Is the Hyperscaling Relation Violated Below the Upper Critical Dimension in Some Particular Cases?

In this review, we show our results with new interpretation on the critical exponents of thin films obtained by high-performance multi-histogram Monte Carlo simulations. The film thickness Nz 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. The large xy plane size, from 702 to 1602, is used for finite-size scaling. The Ising model is used with nearest-neighbor (NN) interaction. When Nz=1, namely the two-dimensional (2D) system, we find strictly the critical exponents given by the renormalization group. While, for Nz>1, the critical exponents calculated with the high-precision multi-histogram technique show that they deviate systematically from the 2D values and get closer to the 3D critical exponents for Nz>13 . If we argue that as long as the thickness Nz is small enough, the correlation length in the z direction is finite, it does not affect the nature of the phase transition, namely it remains in the 2D universality class. This argument is in contradiction with our numerical results which show a systematic deviation from 2D values: If we use these values of critical exponents in the hyperscaling relation with d=2, then the hyperscaling relation is violated. However, if we use the hyperscaling relation and the critical exponents obtained for Nz>1 to calculate the dimension of the system, we find the system dimension between 2 and 3. This can be viewed as an "effective" dimension. More discussion is given in the paper. We also show the cross-over between the first- and second-order transition while varying the film thickness. In addition, we will show evidence that when a 2D system has two order-parameters of different symmetries with a single transition, the critical exponents are new, suggesting a universality class of coupled two-symmetry breakings. In this case, the 2D hyperscaling does not hold. Another case is the 3D Ising model coupled to the lattice vibration: the critical exponents deviate from the 3D Ising ones, the results suggest the violation of the hyperscaling.