
VOLUME 91  ISSUE 4 
PAGE 215

Topological superfluid ^{ 3}HeB in magnetic field and Ising variable

G.E. Volovik
Low Temperature Laboratory, Aalto University, School of Science and Technology, FI00076 AALTO, Finland Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia

Abstract
The topological superfluid ^{3}HeB provides many examples of the
interplay of symmetry and topology. Here we consider the effect of magnetic
field on topological properties of ^{3}HeB. 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}HeB 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}HeB symmetry gives rise to the
Ising variable , which emerges in magnetic field and which
characterizes the states of the surface of ^{3}HeB. 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.


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