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arXiv:2507.02159

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

H. T. Diep, Van Thanh Ngo

Published Jul 2, 2025• 0 citations
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Abstract

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 $N_z$ 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. Large $xy$ plane sizes are used for finite-size scaling. The Ising model is studied with nearest-neighbor (NN) interaction. When $N_z=1$, namely the two-dimensional (2D) system, we find the critical exponents given by the renormalization group. While, for $N_z>1$, the critical exponents calculated with the high-precision multi-histogram technique show that they deviate slightly but systematically from the 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 $N_z>1$ to calculate the dimension of the system, we find the system dimension slightly larger than 2. 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 an antiferromagnetic FCC Ising frustrated thin film. 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.

DOI
10.48550/arxiv.2507.02159

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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 NzN_zNz​ consists of a few layers up to a dozen of layers in the zzz direction. The free boundary condition is applied in this direction while in the xyxyxy plane periodic boundary conditions are used. Large xyxyxy plane sizes are used for finite-size scaling. The Ising model is studied with nearest-neighbor (NN) interaction. When Nz=1N_z=1Nz​=1, namely the two-dimensional (2D) system, we find the critical exponents given by the renormalization group. While, for Nz>1N_z>1Nz​>1, the critical exponents calculated with the high-precision multi-histogram technique show that they deviate slightly but systematically from the 2D values. If we use these values of critical exponents in the hyperscaling relation with d=2d=2d=2, then the hyperscaling relation is violated. However, if we use the hyperscaling relation and the critical exponents obtained for Nz>1N_z>1Nz​>1 to calculate the dimension of the system, we find the system dimension slightly larger than 2. 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 an antiferromagnetic FCC Ising frustrated thin film. 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.
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.
In a previous work, a realistic three-dimensional 3-D Ising model of FeCl2 was studied. It consists of triangular lattice planes stacked along the c axis of the crystal, with the following interactions: a nearest neighbor (nn) ferromagnetic one competing with a next-nearest-neighbor (nnn) antiferromagnetic interaction, which causes some degree of frustration in the plane, the planes being weakly coupled by an antiferromagnetic interaction. In fact, the latter is a superexchange interaction, and we have taken into account all the equivalent superexchange paths in the sense of the Anderson rule, which amounts to 12 neighbors in each one of the adjacent planes. In this work we present the preliminary results of a study of the critical exponents of this model using the Monte Carlo histogram method and finite size scaling. The interest of such study lies in the layered structure of this system, which might have an effect on its effective dimensionality at the transition point, and also in the effects of frustration, as well as those of superexchange interactions on the critical exponents. We also show the results obtained by the same method, on the critical exponents of the diluted system in the zero and low magnetic field.
In addition to the standard scaling rules relating critical exponents at second order transitions, hyperscaling rules involve the dimension of the model. It is well known that in canonical Ising models hyperscaling rules are modified above the upper critical dimension. It was shown by M. Schwartz in 1991 that hyperscaling can also break down in Ising systems with quenched random interactions; Random Field Ising models, which are in this class, have been intensively studied. Here, numerical Ising Spin Glass data relating the scaling of the normalized Binder cumulant to that of the reduced correlation length are presented for dimensions 3, 4, 5, and 7. Hyperscaling is clearly violated in dimensions 3 and 4, as well as above the upper critical dimension D = 6 . Estimates are obtained for the “violation of hyperscaling exponent” values in the various models.
In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced with a modified hyperscaling relation. As a result, standard formulations of finite-size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free-energy cost ΔF of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, ΔF∝L(θ), with θ as the violation of hyperscaling critical exponent and L as the linear extension of the system. This modified behavior facilitates a number of numerical approaches that can be used to locate critical points in random field systems from finite-size simulation data. We test and confirm the approaches on two random field systems in three dimensions, namely, the random field Ising model and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.
In this study, we explored the magnetic coupling in a multilayer system consisting of thin layers separated by a distance ddd. We have employed Monte Carlo (MC) simulations to calculate the thermodynamic quantities such as the magnetization per spin mLμm_{L}^μmLμ​, magnetic susceptibility χLμχ_{L}^μχLμ​, and the reduced fourth-order Binder cumulant ULμU_{L}^μULμ​ as a function of temperature TTT and for several values of lattice size LLL. These quantities were obtained for each layer (μ=l)μ=l)μ=l), for the bulk system (μ=b)μ=b)μ=b), as a function of interlayer distance ddd and the parameter ααα that defines the scale length of the exponential decay of the interaction. Furthermore, we applied the finite-size scaling theory to calculate the critical exponents. Our results reveal that the system exhibits 2D Ising exponents when the layers are sufficiently separated. On the other hand, in the compact limit, where ddd equals the distance between two adjacent sites within the same layer, our bulk results show that the system exhibits 3D Ising critical exponents, provided that the number of layers LzL_{z}Lz​ increases proportionally to the layer sizes. Even with the separation increasing up to d=2.0d=2.0d=2.0, the layers are so correlated that the set of critical exponents retains the values of 3D critical exponents. However, when the number of layers LzL_{z}Lz​ remains fixed, even in the compact limit with periodic boundary conditions, only the exponent βββ aligns closely with the predicted literature values, on the other hand, the other exponents show significant deviations.
We use extensive Monte Carlo transfer-matrix calculations on infinite strips of widths L up to 30 lattice spacing and a finite-size scaling analysis to obtain critical exponents and conformal anomaly number c for the two-dimensional XY Ising model. This model is expected to describe the critical behavior of a class of systems with simultaneous U(1) and Z2{\mathit{Z}}_{2}Z2​ symmetries of which the fully frustrated XY model is a special case. The effective values obtained for c show a significant decrease with L at different points along the line where the transition to the ordered phase takes place in a single transition. Extrapolations based on power-law corrections give values consistent with c=3/2 although larger values cannot be ruled out. Critical exponents are obtained more accurately and are consistent with previous Monte Carlo simulations suggesting critical behavior and with recent calculations for the frustrated XY model.