--- title: Bardasis–Schrieffer Mode in Superconductors url: https://www.emergentmind.com/topics/bardasis-schrieffer-mode type: topic --- # Bardasis–Schrieffer Mode in Superconductors 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 \(2\Delta\) [2011.01042]. 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 [2011.01042]. 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 [2109.11275]. ## 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 \(2\Delta\) [1907.10464]. 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 \(d\)-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 \(d_{x^2-y^2}\) sector when the equilibrium state is extended \(s\)-wave [1907.10464]. In the nematicity-coupled model, the same logic applies to \(s\)- and \(d\)-wave competition on a two-dimensional single-band square lattice, where the mode sits at \(\omega_{\mathrm{BS}}<2\Delta\) and is explicitly associated with the subdominant pairing symmetry [2011.01042]. 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 [1907.10464]. The same softening appears in the \(s\)- versus \(d\)-competition phase diagram of superconductors with competing \(s\)- and \(d\)-wave interactions, where the one-band \(s\)-wave BS mode collapses at the \(s+id\) boundary and evolves into a mixed-symmetry collective mode inside the \(s+id\) phase [1507.07501]. 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 \(1-V_d \Pi_d(\omega)=0\) for an \(s\)-wave ground state [2011.01042]. In the current-enabled optical formalism, the collective frequencies are poles of the effective interaction kernel, \[ \det V_\text{eff}^{-1} \overset{!}{=}0, \] with the single-band \(s\)-phase Bardasis–Schrieffer mode satisfying \[ \frac{1}{V_d}-L_d+\frac{(i\Omega)^2}{2}I_2=0 \] [2504.06642]. In Raman-based multiband formulations, the same physics appears as zeros of denominators such as \([2/\tilde U-\Pi_{22}(\Omega)]\) in the subleading pairing sector [1611.04541]. ## 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 [2107.02834]. In the time-domain studies of unconventional superconductors, the problem is recast in terms of Anderson pseudospins, \[ \mathbf{s}_{\mathbf{k}}=\frac{1}{2}(c^\dagger_{\mathbf{k}\uparrow},c_{-\mathbf{k}\downarrow})\,\boldsymbol{\sigma}\,\begin{pmatrix}c_{\mathbf{k}\uparrow}\\ c^\dagger_{-\mathbf{k}\downarrow}\end{pmatrix}, \] whose dynamics obey \[ \frac{d}{dt}\langle \mathbf{s}_{\mathbf{k}}\rangle=\mathbf{B}_{\mathbf{k}}(t)\times \langle \mathbf{s}_{\mathbf{k}}\rangle \] [1907.10464]. In the nematicity-coupled model, the mean-field Hamiltonian is \[ H = \sum_{k,\sigma} \big(\xi_k + \Delta_n \gamma_k^n\big)\, c_{k\sigma}^\dagger c_{k\sigma} + \sum_k \Big[\Delta_k\, c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger + \text{h.c.}\Big], \] with \[ \Delta_k = \Delta_s \gamma_k^s + e^{i\alpha}\, \Delta_d \gamma_k^d, \] and the choice \(\gamma_k^s=1\), \(\gamma_k^d=\gamma_k^n=\sqrt{2}\cos(2\phi)\) [2011.01042]. Linearizing the pseudospin equations around equilibrium gives \[ \delta \boldsymbol{\Delta}(\omega) = \big(\mathbf{1}-\boldsymbol{\chi}(\omega)\big)^{-1} \big(2\alpha\,\mathcal{F}[A^2](\omega)\big) \sum_k V G_k^T M_k \hat{e}_z, \] and collective modes satisfy \[ \det[\mathbf{1}-\boldsymbol{\chi}(\omega)]=0 \] [2011.01042]. In the Raman-response framework for multicomponent superconductors, one starts from a general BdG Hamiltonian with separable pairing interaction, \[ H_{\mathrm{BdG}}(\mathbf{k})= \begin{bmatrix} H_0(\mathbf{k}) & \Delta(\mathbf{k})\\ \Delta^\dagger(\mathbf{k}) & -H_0^{\mathrm{T}}(-\mathbf{k}) \end{bmatrix}, \quad \Delta(\mathbf{k})=\sum_{\alpha=1}^{N_{\Delta}}\Delta^{\mathrm{MF}}_\alpha \hat{\Delta}_\alpha(\mathbf{k}), \] and derives a gauge-invariant Raman susceptibility in which collective modes appear as poles of an effective fluctuation kernel \(U_{\mathrm{eff},\phi}\) [2602.05607]. 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 [2602.05607]. ## 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 \(\omega_{\mathrm{BS}}<\omega_H\), where \(\omega_H=2\Delta_{l_0,\max}\) is the Higgs energy for the dominant gap [1907.10464]. In rotationally symmetric \(s\)-wave states, the mode is undamped because it sits below the continuum. In nodal \(d\)-wave states, however, the analogous subdominant-channel oscillation is strongly damped by nodal quasiparticles [1907.10464]. The frequency tracks the competition between pairing channels. In the one-band \(s\)-wave phase of the \(s\)- versus \(d\)-competition model, the phase-sector equation for the BS mode is \[ -\left(\frac{\Omega}{2\Delta_0^s}\right)^2 I_2(\Omega)=\frac{2}{u_d}-\frac{2}{u_s}, \] and the solution softens as \(u_d\to u_s\), marking the approach to the \(s+id\) phase boundary [1507.07501]. In the mixed \(s+id\) regime the mode no longer remains a pure \(d\)-channel phase oscillation, but becomes a “mixed-symmetry Bardasis–Schrieffer mode” determined by a coupled equation involving \(I_0\), \(I_2\), and \(I_4\) [1507.07501]. The multiband case adds further structure. In the three-pocket model for hole-doped Fe-based superconductors, an \(s\)-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 [1507.07501]. These coexist with a damped Leggett mode between the two gap scales [1507.07501]. 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 [2602.05607]. 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 \(s+d\)-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 [2507.14466]. 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 \(s\)-wave ground state the \(d\)-wave Bardasis–Schrieffer fluctuation hybridizes with the nematic collective mode, producing a single in-gap excitation of mixed nematic–BS character [2011.01042]. The relevant \(2\times2\) kernel couples the imaginary \(d\)-wave component \(\Delta_d''\) and the nematic amplitude \(\Delta_n\), \[ \chi_{\mathrm{N\text{-}BS}}(\omega)= \begin{pmatrix} -2V_d(I_1+I_2) & iV_d\,\omega\,I_2/\Delta \\ -2iV_n\,\omega\,I_2/\Delta & -4V_n I_2 \end{pmatrix}, \] and only one in-gap solution exists for \(0