A scaling approach to critical exponent calculations for the 2D Ising model

Shuo Liu, E. W. Carlson, Karin A. Dahmen

Scanning probes reveal complex, inhomogeneous patterns on the surface of many condensed matter systems. In some cases, the patterns form self-similar, fractal geometric clusters. In this paper, we advance the theory of criticality as it pertains to those geometric clusters (defined as connected sets of nearest-neighbor aligned spins) in the context of Ising models. We show how data from surface probes can be used to distinguish whether electronic patterns observed at the surface of a material are confined to the surface, or whether the patterns originate in the bulk. Whereas thermodynamic critical exponents are derived from the behavior of Fortuin–Kasteleyn (FK) clusters, critical exponents can be similarly defined for geometric clusters. We find that these geometric critical exponents are not only distinct numerically from the thermodynamic and uncorrelated percolation exponents, but that they separately satisfy scaling relations at the critical fixed points discussed in the text. We furthermore find that the two-dimensional (2D) cross-sections of geometric clusters in the three-dimensional (3D) Ising model display critical scaling behavior at the bulk phase transition temperature. In particular, we show that when considered on a 2D slice of a 3D system, the pair connectivity function familiar from percolation theory displays more robust critical behavior than the spin-spin correlation function, and we calculate the corresponding critical exponent. We discuss the implications of these two distinct length scales in Ising models. We also calculate the pair connectivity exponent in the clean 2D case. These results extend the theory of geometric criticality in the clean Ising universality classes, and facilitate the broad application of geometric cluster analysis techniques to maximize the information that can be extracted from scanning image probe data in condensed matter systems.

Nevzat AKTEKİN

Aleksandr Pishtshev

H. T. Diep, Van Thanh Ngo

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.

Simran Singh, Massimo Cipressi, Francesco Di Renzo

We present a numerical calculation of the Lee-Yang and Fisher zeros of the 2D Ising model using multipoint Padé approximants. We perform simulations for the 2D Ising model with ferromagnetic couplings both in the absence and in the presence of a magnetic field using a cluster spin-flip algorithm. We show that it is possible to extract genuine signature of Lee-Yang and Fisher zeros of the theory through the poles of magnetization and specific heat, using the multipoint Padé method. We extract the poles of magnetization using Padé approximants and compare their scaling with known results. We verify the circle theorem associated to the well known behavior of Lee-Yang zeros. We present our finite volume scaling analysis of the zeros done at T=Tc for a few lattice sizes, extracting to a good precision the (combination of) critical exponents βδ . The computation at the critical temperature is performed after the latter has been determined via the study of Fisher zeros, thus extracting both βc and the critical exponent ν . Results already exist for extracting the critical exponents for the Ising model in two and three dimensions making use of Fisher and Lee-Yang zeros. In this work, multipoint Padé is shown to be competitive with this respect and thus a powerful tool to study phase transitions. Published by the American Physical Society 2024

Ziya Merdan, Nida Aras, Cihan Kürkçü