The theoretical study of topological superfluids and superconductors has so far been carried out largely as a translation of the theory of noninteracting topological insulators into the superfluid language, whereby one replaces electrons by Bogoliubo
v quasiparticles and single-particle band Hamiltonians by Bogoliubov-de Gennes Hamiltonians. Band insulators and superfluids are, however, fundamentally different: while the former exist in the absence of inter-particle interactions, the latter are broken symmetry states that owe their very existence to such interactions. In particular, unlike the static energy gap of a band insulator, the gap in a superfluid is due to a dynamical order parameter that is subject to both thermal and quantum fluctuations. In this work, we explore the consequences of bulk quantum fluctuations of the order parameter in the $B$ phase of superfluid $^3$He on the topologically protected Majorana surface states. Neglecting the high-energy amplitude modes, we find that one of the three spin-orbit Goldstone modes in $^3$He-$B$ couples to the surface Majorana fermions. This coupling in turn induces an effective short-range two-body interaction between the Majorana fermions, with coupling constant inversely proportional to the strength of the nuclear dipole-dipole interaction in bulk $^3$He. A mean-field theory estimate of the value of this coupling suggests that the surface Majorana fermions in $^3$He-$B$ are in the vicinity of a quantum phase transition to a gapped time-reversal symmetry breaking phase.
The discontinuity of guiding-center Hall viscosity (a bulk property) at edges of incompressible quantum Hall fluids is associated with the presence of an intrinsic electric dipole moment on the edge. If there is a gradient of drift velocity due to a
non-uniform electric field, the discontinuity in the induced stress is exactly balanced by the electric force on the dipole. The total Hall viscosity has two distinct contributions: a trivial contribution associated with the geometry of the Landau orbits, and a non-trivial contribution associated with guiding-center correlations. We describe a relation between the guiding-center edge-dipole moment and momentum polarization, which relates the guiding-center part of the bulk Hall viscosity to the orbital entanglement spectrum(OES). We observe that using the computationally-more-onerous real-space entanglement spectrum (RES) just adds the trivial Landau-orbit contribution to the guiding-center part. This shows that all the non-trivial information is completely contained in the OES, which also exposes a fundamental topological quantity $gamma$ = $tilde c- u$, the difference between the chiral stress-energy anomaly (or signed conformal anomaly) and the chiral charge anomaly. This quantity characterizes correlated fractional quantum Hall fluids, and vanishes in uncorrelated integer quantum Hall fluids.
