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 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 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. Large xy plane sizes are used for finite-size scaling. The Ising model is studied with nearest-neighbor (NN) interaction. When Nz=1, namely the two-dimensional (2D) system, we find 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 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 Nz>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 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.
We study the phase transition of thin films in the three-dimensional XY universality class. To this end, we perform a Monte Carlo study of the improved two-component ϕ 4 model, the improved dynamically diluted XY model and the standard XY model on the simple cubic lattice. We study films of a thickness up to L 0 = 32 lattice spacings. In the short direction of the lattice, free boundary conditions are employed. Using a finite size scaling (FSS) method, proposed recently, we determine the transition temperature with high accuracy. The effectively two-dimensional finite size scaling behaviour of the Binder cumulant U 4 , the second-moment correlation length over the lattice size ξ 2nd / L , the ratio of the partition functions with anti-periodic and periodic boundary conditions Z a / Z p and the helicity modulus ϒ clearly confirm the Kosterlitz–Thouless nature of the transition. We analyse the scaling of the transition temperature with the thickness L 0 of the film. We compute the universal ratio of the thickness of the film L 0 and the transverse correlation length ξ T in the three-dimensional thermodynamic limit at the Kosterlitz–Thouless transition temperature of a film of thickness L 0 : [ L 0,KT /ξ T ] * = 1.595(7). This results can be compared with experimental results on thin films of 4 He near the λ-transition.
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 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.
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems of square geometry with periodic boundaries oriented either along <10> or along <11> direction have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction proportional to 1/L^0.25 of a very small magnitude also has been detected in agreement with predictions of our recently developed GFD (grouping of Feynman diagrams) theory. A refined analysis of the recent MC data for 3D Ising, phi^4, and XY lattice models has been performed. It includes an analysis of the partition function zeros of 3D Ising model, an estimation of the correction-to-scaling exponent omega from the Binder cumulant data near criticality, as well as a study of the effective critical exponent eta and the effective amplitudes in the asymptotic expansion of susceptibility at the critical point. In all cases a refined analysis is consistent with our (GFD) asymptotic values of the critical exponents (nu=2/3, omega=1/2, eta=1/8 for 3D Ising model and omega=5/9 for 3D XY model), while the actually accepted "conventional" exponents are, in fact, effective exponents which are valid for approximation of the finite--size scaling behavior of not too large systems.