Intraband Leggett Mode in Superconductors

• Intraband Leggett mode is a collective oscillation between superconducting condensate components within a single band or orbital sector, characterized by a finite gap and neutral response.
• It arises from internal Josephson-like coupling in multiband superconductors, with its observability determined by symmetry selection rules and its position below the two-particle continuum.
• Realizations in systems like MgB2, iron pnictides, and non-centrosymmetric superconductors allow probing via Raman and THz spectroscopy, offering insights into pairing dynamics and phase fluctuations.

Searching arXiv for relevant papers on intraband Leggett modes and related multiband superconductivity. The intraband Leggett mode is a massive collective excitation corresponding to an out-of-phase oscillation between superconducting condensate components that reside within the same band, orbital manifold, or symmetry-decomposed pairing sector. In contrast to the more familiar interband Leggett mode of multiband superconductors, the intraband variant arises when more than one superconducting component is supported on a single Fermi surface, orbital sector, or helicity-split branch, so that the relevant dynamical variable is a relative phase internal to that subsystem. Across microscopic BCS treatments, strong-coupling orbital models, gauge-invariant Raman theories, and effective time-dependent Ginzburg–Landau descriptions, the mode is characterized by a finite gap at zero momentum generated by a Josephson-like coupling between components, by a neutral relative-phase character, and by observability conditions controlled by its position relative to the two-particle continuum (Cea et al., 2016, Burnell et al., 2010, Grigorishin, 2021, Matsumoto et al., 19 Jul 2025).

1. Definition and conceptual scope

The defining feature of any Leggett mode is a relative-phase oscillation between two superconducting components. In the standard two-band case this means oscillations between order parameters on distinct bands; the corresponding frequency is set by the inter-condensate coupling and the condensate inertias. The intraband Leggett mode generalizes this notion to situations in which the two oscillating components are not separate bands in the ordinary sense, but components within a single band-sector description, such as orbital condensates, mixed symmetry channels on one pocket, or helicity-resolved gaps in non-centrosymmetric systems (Burnell et al., 2010, Bittner et al., 2015, Matsumoto et al., 19 Jul 2025).

Several realizations appear in the literature. In the two-orbital strong-coupling model for iron arsenides, the relevant components are the $d_{xz}$ and $d_{yz}$ orbital condensates, and the mode is therefore explicitly described as an “orbital” Leggett mode (Burnell et al., 2010). In non-centrosymmetric superconductors, Rashba spin–orbit coupling splits the system into helicity branches $\mu=\pm$, each carrying a mixed singlet–triplet gap $\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$, and the Leggett excitation is the out-of-phase oscillation of these intraband condensates (Bittner et al., 2015). In FeSe in the nematic phase, the hole pocket supports two mixed pairing channels, so that the relative-phase oscillation between the $s$-like and $d$-like components is described as an intraband Leggett mode and, simultaneously, as a Bardasis–Schrieffer-like excitation (Matsumoto et al., 19 Jul 2025).

This usage distinguishes the intraband Leggett mode from a pure pair-breaking feature and from the global Goldstone mode. The Goldstone mode is the common phase oscillation and, in a charged system, is shifted to the plasma frequency by the Anderson–Higgs mechanism; the Leggett mode is neutral and survives as a finite-frequency excitation (Bittner et al., 2015, Grigorishin, 2021). In Raman and THz probes, this distinction is central because a subgap resonance can either signal a genuine relative-phase collective mode or simply reflect continuum physics. The gauge-invariant analysis of the $A_{1g}$ Raman response demonstrates that identifying the collective mode requires simultaneous treatment of density and phase fluctuations (Cea et al., 2016).

2. Microscopic formulation of relative-phase dynamics

A standard starting point is a two-component superconducting Hamiltonian with attractive intracomponent couplings and a coupling that transfers Cooper pairs between components. In the two-band model of one hole and one electron band, the Hamiltonian is

$$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$

with intraband couplings $U_{11},U_{22}$ and interband coupling $U_{12}=U_{21}$ (Cea et al., 2016). At mean field the gaps satisfy coupled equations,

$d_{yz}$0

where

$d_{yz}$1

The relative-phase mode is obtained by writing phase fluctuations $d_{yz}$2 of the two order parameters and forming the antisymmetric combination $d_{yz}$3 (Cea et al., 2016).

The same structure appears in different languages across models. In the strong-coupling two-orbital pnictide model, one writes

$d_{yz}$4

and introduces the symmetric and antisymmetric combinations

$d_{yz}$5

with $d_{yz}$6 identified as the Leggett variable (Burnell et al., 2010). In the ETDGL framework, the phase-only sector follows from the Lorentz-invariant Lagrangian density for two complex order parameters $d_{yz}$7, coupled by the inter-band proximity coupling $d_{yz}$8 and drag coefficient $d_{yz}$9. Linearization yields coupled phase equations in $\mu=\pm$0, whose eigenvectors are the common and anti-phase branches (Grigorishin, 2021).

A broader criterion emerges from the Hubbard–Stratonovich analysis of multiband systems. Diagonalizing the pairing matrix into bonding and antibonding channels shows that a Leggett mode requires a second bonding channel with positive stiffness. In a two-band system with dominant interband pairing and $\mu=\pm$1, one eigenvalue is negative, the second channel is antibonding, and no underdamped Leggett resonance exists below the continuum (Marciani et al., 2013). This result is particularly important for iron-based superconductors, where the distinction between intraband-dominated and interband-dominated pairing qualitatively changes the collective-mode spectrum (Cea et al., 2016, Marciani et al., 2013).

3. Leggett gap, mass, and dispersion

In the intraband-dominated two-band limit, the Leggett frequency is determined by the phase-sector determinant. Defining

$\mu=\pm$2

the collective mode is obtained from the vanishing of the relevant determinant, and for small $\mu=\pm$3 one may approximate $\mu=\pm$4, giving the classic result

$\mu=\pm$5

which lies between $\mu=\pm$6 and $\mu=\pm$7 (Cea et al., 2016). An equivalent form, written in terms of $\mu=\pm$8, is

$\mu=\pm$9

again emphasizing that positive relative-phase stiffness requires $\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$0 (Marciani et al., 2013).

In the two-orbital strong-coupling model, the long-wavelength Leggett gap is written as

$\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$1

with $\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$2 arising from the orbital hybridization term $\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$3 and $\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$4 the antisymmetric phase kernel (Burnell et al., 2010). In the simplified two-band language used there,

$\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$5

showing explicitly how the mode is controlled by both the gaps and the coupling matrix (Burnell et al., 2010).

Within ETDGL, the anti-phase branch has dispersion

$\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$6

with

$\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$7

and $\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$8 (Grigorishin, 2021). In the symmetric limit $\Delta_\mu=\Delta_s+\mu \Delta_t f_{\mathbf k}$9, $s$0, and $s$1,

$s$2

Here the finite gap reflects the restoring force due to the interband coupling $s$3, and the mode remains massive as $s$4 (Grigorishin, 2021).

In non-centrosymmetric superconductors, the corresponding mass is

$s$5

or, in the microscopic two-band formalism,

$s$6

with a small-$s$7 dispersion

$s$8

(Bittner et al., 2015). In FeSe, in the weak-coupling, small-nematic-splitting limit, the corresponding intraband Leggett frequency between the $s$9-like and $d$0-like hole-pocket components is

$d$1

which can be recast, using the gap equations, as

$d$2

or

$d$3

These expressions identify nematicity-induced mixing $d$4 or $d$5 as the phase-locking scale (Matsumoto et al., 19 Jul 2025).

4. Damping, continuum thresholds, and conditions for observability

A Leggett mode is observable as a sharp resonance only when it lies below the two-particle continuum. In the gauge-invariant Raman treatment, if $d$6 and $d$7, the mode is undamped and yields a sharp pole in the response (Cea et al., 2016). More specifically, because $d$8, a pole at $d$9 is a sharp undamped resonance (Cea et al., 2016).

For MgB$A_{1g}$0, the observability conditions are stated explicitly. A well-defined intraband Leggett mode requires: $A_{1g}$1 so that pairing is intraband dominated, distinct gaps $A_{1g}$2 so that $A_{1g}$3 falls in the clean subgap window $A_{1g}$4, low quasiparticle damping, and a nonzero relative Raman vertex $A_{1g}$5 (Cea et al., 2016). This set of conditions has become the canonical criterion for a sharp Raman-active Leggett resonance.

The strong-coupling orbital model for pnictides reaches the same conclusion in a different parameterization. For the extended $A_{1g}$6 gap $A_{1g}$7, the criterion $A_{1g}$8 is satisfied only in a narrow window of electron doping $A_{1g}$9–$$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$0 and for moderate gap sizes $$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$1–$$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$2 eV; outside that window, and for nodal gaps such as $$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$3 or $$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$4, the mode lies above $$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$5 and is overdamped by quasiparticles (Burnell et al., 2010). The orbital character therefore does not, by itself, guarantee visibility; the subgap placement remains decisive.

Cold-atom realizations exhibit the same threshold structure. In the dipolar two-component Fermi superfluid, the mode is undamped only if

$$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$6

and the Leggett peak in the Cooper-pair spectral function broadens and disappears once the mode enters the continuum (Mulkerin et al., 2019). In the finite-temperature Gaussian-pair-fluctuation treatment of two-band Fermi gases, the lower pair-breaking threshold is

$$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$7

and above $$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$8 the analytically continued determinant acquires an imaginary part, leading to a damping rate

$$H = \sum_{i=1,2}\sum_{k\sigma} \xi_{i,k} c^\dagger_{i,k,\sigma} c_{i,k,\sigma} -\sum_{i,j=1,2} U_{ij}\sum_{k,k'} c^\dagger_{i,k,\uparrow} c^\dagger_{i,-k,\downarrow} c_{j,-k',\downarrow} c_{j,k',\uparrow} +\text{Coulomb terms},$$9

This provides a nonperturbative description of damping when the collective excitation reaches the pair-breaking edge (Klimin et al., 2019).

A further qualification arises in ETDGL. The anti-phase mode exists only for $U_{11},U_{22}$0 and $U_{11},U_{22}$1; for the special choice

$U_{11},U_{22}$2

the coefficient of the relative-phase eigenvalue vanishes and the Leggett mode is removed altogether (Grigorishin, 2021). This is not a damping effect but a structural elimination of the anti-phase branch.

5. Spectroscopic manifestations

The gauge-invariant $U_{11},U_{22}$3 Raman response provides a direct route to the Leggett resonance in two-band superconductors. Including both density and phase fluctuations, the full susceptibility is

$U_{11},U_{22}$4

and for parabolic bands with opposite effective masses one has $U_{11},U_{22}$5, so the density-only screening cancels while the $U_{11},U_{22}$6-term survives (Cea et al., 2016). The denominator is identical to the phase–phase determinant, hence the Raman response has a pole at $U_{11},U_{22}$7. Under further simplifications $U_{11},U_{22}$8 and $U_{11},U_{22}$9,

$U_{12}=U_{21}$0

so that $U_{12}=U_{21}$1, the sharp Leggett peak (Cea et al., 2016).

In orbital pnictide models, the selection rules differ. Because the relative phase corresponds to oscillations of $U_{12}=U_{21}$2, the operator transforms as $U_{12}=U_{21}$3 under $U_{12}=U_{21}$4, and the orbital Leggett mode is expected most strongly in the $U_{12}=U_{21}$5 Raman channel rather than $U_{12}=U_{21}$6 (Burnell et al., 2010). This contrasts explicitly with MgB$U_{12}=U_{21}$7, where the Leggett mode is $U_{12}=U_{21}$8-active (Burnell et al., 2010, Cea et al., 2016).

THz nonlinear spectroscopy provides another route, particularly when the mode is subgap and weak in linear optical response. The Raman analysis of multiband superconductors already notes implications for the non-linear optical response probed by intense THz fields (Cea et al., 2016). In FeSe, terahertz third-harmonic generation reveals a resonance when the drive satisfies $U_{12}=U_{21}$9: the third-harmonic intensity $d_{yz}$00 peaks and the relative phase $d_{yz}$01 jumps by approximately $d_{yz}$02 (Matsumoto et al., 19 Jul 2025). For $d_{yz}$03 THz and $d_{yz}$04 THz, clear peaks occur below $d_{yz}$05 K, with $d_{yz}$06 THz ($d_{yz}$07 meV), well below the smallest gap $d_{yz}$08 meV, while no resonance is seen for $d_{yz}$09 THz (Matsumoto et al., 19 Jul 2025).

In ultracold Fermi gases, the in-medium Cooper-pair propagator $d_{yz}$10 yields a spectral function

$d_{yz}$11

with two sharp low-$d_{yz}$12 features: a gapless phonon and a finite-frequency Leggett peak. Bragg spectroscopy is therefore proposed as a direct probe (Mulkerin et al., 2019). This suggests that the spectroscopic signature of an intraband Leggett mode is not tied to one specific experimental channel, but depends on how the relative phase couples to the external perturbation.

6. Materials, model systems, and open distinctions

MgB$d_{yz}$13 remains the standard benchmark for a sharp Leggett resonance in an intraband-dominated multiband superconductor. With $d_{yz}$14 meV, $d_{yz}$15 meV, and intraband couplings $d_{yz}$16 greatly exceeding $d_{yz}$17, one has $d_{yz}$18 and $d_{yz}$19; the measured $d_{yz}$20 Raman peak at $d_{yz}$21 meV lies between $d_{yz}$22 and $d_{yz}$23, as predicted (Cea et al., 2016). In this setting the resonance is a conventional Leggett mode between distinct band condensates, but it also furnishes the template for identifying relative-phase modes more generally.

Iron-based superconductors present a more differentiated situation. In the one-hole/one-electron-band model with dominant interband pairing, the Leggett resonance is pushed to twice the largest gap, resembling a pair-breaking peak rather than a sharp subgap mode, and this behavior is argued to be in very good agreement with experimental data in iron-based superconductors (Cea et al., 2016). A separate line of work shows that in the usual two-band description of pnictides with dominant interband pairing the Leggett mode is absent because the second pairing channel is antibonding; only in a more general three-band model can a Leggett mode appear, and it softens to zero at the onset of an $d_{yz}$24 time-reversal-symmetry-breaking state (Marciani et al., 2013). These results are complementary rather than identical: one concerns the Raman response of a two-band model including phase fluctuations, the other the existence of a genuine underdamped Leggett resonance below the continuum.

The orbital strong-coupling model adds another possibility. There the Leggett excitation is not primarily between conventional bands but between $d_{yz}$25 and $d_{yz}$26 orbital condensates, with $d_{yz}$27 symmetry and visibility confined to a narrow parameter window (Burnell et al., 2010). This establishes that “intraband” can mean internal to an orbital sector, not necessarily restricted to a single geometrical Fermi pocket.

FeSe in the nematic phase sharpens the distinction between Bardasis–Schrieffer and Leggett language. The observed low-energy mode is attributed to a collective fluctuation between the $d_{yz}$28-wave-like ground state and the subleading pairing channel on the hole pocket; this corresponds to a Bardasis–Schrieffer mode but also resembles an intraband Leggett mode (Matsumoto et al., 19 Jul 2025). The paper states that in FeSe’s $d_{yz}$29 phase “the BS and Leggett language merge”: the mode is simultaneously an intraband “BS” exciton and a two-channel “Leggett” phase oscillation within the same Fermi pocket (Matsumoto et al., 19 Jul 2025). A plausible implication is that the taxonomy of subgap collective modes becomes basis-dependent whenever the superconducting state is multicomponent within one pocket or one orbital sector.

Non-centrosymmetric superconductors supply still another realization. Because the relevant condensates live on helicity branches with mixed singlet–triplet character, the Leggett mode is an intraband relative-phase oscillation whose mass is unaffected by the Anderson–Higgs mechanism, while only its dispersion is slightly modified by Coulomb effects (Bittner et al., 2015). Cold atomic Fermi superfluids extend the same structure beyond electronic superconductors: dipolar interactions or orbital Feshbach settings produce multiple order parameters and an out-of-phase massive mode whose damping can be tracked across the BCS–BEC crossover (Mulkerin et al., 2019, Klimin et al., 2019).

Taken together, these studies establish the intraband Leggett mode as a neutral, finite-frequency collective excitation of internal superconducting phase structure. Its precise realization depends on the microscopic decomposition of the condensate—band, orbital, helicity, or symmetry channel—but its defining signature remains the same: a relative-phase oscillation with a mass generated by internal Josephson coupling and observability controlled by symmetry selection rules and proximity to the two-particle continuum (Burnell et al., 2010, Bittner et al., 2015, Matsumoto et al., 19 Jul 2025).