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The nucleon-nucleon correlation between nucleons leads to the Fermi surfacedepletion measured by a $Z$-factor in momentum distribution of dense nuclearmatter. The roles of the Fermi surface depletion effect ($Z$-factor effect) andits quenched neutron triplet superfluidity of nuclear matter in viscosity andhence in the gravitational-wave-driven $r$-mode instability of neutron stars(NSs) are investigated. The bulk viscosity is reduced by both the two effects,especially the superfluid effect at low temperatures which is also able toreduce the inferred core temperature of NSs. Intriguingly, due to the neutronsuperfluidity, the core temperature of the NSs in known low-mass X-ray binaries(LMXBs) are found to be clearly divided into two groups: high and lowtemperatures which correspond to NSs with short and long recurrence times fornuclear-powered bursts respectively. Yet, a large number of NSs in these LMXBsare still located in the $r$-mode instability region. If the density-dependentsymmetry energy is stiff enough, the occurence of direct Urca process reducesthe inferred core temperature by about one order of magnitude. Accordingly, thecontradiction between the predictions and observations is alleviated to someextent, but some NSs are still located inside the unstable region.

r-mode instability of neutron stars in low-mass X-ray binaries: effects of Fermi surface depletion and superfluidity of dense matter