  The mechanism of avian navigation is an important question for global
scientists. One of the most famous candidates is the radical pair mechanism
(RPM), which shows that avian navigation is achieved by detecting the amount of
products from chemical reactions, which exhibit different properties for
singlet and triplet spin states. Based on the theory, we explore the amount of
the the singlet recombination product by changing the coupling strength and
direction of the magnetic field. The radical pair in our model is coupled with
a Lipkin-Meshkov-Glick (LMG) bath, which proposes attractive results at quantum
phase transition (QPT) point, especially when the magnetic field is parallel
with the original direction of the pair. We find weakest navigation ability in
QPT point according to one of our proposing methods. And we find invariant
subspace in our Hamiltonian when the magnetic field is vertical to the plane of
the radical pair and the bath, which will greatly speed up our calculation. Our
results could help the exploration of the design principle of artificial avian
navigation.
