  There are two discrepant experimental results on the magnetic ground state of
Y$_{2}$Mn$_{2}$O$_{7}$, one study proposes a spin glass state, while another
introduces the material as a ferromagnet. In this study, we attempt to resolve
this issue by employing density functional theory and Monte Carlo simulations.
We derive different spin models by varying the Hubbard $U$ parameter in ab
initio GGA+$U$ calculations. For the most range of Hubbard $U$, We obtain that
the leading terms in the spin Hamiltonian are bi-quadratic and the nearest
neighbor Heisenberg exchange interactions. By comparing Monte Carlo simulations
of these models with the experiments, we find a ferromagnetic ground state for
Y$_{2}$Mn$_{2}$O$_{7}$ as the most compatible with experiments. We also
consider Y$_{2}$Mo$_{2}$O$_{7}$ as a prototype of the defect-free pyrochlore
system with spin-glass behavior and compare it with Y$_{2}$Mn$_{2}$O$_{7}$. The
orbital degrees of freedom are considered as a leading factor in converting a
defect-free pyrochlore such as Y$_{2}$Mn$_{2}$O$_{7}$ to a spin glass system.
By changing the $d$ orbital occupations of Mo atoms, our GGA+$U$ calculations
for Y$_{2}$Mo$_{2}$O$_{7}$ indicate many nearly degenerate states with
different $d$ orbital orientations which reveals $d$ orbital degrees of freedom
in this material. While for Y$_{2}$Mn$_{2}$O$_{7}$, we find a single ground
state with a fixed orbital orientation. Consequently, all of our ab initio
approaches confirm Y$_{2}$Mn$_{2}$O$_{7}$ as a ferromagnetic system.
