---
title: Bardasis-Schrieffer Modes in Superconductors
url: https://www.emergentmind.com/topics/bardasis-schrieffer-modes
type: topic
---

# Bardasis-Schrieffer Modes in Superconductors

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 [1807.06601]. 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 [2001.03704].

## 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
\[
1 = g_s\,\Pi_s(0), \qquad
\Pi_s(0)=\sum_k\frac{1}{2E_k}\tanh\!\frac{\beta E_k}{2},
\quad
E_k=\sqrt{\xi_k^2+\Delta_s^2},
\]
while fluctuations in the \(d\)-wave Cooper channel form a bound state when
\[
1 - g_d\,\Pi_d(\Omega_{\rm BS}) = 0,
\qquad
\Pi_d(\Omega) =\sum_k f_d^2(\phi_k)\,
\frac{2E_k}{(2E_k)^2-\Omega^2}\,
\tanh\!\frac{\beta E_k}{2}.
\]
The solution \(\Omega_{\rm BS}<2\Delta_s\) is the Bardasis–Schrieffer mode [1807.06601].

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 \(s\)-wave gap \(\Delta_0\) gives a Gaussian fluctuation action containing the \(U(1)\) phase \(\theta\), the amplitude or Higgs mode \(\delta\Delta\), and subdominant-pairing Bardasis–Schrieffer fields \(\delta\Delta_l\),
\[
S_{\rm fl}=S_\theta+S_H+S_{BS}+S_c,
\]
with
\[
S_{BS}=\frac12\sum_q G_{BS,l}^{-1}(q)\,\delta\Delta_{l,-q}\,\delta\Delta_{l,q},
\]
and mixed vertices \(B_l^\mu\) coupling the BS field to \(\partial_\mu\theta+eA_\mu\). In this formulation the Bardasis–Schrieffer mode is a pole of \(G_{BS,l}(q)\), and its electromagnetic visibility is controlled by the loop-generated coupling structure \(B_l^\mu\) rather than by the existence of the pole alone [2001.03704].

The same logic extends beyond a single-band \(s\)-\(d\) 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
\[
\det\bigl[g_o^{-1}\,\mathbf{1}+\boldsymbol{\Pi}^{(o)}(\omega+i0^+)\bigr]=0,
\]
and the dominant root is controlled by the “phase” channel kernel \(\Pi^{(o)}_{22}(\omega)\) [2212.13722]. In multicomponent Raman formalisms, the same BS pole appears through \(\det[U_{\rm eff}^{-1}(\omega)]=0\), with the subdominant channel satisfying \(1-V_d\,\Pi_{dd}(\omega)=0\) [2602.05607].

## 2. Pole structure, dispersion, and softening

For the \(d_{x^2-y^2}\) Bardasis–Schrieffer mode in an \(s\)-wave superconductor, the inverse propagator at \(T=0\) in 2D and small \(q\) is
\[
G_{BS}^{-1}(\omega,q)
= \frac1{g_d} + \chi_{f_d\sigma_2,f_d\sigma_2}(\omega,q)
= \Bigl(\frac1{g_d}-\frac12g^{-1}\Bigr)
-\frac12\omega^2 F(\omega)
+\frac1{16}\frac{\nu}{\Delta^2}v_F^2q^2.
\]
The zero-momentum pole is determined by \(G_{BS}^{-1}(\omega,0)=0\). In the weak-fluctuation limit \(g_d\ll2g\),
\[
\omega_{BS}(0) \simeq 2\Delta
\Bigl[1-\frac{\pi^2}{32}(\nu g_d)^2\Bigr],
\]
and for finite momentum
\[
\omega_{BS}(q)^2 \simeq \omega_{BS}(0)^2 + \frac12 v_F^2q^2,
\qquad
(\omega_{BS}\ll2\Delta,\; q\ll\xi^{-1},\; \xi=v_F/\Delta).
\]
This is the standard subgap excitonic mode with a quadratic long-wavelength dispersion [2001.03704].

A second generic feature is softening near a degeneracy of pairing channels. In the one-band \(s\)-\(d\) problem the pure-\(s\) BS condition can be written as
\[
1-U_d\,\Pi_d(\Omega)=0,
\]
and as \(\lambda_d\to\lambda_s\) the Bardasis–Schrieffer mode softens, \(\Omega_{BS}\to0\), signaling the \(s+id\) instability. In the \(s+id\) phase itself, the pure-\(s\) BS excitation evolves into a mixed-symmetry mode, continuously connecting the pure-\(s\) BS mode at the \(s\to s+id\) boundary to the pure-\(d\) zero mode at the \(s+id\to d\) boundary [1507.07501].

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
\[
\frac{2y\arcsin y}{\sqrt{1-y^2}}
=
\ln\!\Bigl(\tfrac{T_{c,e}}{T_{c,o}}\Bigr),
\qquad y=\frac{\omega}{2\Delta_e},
\]
so the mode lies below the pair-breaking continuum \(2\Delta_e\) and collapses to zero as \(T_{c,o}\to T_{c,e}\) [2212.13722].

In systems with strong spin-orbit structure, the dispersion can be less conventional. In the \(j=3/2\) Luttinger–Kohn model, fluctuations from a fully gapped singlet state into the quintet (\(J=2\)) channel obey
\[
D_{d,a}^{-1}(i\omega,\mathbf q) \equiv g_d^{-1}+\chi_{d,a}(i\omega,\mathbf q),
\]
and the BS mode is a pole of \(D_{d,a}\). The small-\(q\) expansion
\[
\Pi_{J=2}(\omega,\mathbf q)\approx \Pi_0 + A q^2 - B\omega^2+\cdots
\]
gives
\[
\omega_{\rm BS}^2(\mathbf q)=\Delta_{\rm BS}^2+c^2q^2,
\]
but \(A\) can change sign for intermediate SOC, leading to a local minimum of \(\omega_{\rm BS}(q)\) at a finite momentum \(q_0\). In that regime the singlet state is unstable to the formation of a finite-\(\mathbf q\) quintet condensate, which the paper describes as “hinting at Fulde-Ferrell-Larkin-Ovchinnikov physics” [2405.06111].

## 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 \(d\)-wave phase fluctuation and the vector potential \(A\), so the BS mode does not absorb at linear order” [1807.06601]. 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 \(s\)-wave ground state and a close \(d\)-wave competitor, the non-resonant \(B_{2g}\) Raman operator couples to the \(d\)-wave density, and the Raman susceptibility contains a Cooper-channel pole at
\[
1+\frac{u_d}{2}\Pi_{22}(\Omega_{\rm BS})=0.
\]
In the same symmetry channel there can also be a particle-hole exciton from
\[
1+\frac{u_\rho}{2}\Pi_{33}(\Omega_{\rm ex})=0.
\]
When \(\Pi_{23}=0\) they appear independently; finite \(\Pi_{23}\) produces level repulsion, and at finite impurity damping the two modes merge into one broad peak [1405.6246]. In more general multiband Raman theory, each attractive eigenchannel \(\ell\) in the subleading irreducible representation produces its own pole, and the spectral weight is set by the overlap \(\tilde c_\ell=\langle\gamma\,\psi_\ell\rangle_{FS}\) between the Raman vertex and the subleading gap function [1611.04541].

At finite momentum, THz near-field probes activate BS modes through the mixed phase–BS vertex. Because the BS fluctuation \(\delta\Delta_l\) carries angular momentum \(\ell=2\), it cannot couple to a uniform \(q=0\) electric field. At finite in-plane momentum, however, the linear phase–BS vertex is nonzero even in particle-hole symmetric models:
\[
B^\mu(q) = (B_0\,\omega q^2,\; B_i\,(q_x,-q_y)),
\qquad
B_i(\omega)= i\,\pi\,\Delta\,v_F^2\,F(\omega).
\]
After integrating out \(\theta\) and \(\delta\Delta_l\), the longitudinal conductivity becomes
\[
\sigma(\omega,q)=i(D_s/\pi)/\omega
\times
\left\{
1-\frac{v_g^2q^2}{\omega^2}
\left[
1+\frac{\kappa_{BS}^2v_g^2q^2}{\omega^2-\omega_{BS}^2}
\right]^{-1}
\right\}^{-1},
\]
and the \(p\)-polarized near-field reflection coefficient
\[
R_p(\omega,q)=1-\frac1{\epsilon_{2D}(\omega,q)},
\qquad
\epsilon_{2D}=1+\frac{i2\pi q}{\omega}\sigma(\omega,q),
\]
shows the BS mode as a resonance-dip or anti-crossing in \(|R_p|\) at \(\omega\approx\omega_{BS}(q)\) [2001.03704].

Locally non-centrosymmetric bilayers provide a different activation route. There, the current operator has nonzero interband matrix elements, and mixed diagrams \(L_{z,2}\) and \(R_{2z}\) remain nonzero down to low frequency. The renormalized current response
\[
\tilde K_{zz}(\omega)=K_{zz}^{(0)} - L_{z,2}\,[g_o^{-1}+\Pi_{22}^{(o)}(\omega)]^{-1}R_{2,z}
\]
therefore picks up a pole at \(\omega_{BS}\). The resulting linear conductivity
\[
\sigma_{zz}(\omega)=\frac{i}{\omega}\tilde K_{zz}(\omega)
\]
contains a delta-function resonance at \(\omega=\omega_{BS}\), and the selection rule is explicit: only \(E\parallel \hat z\) couples to the odd-parity BS mode; in-plane fields give no linear peak [2212.13722].

A further route is inversion breaking by a supercurrent or by Rashba spin–orbit coupling. With a finite supercurrent \(q\), inversion is broken, \(Q_a\neq0\), and the collective-mode term in the optical conductivity acquires poles at \(\Omega_{BS}\). In Rashba systems, Bardasis–Schrieffer modes of opposite parity to the ground state can be linearly active even at zero supercurrent [2504.06642].

| Probe | Activation mechanism | Hallmark |
|---|---|---|
| Raman | Symmetry projection onto subleading channel | In-gap pole below \(2\Delta\) |
| THz near field | Finite-\(q\) BS–phase vertex | Resonance-dip or anti-crossing in \(|R_p|\) |
| Microwave \(zz\) conductivity | Interband current matrix elements in a bilayer | Delta-function resonance at \(\omega_{BS}\) |
| Linear optical conductivity | Supercurrent or Rashba SOC breaks inversion | Subgap conductivity peak |
| Cavity photons | Applied supercurrent generates \(d\)-wave–photon coupling | Hybrid BS-polariton branches |

The cavity case makes the activation mechanism especially explicit. A uniform supercurrent \(\mathbf v_S\) generates a linear coupling
\[
H_{\rm int}=g_\mathbf q\,(a_\mathbf q\,b^\dagger_\mathbf q+a^\dagger_\mathbf q\,b_\mathbf q),
\]
between a photon mode \(a_\mathbf q\) and a BS boson \(b_\mathbf q\). The corresponding effective Hamiltonian
\[
H^{\rm eff}_\mathbf q=
\begin{pmatrix}
\Omega_{\rm BS} & g_\mathbf q\\
g_\mathbf q & \omega_\mathbf q+\Pi^S_\mathbf q
\end{pmatrix}
\]
yields two polariton branches \(\Omega_\pm(\mathbf q)\), and condensation of the lower branch at \(\mathbf q=0\) would imply a finite \(d\)-wave component with phase \(\pi/2\) relative to the \(s\)-wave gap, i.e. an \(s\pm i d\) state [1807.06601].

## 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 \(A^2\) even when linear dipole coupling is weak. In third-harmonic generation, the full kernel decomposes into quasiparticle, density, Higgs, phase, and BS contributions,
\[
K=K_0+K_\rho+K_{\Delta_s}+K_\theta+K_{\Delta'_d}+K_{\Delta''_d},
\]
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)

Source: https://www.emergentmind.com/topics/bardasis-schrieffer-modes