  We study the spontaneous parity breaking and generating of Hall viscosity and
angular momentum in holographic p+ip model, which can describe strongly-coupled
chiral superfluid states in many quantum systems. The dual gravity theory, an
SU(2) gauge field minimally coupled to Einstein gravity, is parity-invariant
but allows a black hole solution with vector hair corresponding to a
parity-broken superfluid state. We show that this state possesses a
non-vanishing parity-odd transport coefficient -- Hall viscosity -- and an
angular momentum density. We first develop an analytic method to solve this
model near the critical regime and to take back-reactions into account. Then we
solve the equation for the tensor mode fluctuations and obtain the expression
for Hall viscosity via Kubo formula. We also show that a non-vanishing angular
momentum density can be obtained through the vector mode fluctuations and the
corresponding boundary action. We give analytic results of both Hall viscosity
and angular momentum density near the critical regime in terms of physical
parameters. The near-critical behavior of Hall viscosity is different from that
obtained from a gravitational Chern-Simons model. We find that the magnitude of
Hall viscosity to angular momentum density ratio is numerically consistent with
being equal to 1/2 at large SU(2) coupling corresponding to the probe limit, in
agreement with previous results obtained for various quantum fluid systems and
from effective theory approaches. In addition, we find the shear viscosity to
entropy density ratio remains above the universal bound.
