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

¹ Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, CNRS, UMR 8089 2, Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France.
² Institute of Scientific Data and Information, Vietnam Academy of Science and Technology (VAST) 18 Hoang Quoc Viet, Hanoi 10000, Vietnam.

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Published online: 29 April 2026

Abstract

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.