Bardasis-Schrieffer Modes in Superconductors

• Bardasis–Schrieffer modes are collective excitations that emerge from fluctuations in a subdominant pairing channel, manifesting as sharp excitonic bound states below the 2Δ threshold.
• They are microscopically described using Hubbard–Stratonovich decoupling and pole identification in effective propagators, revealing the interplay between competing pairing channels.
• Activation via Raman, THz near-field, and other optical probes underscores their significance in diagnosing pairing symmetry and nonlinear superconducting dynamics.

Bardasis–Schrieffer modes are collective excitations associated with fluctuations of the order parameter in a subdominant pairing channel. In an $s$-wave superconductor with a weaker residual attraction in the $d$-wave channel, they appear as sharp particle-particle excitonic bound states below the pair-breaking continuum $2\Delta$, and more generally they arise whenever closely competing pairing channels coexist in the fluctuation spectrum (Allocca et al., 2018). Their microscopic description is usually formulated through a Hubbard–Stratonovich decoupling of multiple pairing channels and the identification of poles in an effective fluctuation propagator or Cooper-channel $T$-matrix, while their observability depends strongly on symmetry, momentum transfer, Coulomb screening, multiband structure, and inversion breaking (Sun et al., 2020).

1. Microscopic origin and field-theoretic formulation

A standard starting point is a BCS-type interaction with a dominant and a subdominant pairing channel. In the simplest $s$-$d$ model,

$$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$

with $f_s=1$ and $f_d(\phi)=\sqrt2\cos2\phi$. The $s$-wave saddle point satisfies

$d$0

while fluctuations in the $d$1-wave Cooper channel form a bound state when

$d$2

The solution $d$3 is the Bardasis–Schrieffer mode (Allocca et al., 2018).

In the path-integral formulation, one introduces Hubbard–Stratonovich fields for the dominant and subdominant channels and integrates out the fermions. Expanding about the saddle-point $d$4-wave gap $d$5 gives a Gaussian fluctuation action containing the $d$6 phase $d$7, the amplitude or Higgs mode $d$8, and subdominant-pairing Bardasis–Schrieffer fields $d$9,

$2\Delta$0

with

$2\Delta$1

and mixed vertices $2\Delta$2 coupling the BS field to $2\Delta$3. In this formulation the Bardasis–Schrieffer mode is a pole of $2\Delta$4, and its electromagnetic visibility is controlled by the loop-generated coupling structure $2\Delta$5 rather than by the existence of the pole alone (Sun et al., 2020).

The same logic extends beyond a single-band $2\Delta$6-$2\Delta$7 problem. In a globally centrosymmetric bilayer with locally non-centrosymmetric layers, a subleading odd-parity channel produces a particle-particle excitonic collective mode below the pair-breaking edge, with collective-mode condition

$2\Delta$8

and the dominant root is controlled by the “phase” channel kernel $2\Delta$9 (Lee et al., 2022). In multicomponent Raman formalisms, the same BS pole appears through $T$0, with the subdominant channel satisfying $T$1 (Yamazaki et al., 5 Feb 2026).

2. Pole structure, dispersion, and softening

For the $T$2 Bardasis–Schrieffer mode in an $T$3-wave superconductor, the inverse propagator at $T$4 in 2D and small $T$5 is

$T$6

The zero-momentum pole is determined by $T$7. In the weak-fluctuation limit $T$8,

$T$9

and for finite momentum

$s$0

This is the standard subgap excitonic mode with a quadratic long-wavelength dispersion (Sun et al., 2020).

A second generic feature is softening near a degeneracy of pairing channels. In the one-band $s$1-$s$2 problem the pure-$s$3 BS condition can be written as

$s$4

and as $s$5 the Bardasis–Schrieffer mode softens, $s$6, signaling the $s$7 instability. In the $s$8 phase itself, the pure-$s$9 BS excitation evolves into a mixed-symmetry mode, continuously connecting the pure-$d$0 BS mode at the $d$1 boundary to the pure-$d$2 zero mode at the $d$3 boundary (Maiti et al., 2015).

An analogous collapse occurs in parity-switching bilayers. For an even-parity ground state with a subleading odd-parity channel, the BS-mode energy at zero momentum obeys

$d$4

so the mode lies below the pair-breaking continuum $d$5 and collapses to zero as $d$6 (Lee et al., 2022).

In systems with strong spin-orbit structure, the dispersion can be less conventional. In the $d$7 Luttinger–Kohn model, fluctuations from a fully gapped singlet state into the quintet ($d$8) channel obey

$d$9

and the BS mode is a pole of $$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$0. The small-$$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$1 expansion

$$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$2

gives

$$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$3

but $$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$4 can change sign for intermediate SOC, leading to a local minimum of $$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$5 at a finite momentum $$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$6. In that regime the singlet state is unstable to the formation of a finite-$$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$7 quintet condensate, which the paper describes as “hinting at Fulde-Ferrell-Larkin-Ovchinnikov physics” (Li et al., 2024).

3. Linear spectroscopy, Raman selection rules, and optical activation

In a clean equilibrium single-band superconductor, the Bardasis–Schrieffer mode is often “dark.” In the cavity formulation, “in the clean equilibrium superconductor there is no bilinear term mixing the $$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$8-wave phase fluctuation and the vector potential $$H_{\rm int} = -\sum_{\ell=s,d} g_\ell \sum_{k,k'} f_\ell(\phi_k)\,f_\ell(\phi_{k'})\, c^\dagger_{k\uparrow} c^\dagger_{-k\downarrow}\, c_{-k'\downarrow}c_{k'\uparrow}, \qquad g_s>g_d>0,$$9, so the BS mode does not absorb at linear order” (Allocca et al., 2018). This statement, however, is not universal: several distinct mechanisms render BS modes linearly visible.

In Raman scattering, the relevant mechanism is symmetry projection rather than dipole activity. In alkali-intercalated iron selenides with an $f_s=1$0-wave ground state and a close $f_s=1$1-wave competitor, the non-resonant $f_s=1$2 Raman operator couples to the $f_s=1$3-wave density, and the Raman susceptibility contains a Cooper-channel pole at

$f_s=1$4

In the same symmetry channel there can also be a particle-hole exciton from

$f_s=1$5

When $f_s=1$6 they appear independently; finite $f_s=1$7 produces level repulsion, and at finite impurity damping the two modes merge into one broad peak (Khodas et al., 2014). In more general multiband Raman theory, each attractive eigenchannel $f_s=1$8 in the subleading irreducible representation produces its own pole, and the spectral weight is set by the overlap $f_s=1$9 between the Raman vertex and the subleading gap function (Maiti et al., 2016).

At finite momentum, THz near-field probes activate BS modes through the mixed phase–BS vertex. Because the BS fluctuation $f_d(\phi)=\sqrt2\cos2\phi$0 carries angular momentum $f_d(\phi)=\sqrt2\cos2\phi$1, it cannot couple to a uniform $f_d(\phi)=\sqrt2\cos2\phi$2 electric field. At finite in-plane momentum, however, the linear phase–BS vertex is nonzero even in particle-hole symmetric models: $f_d(\phi)=\sqrt2\cos2\phi$3 After integrating out $f_d(\phi)=\sqrt2\cos2\phi$4 and $f_d(\phi)=\sqrt2\cos2\phi$5, the longitudinal conductivity becomes

$f_d(\phi)=\sqrt2\cos2\phi$6

and the $f_d(\phi)=\sqrt2\cos2\phi$7-polarized near-field reflection coefficient

$f_d(\phi)=\sqrt2\cos2\phi$8

shows the BS mode as a resonance-dip or anti-crossing in $f_d(\phi)=\sqrt2\cos2\phi$9 at $s$0 (Sun et al., 2020).

Locally non-centrosymmetric bilayers provide a different activation route. There, the current operator has nonzero interband matrix elements, and mixed diagrams $s$1 and $s$2 remain nonzero down to low frequency. The renormalized current response

$s$3

therefore picks up a pole at $s$4. The resulting linear conductivity

$s$5

contains a delta-function resonance at $s$6, and the selection rule is explicit: only $s$7 couples to the odd-parity BS mode; in-plane fields give no linear peak (Lee et al., 2022).

A further route is inversion breaking by a supercurrent or by Rashba spin–orbit coupling. With a finite supercurrent $s$8, inversion is broken, $s$9, and the collective-mode term in the optical conductivity acquires poles at $d$00. In Rashba systems, Bardasis–Schrieffer modes of opposite parity to the ground state can be linearly active even at zero supercurrent (Niederhoff et al., 9 Apr 2025).

Probe	Activation mechanism	Hallmark
Raman	Symmetry projection onto subleading channel	In-gap pole below $d$01
THz near field	Finite-$d$02 BS–phase vertex	Resonance-dip or anti-crossing in $d$03
Microwave $d$04 conductivity	Interband current matrix elements in a bilayer	Delta-function resonance at $d$05
Linear optical conductivity	Supercurrent or Rashba SOC breaks inversion	Subgap conductivity peak
Cavity photons	Applied supercurrent generates $d$06-wave–photon coupling	Hybrid BS-polariton branches

The cavity case makes the activation mechanism especially explicit. A uniform supercurrent $d$07 generates a linear coupling

$d$08

between a photon mode $d$09 and a BS boson $d$10. The corresponding effective Hamiltonian

$d$11

yields two polariton branches $d$12, and condensation of the lower branch at $d$13 would imply a finite $d$14-wave component with phase $d$15 relative to the $d$16-wave gap, i.e. an $d$17 state (Allocca et al., 2018).

4. Nonlinear response, pump–probe dynamics, and nematic mixing

The nonlinear optical response provides a complementary window because the Bardasis–Schrieffer mode can couple strongly to $d$18 even when linear dipole coupling is weak. In third-harmonic generation, the full kernel decomposes into quasiparticle, density, Higgs, phase, and BS contributions,

$d$19

and the BS part is [

K_{ij}^{BS}(2\Omega)

-\,\frac{16\Delta^4 \bigl[\sum_k d_k(\partial_{k_i}^2\xi_k)