Upper critical dimension

Remarks

Example: for the Ising model, mean-field theory gives $\beta = 1/2$, $\gamma = 1$, $\delta = 3$, $\nu = 1/2$ and $\eta = 0$, and $d_u = 4$. A heuristic check is the hyperscaling relation $d\nu = 2 - \alpha$: with $\nu = 1/2$ and $\alpha = 0$ it holds exactly at $d = 4$.

Non-example: $d_u = 3$ fails for the Ising family, since the three-dimensional Ising model has $\beta \approx 0.326 \ne 1/2$.

The lower critical dimension $d_l$ is the largest $d$ at which there is no ordered phase at any positive temperature. It is $1$ for the Ising model and $2$ for models with a continuous symmetry, by the Mermin–Wagner theorem. Percolation has $d_u = 6$.