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.
Numerical data on scaling of the normalized Binder cumulant and the normalized correlation length are shown for the Thermodynamic limit regime, first for canonical Ising ferromagnet models and then for a range of Ising spin glass models. A fundamental Renormalization Group Theory rule linking the critical exponents for the two observables is well obeyed in the Ising models, but not for the Ising spin glasses in dimensions three and four. We conclude that there is a violation of a standard Josephson hyperscaling rule in Ising spin glasses.
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.
We performed extensive simulations of the random-bond Ising model confined between walls where competitive surface fields act. By properly taking the thermodynamic limit we unambiguously determined wetting transition points of the system, as extrapolation of localization-delocalization transitions of the interface between domains of different orientation driven by the respective fields. The finite-size scaling theory for wetting with short-range fields [E. V. Albano and K. Binder, Phys. Rev. Lett. 109, 036101 (2012)] establishes that the average magnetization of the sample, with critical exponent $\ensuremath{\beta}$, is the proper order parameter for the study of wetting. While the hyperscaling relationship given by $\ensuremath{\gamma}+2\ensuremath{\beta}={\ensuremath{\nu}}_{\ensuremath{\parallel}}+{\ensuremath{\nu}}_{\ensuremath{\perp}}$ requires $\ensuremath{\beta}=1/2$ ($\ensuremath{\gamma}=4, {\ensuremath{\nu}}_{\ensuremath{\parallel}}=3$, and ${\ensuremath{\nu}}_{\ensuremath{\perp}}=2$), the thermodynamic scaling establishes that ${\mathrm{\ensuremath{\Delta}}}_{s}=\ensuremath{\gamma}+\ensuremath{\beta}$, which in contrast requires $\ensuremath{\beta}=0$ (${\mathrm{\ensuremath{\Delta}}}_{s}=4$), where $\ensuremath{\gamma}, {\ensuremath{\nu}}_{\ensuremath{\parallel}}, {\ensuremath{\nu}}_{\ensuremath{\perp}}$, and ${\mathrm{\ensuremath{\Delta}}}_{s}$ are the critical exponents of the susceptibility, the correlation lengths parallel and perpendicular to the interface, and the gap exponent, respectively. So, we formulate a finite-size scaling theory for wetting without hyperscaling and perform numerical simulations that provide strong evidence of hyperscaling violation (i.e., $\ensuremath{\beta}=0$) and a direct measurement of the susceptibility critical exponent $\ensuremath{\gamma}/{\ensuremath{\nu}}_{\ensuremath{\perp}}=2.0\ifmmode\pm\else\textpm\fi{}0.2$, in agreement with theoretical results for the strong fluctuation regime of wetting transitions with quenched noise.
Extensive simulations are made on Ising Spin Glasses (ISG) with Gaussian, Laplacian and bimodal interaction distributions in dimension four. Standard finite size scaling analyses near and at criticality provide estimates of the critical inverse temperatures
βc, critical exponents, and critical values of a number of dimensionless parameters. Independent estimates are obtained for
βc and the exponent
ν from thermodynamic derivative peak data. A detailed explanation is given of scaling in the thermodynamic limit with the ISG scaling variable
τ=1−β2/βc2 and the appropriate scaling expressions. Data over the entire paramagnetic range of temperatures are analysed in order to obtain further estimates of the critical exponents together with correction to scaling terms. The Privman-Fisher ansatz then leads to compact scaling expressions for the whole paramagnetic regime and for all sample sizes L. Comparisons between the 4d ISG models show that the critical dimensionless parameters characteristic of a universality class, and the susceptibility and correlation length critical exponents
γ and
ν, depend on the form of the interaction distribution. From these observations it can be deduced that critical exponents are not universal in ISGs, at least in dimension four.
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.
The critical behavior of the Binder cumulant for Ising spin glasses in dimension four is studied through simulation measurements. Data for the bimodal interaction model are compared with those for the Laplacian interaction model. Special attention is paid to scaling corrections. The limiting infinite size value at criticality for this dimensionless variable is a parameter characteristic of a universality class. This critical limit is estimated to be equal to 0.523(3) in the bimodal model and to 0.473(3) in the Laplacian model.