  The neutron star starts to cool down shortly after its birth by emitting
neutrinos. When it is cold enough, the Cooper pairs of neutrons are formed,
which triggers a superfluid transition. Previous works on neutron superfluidity
are focused on the finite-temperature transition. Little attention is paid to
the potentially important quantum critical phenomena associated with
superfluidity. Here, we provide the first theoretical analysis of superfluid
quantum criticality, concentrating on its impact on neutron star cooling.
Extensive calculations found that superfluidity occurs within a finite range of
nuclear density $\rho$. The density serves as a non-thermal parameter for
superfluid quantum phase transition. In a broad quantum critical region, there
is a strong coupling between the gapless neutrons and the quantum critical
fluctuations of superfluid order parameter. We handle this coupling by using
perturbation theory and renormalization group method, and show that it leads to
non-Fermi liquid behavior, which yields a logarithmic $T\ln(1/T)$ correction to
the Fermi-liquid specific heat $c_{n}\propto T$ and also markedly alters the
neutrino emissivity. Quantum critical phenomena emerge at a time much earlier
than the onset of superfluidity and persist throughout almost the entire life
of a neutron star. At a low $T$, such phenomena coexist with superfluidity in
the neutron star interior, yet occupying different layers. We include
superfluid quantum criticality into the theoretical description of neutron star
cooling and find that it obviously prolongs the thermal relaxation time. By
varying the strength of superfluid fluctuations and other quantities, we obtain
an excellent fit to the observed cooling data of a variety of neutron stars.
Our results indicate the existence of an intriguing correlation between
superfluid quantum criticality and the thermal evolution of neutron stars.
