Leggett modes and multi-band superconductivity in Sr2RuO4

Sr$_2$RuO$_4$ is a prototypical multi-band superconductor with three bands crossing the Fermi level. These bands exhibit distinct dimensional characteristics, with one quasi-2D $\gamma$-band and two quasi-1D $\alpha$- and $\beta$-bands. This leads to the expectation that the superconductivity on the $\gamma$-band may be only weakly Josephson-coupled to that on the other two bands. Based on an explicit microscopic weak coupling calculation appropriate for Sr$_2$RuO$_4$, we study the collective Leggett modes associated with the relative phase oscillations between the bands and show that a relatively soft Leggett mode exists due to the comparatively weaker inter-band Josephson coupling. These calculations also provide insight into why the superconducting gap magnitudes may be comparable on all three bands, despite the noticeable differences between the $\gamma$ and $\alpha, \beta$ bands. The analyses can be readily applied to other multi-band superconductors.