Dimension-Dependent Critical Scaling Analysis and Emergent Competing Interaction Scales in a 2D Van der Waals magnet Cr$_{2}$Ge$_{2}$Te$_{6}$

We investigate thickness-dependent transformation from a paramagnetic to ferromagnetic phase in Cr$_{2}$Ge$_{2}$Te$_{6}$ (CGT) in bulk and few-layer flake forms. 2D Ising-like critical transition in bulk CGT occurs at $T_{c}$ = 67 K with out-of-plane magnetic anisotropy. Few-layer CGT on hBN/SiO$_{2}$/Si substrate displays the same $T_{c}$ but also exhibits a new critical transition at $T^{\prime}_c$ = 14.2 K. Here, critical scaling analysis reveals the critical exponents differ significantly from those in bulk and do not align with the known universality classes. Our Density Functional Theory (DFT) and classical calculations indicate competition between magnetocrystalline and dipolar anisotropy emerges with reduced dimensions. The observed behavior is due to minor structural distortions in low dimensional CGT, which modify the balance between spin-orbit coupling, exchange interactions and dipolar anisotropy. This triggers a critical crossover at $T^{\prime}_c$. Our study shows the emergence of a complex interplay of short- and long-range interactions below $T^{\prime}_c$ as CGT approaches the 2D limit.