Symmetry Protected Topological Order in Superfluid $^3$He-B

The superfluid $^{3}$He-B has been recognized as a concrete example of topological superconductors, where the time-reversal symmetry ensures a nontrivial topological number and the existence of helical Majorana fermions. This may indicate that any time-reversal breaking disturbance wipe out the topological nature. In this talk, I will demonstrate that the B phase under a magnetic field in a particular direction stays topological due to a discrete symmetry, that is, in a symmetry protected topological order [1]. Due to the symmetry protected topological order, helical surface Majorana fermions in the B phase remain gapless and their Ising spin character persists. I unveil that the competition between the Zeeman magnetic field and dipole interaction involves an anomalous quantum phase transition where a topological phase transition takes place together with spontaneous breaking of symmetry. Based on the quasiclassical theory, I illustrate that the phase transition is accompanied by anisotropic quantum criticality of spin susceptibilities on the surface, which is detectable in NMR experiments~[1,2].\\[4pt] [1] T. Mizushima, M. Sato, and K. Machida, Phys. Rev. Lett. \textbf{109}, 165301 (2012).\\[0pt] [2] T. Mizushima, Phys. Rev. B \textbf{86}, 094518 (2012).