In this paper, the concept of $\varphi$-mixing is extended to random fields, and a central limit theorem analogous to Theorem 20.1 of Billingsley (Convergence of Probability Measures, Wiley (1968)) is obtained for stationary, $\varphi$-mixing random fields.
"A Functional Central Limit Theorem for Stationary Random Fields." Ann. Probab. 3 (4) 708 - 715, August, 1975. https://doi.org/10.1214/aop/1176996310