Leggett modes in the multi-band superconductor 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. As a consequence, the superconducting order parameter on the $\gamma$- and $\alpha/\beta$-bands may only be weakly Josephson-coupled, in contrast to the stronger coupling between the quasi-1D bands. In this work, we study the Leggett modes associated with the relative phase fluctuations between the bands. We show that a soft Leggett mode exists in the case of comparatively weaker inter-band Josephson coupling between the $\gamma$- and $\alpha/\beta$-bands. We further analyze the dependence of the inter-band Josephson coupling on spin-orbit coupling, and discuss the possibility of an exotic time-reversal symmetry breaking phase when the Josephson coupling is comparable between all pairs of bands.