Abstract An extension of a heuristic treatment of the critical behavior of the Ising model (N = 1) developed by Thompson, to the general N‐vector model is made. Two approximations for the correlation length critical exponent v as a function of the lattice spatial dimension d and of the number of components of the order parameter N are obtained. Using the hyperscaling relation, the specific heat critical exponent α is also evaluated. Finally, diagrams are constructed displaying in the plane (d, N) various regions, namely: classical critical indexes, non‐classical critical indexes, and no‐phase transition regions or regions where the critical temperature goes to zero. These diagrams are also used to separate regions with divergent specific heat (α > 0) from the ones where it has a cusp‐like behavior (α < 0).
Abstract An extension of a heuristic treatment of the critical behavior of the Ising model (N = 1) developed by Thompson, to the general N‐vector model is made. Two approximations for the correlation length critical exponent v as a function of the lattice spatial dimension d and of the number of components of the order parameter N are obtained. Using the hyperscaling relation, the specific heat critical exponent α is also evaluated. Finally, diagrams are constructed displaying in the plane (d, N) various regions, namely: classical critical indexes, non‐classical critical indexes, and no‐phase transition regions or regions where the critical temperature goes to zero. These diagrams are also used to separate regions with divergent specific heat (α > 0) from the ones where it has a cusp‐like behavior (α < 0).
Abstract Upper and lower bounds are estimated of the two‐point correlation function exponent, η, of the Ising model, as a function of the lattice dimensionality, d. The average value of these two bounds together with the correlation length critical exponent v(d) evaluated by Thompson, lead to a set of critical indexes which satisfy the hyperscaling relation. This set of exponents also shows an adequate agreement with the results obtained by Le‐Guillou and Zinn‐Justin for the Ising model in non‐integer dimensions, in a large range of the parameter d.
The order parameter for a continuous transition shows diverging fluctuation near the critical point. Here we show, through numerical simulations and scaling arguments, that the inequality (or variability) between the values of an order parameter, measured near a critical point, is independent of the system size. Quantification of such variability through the Gini index (g) therefore leads to a scaling form g=G[|F-F_{c}|N^{1/dν}], where F denotes the driving parameter for the transition (e.g., temperature T for ferromagnetic to paramagnetic transition, or lattice occupation probability p in percolation), N is the system size, d is the spatial dimension and ν is the correlation length exponent. We demonstrate the scaling for the Ising model in two and three dimensions, site percolation on square lattice, and the fiber bundle model of fracture.
The Migdal-Kadanoff position-space renormalization-group scheme is used to study the critical behavior of the isotropic n-component-vector model in the previously unexplored region, 1<n<2, 1<d<2, of the n-d plane (d is the dimensionality of the space). We find a continuous phase transition at a finite temperature if d\ensuremath{\ge}dl(n). The lower critical dimension dl(n) increases continuously, but nonlinearly from 1 to 2 as n changes from 1 to 2. For dl(n)\ensuremath{\le}d2, the low-temperature phase is characterized by a power-law decay of the two-point correlation function with a temperature-independent exponent.
The critical behavior of the two-dimensional Ising Model with non-conserved order parameter in steady-state shear is studied by large-scale Monte Carlo simulations. Studying the structure factor S ( q x , q y ) in the disordered phase, the ratio of correlation length exponents ν x /ν y in the two lattice directions ( x , y ) is estimated, and the critical temperature is determined as a function of the shear rate as T c ( ) − T c (0) ∝ ϕ with ϕ≈0.45. Critical exponents β≈0.37, γ≈1.1, ; ν ⊥ ≈0.46, ν ∥ ≈1.38 are roughly compatible with anisotropic hyperscaling.
The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ, and δ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.
Using results we have previously derived for the high‐temperature susceptibility expansion of classical models as a closed‐form function of lattice dimension, we study the dimensional dependence of the critical exponent γ and critical temperature, as well as the correction‐to‐scaling exponent Δ1, for Ising‐like (n=1) models. The numerical results are obtained by extrapolation of 10‐th order series on loose‐packed hypercubical lattices, and 8‐th order series on close‐packed hypertriangular lattices. The critical exponent γ increases monotonically with decreasing dimension, d, for d<4, and apparently tends to infinity at d=1; while the critical temperature decreases monotonically and smoothly to zero at d=1. Detailed contact is made with the ε‐expansion estimates for critical exponents obtained in the context of renormalization group theory.