Topological superfluid $^3$He-B in magnetic field and Ising variable

The topological superfluid $^3$He-B provides many examples of the interplay of symmetry and topology. Here we consider the effect of magnetic field on topological properties of $^3$He-B. Magnetic field violates the time reversal symmetry. As a result, the topological invariant supported by this symmetry ceases to exist; and thus the gapless fermions on the surface of $^3$He-B are not protected any more by topology: they become fully gapped. Nevertheless, if perturbation of symmetry is small, the surface fermions remain relativistic with mass proportional to symmetry violating perturbation – magnetic field. The $^3$He-B symmetry gives rise to the Ising variable $I=\pm1$, which emerges in magnetic field and which characterizes the states of the surface of $^3$He-B. This variable also determines the sign of the mass term of surface fermions and the topological invariant describing their effective Hamiltonian. The line on the surface, which separates the surface domains with different $I$, contains $1+1$ gapless fermions, which are protected by combined action of symmetry and topology.