Massless Leggett Mode in Three-Band Superconductors with Time-Reversal-Symmetry Breaking

The Leggett mode associated with out-of-phase oscillations of the superconducting phase in multiband superconductors usually is heavy due to interband coupling, which makes its excitation and detection difficult. We report on the existence of a massless Leggett mode in three-band superconductors with time-reversal-symmetry breaking. The mass of this Leggett mode is small close to the time-reversal-symmetry-breaking transition and vanishes at the transition point, and thus locates within the smallest superconducting energy gap, which makes it stable and detectable, e.g., by means of the Raman spectroscopy. The thermodynamic consequences of this massless mode and possible realization in iron-based superconductors are also discussed.

Figure 1

Frustrated interband scatterings force Cooper pairs in different bands to carry different phases, which results in interband Josephson currents. Two dynamical modes associated with superconducting phases in three-band superconductors: the LM, where two phases oscillate out-of-phase while the third one stays unchanged, becomes massless at the TRSB transition (left), and the BAG mode, where all the three phases rotate in the same direction during the propagation of plasma wave in space (right).

Figure 2

Amplitudes and phases of order parameters at the TRSB phase transition, in (a) and (b) as a function of $\eta$, and in (c) and (d) as a function of DOS $N_{1}V$ of the first component. ${\Delta }_1$ is taken as real and positive. In (a) and (b), an identical DOS $NV=0.5$ is taken for the three bands and $\alpha =2$ in Eq. (5). In (c) and (d), $N_{2}V=0.5$ and $N_{3}V=0.4$, $\alpha =2$ and $\eta =1$ [see also Eq. (17)]. In the TRSB regime, there are two degenerate ground states (${\Delta }_1,{\Delta }_2,{\Delta }_3$) (solid lines) and (${\Delta }_1^{*},{\Delta }_2^{*},{\Delta }_3^{*}$) (dashed lines). The two solid lines for $N_{1}V>0.64$ in (d) refer to the same state without TRSB.

Figure 3

Dependence of the masses of LMs on the interband coupling $\eta$. Here $NV=0.5$ and $\alpha =2$, and the masses are in units of ${\omega }_c$.

Figure 4

Schematic view of the Raman response in three-band superconductors with TRSB. The finite linewidth of peaks is due to the damping and interaction between Leggett bosons. The background at an energy larger than $2{\Delta }_1$ is due to the quasiparticle excitations.