Collective modes in multiband superfluids and superconductors: Multiple dynamical classes

One important way to characterize the states having a gauge symmetry spontaneously broken over multibands is to look at their collective excitation modes. We find that a three-band system has multiple Leggett modes with significantly different masses, which can be classified into different dynamical classes according to whether multiple interband Josephson currents add or cancel. This provides a way to dynamically characterize multiband superconductivity while the pairing symmetry is a static property.

Figure 1

Schematic diagrams for $N=3$ for (a) the differences in the superfluid phases in equilibrium ${\phi }_0^{(i)}$ and (b) the interband Josephson coupling for the phase fluctuation ${\phi }^{(i)}$ around the equilibrium (where arrows represent the phase difference associated with the interband Josephson current).

Figure 2

Dispersion relations for ${\omega }_{\mathrm{L}+}$ (red squares), ${\omega }_{\mathrm{L}-}$ (blue circles), and ${\omega }_0$ (green triangles) for (a) the class even with the values of ${\mathrm{g̃}}_{\mathit{ij}}$ displayed for which ${\Delta }_2/{\Delta }_1=-1.00$ and ${\Delta }_3/{\Delta }_1=-1.01$, and (b) the class odd with ${\Delta }_2/{\Delta }_1=-0.73$, ${\Delta }_3/{\Delta }_1=-0.38$. We set ${\omega }_{\mathrm{c}}/k_{\mathrm{B}}{T}_{\mathrm{c}}=5000$ and ${\mathrm{g̃}}_{\mathit{ii}}=\frac{1}{3}$. The DOS ($N_i$) and the Fermi velocity are assumed to be identical between the bands.

Figure 3

Color-coded mass difference ($\propto \sqrt{1-D}$) in the multiple Leggett’s modes plotted on the ${\mathrm{g̃}}_{12}$-${\mathrm{g̃}}_{31}$ plane for ${\mathrm{g̃}}_{\mathit{ii}}=\frac{1}{3}$. We assume identical normal-state properties between the three bands. (a) Illustrates the case of two negative (repulsive) and one positive (attractive) interband couplings. The yellow-dotted lines represent the boundaries between the even and odd classes. (b) Illustrates the case of all the three interband couplings negative (repulsive), where only the class odd is allowed. In the blank area, the Leggett’s modes become unstable.