Lower critical dimension ($d_{\mathrm{l}}$)

The lower critical dimension is the spatial dimension at or below which fluctuations prevent the ordered phase associated with a proposed finite-temperature transition. A short-range model with discrete symmetry normally has $d_{\mathrm{l}}=1$, whereas continuous $O(2)$ symmetry has $d_{\mathrm{l}}=2$.

The Berezinskii–Kosterlitz–Thouless transition in the two-dimensional XY model separates a low-temperature phase with algebraically decaying correlations from a high-temperature phase with unbound vortices and exponentially decaying correlations. It does not produce conventional long-range order.