Is the hyperscaling relation violated below the upper critical dimension in some particular cases?

In this paper, we review our recent results on the critical exponents of thin films obtained by high-performance multi-histogram Monte Carlo simulations. This review shows that the critical exponents do not satisfy the hyperscaling relation. The film thickness N z consists of a few layers up to a dozen of layers in the z direction. The free boundary condition is applied in this direction while in the xy plane periodic boundary conditions are used. We also show the cross-over between the first- and second-order transition while decreasing the film thickness in a frustrated thin film. In the third case, we show evidence that when a 2D system has two order parameters of different symmetries, the critical exponents break the hyperscaling. The last case is the 3D Ising model coupled to the lattice vibration: the results also suggest the violation of the hyperscaling.