in current work is that the Bardasis–Schrieffer mode is best understood not as a single universal spectral line, but as a class of subdominant-channel collective excitations whose detailed manifestation depends on nodal structure, multiband composition, symmetry lowering, and coupling to additional bosonic sectors. This suggests that identifying a BS mode experimentally requires combined information from symmetry selection rules, location relative to $s$03, polarization dependence, and evolution with tuning parameters such as doping, pressure, nematicity, or supercurrent (Müller et al., 2020).