  In this work we study and establish some quenched functional Central Limit
Theorems (CLTs) for stationary random fields under a projective criteria. These
results are functional generalizations of the theorems obtained by Zhang et al.
(2020) and of the quenched functional CLTs for ortho-martingales established by
Peligrad and Voln{\'y} (2020) to random fields satisfying a Hannan type
projective condition. In the work of Zhang et al. (2020), the authors have
already proven a quenched functional CLT however the assumptions were not
optimal as they require the existence of a 2 + $\delta$-moment. In this
article, we establish the results under weaker assumptions, namely we only
require an Orlicz space condition to hold. The methods used to obtain these
generalizations are somewhat similar to the ones used by Zhang et al. (2020)
but we improve on them in order to obtain results within the functional
framework. Moreover, a Rosenthal type inequality for said Orlicz space is also
derived and used to obtain a sufficient condition analogous to that of Theorem
4.4 in the work of Zhang et al. (2020). Finally, we apply our new results to
derive some quenched functional CLTs under weak assumptions for a variety of
stochastic processes.
