scholarly journals Anisotropic scaling of the two-dimensional Ising model I: the torus

2019 ◽  
Vol 7 (3) ◽  
Author(s):  
Hendrik Hobrecht ◽  
Fred Hucht
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Boundary Conditions ◽  
Scaling Limit ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Anisotropic Scaling

We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and – if present – the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.

scholarly journals Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields

2020 ◽  
Vol 8 (3) ◽  
Author(s):  
Hendrik Hobrecht ◽  
Fred Hucht
Keyword(s):  
Ising Model ◽  
Boundary Conditions ◽  
Scaling Limit ◽  
Finite Size ◽  
Square Lattice ◽  
Conformal Field ◽  
Anisotropic Scaling ◽  
Finite Lattices ◽  
Antiferromagnetic Ising Model ◽  
Boundary Fields

Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we independently apply boundary fields on both sides which can be either homogeneous or staggered, representing different combinations of boundary conditions. We confirm several predictions from scaling theory, conformal field theory and renormalisation group theory: we explicitly show that anisotropic couplings enter the scaling functions through a generalised aspect ratio, and demonstrate that open and staggered boundary conditions are asymptotically equal in the scaling regime. Furthermore, we examine the emergence of the surface tension due to one antiperiodic boundary in the system in the presence of symmetry breaking boundary fields, again for finite systems as well as in the scaling limit. Finally, we extend our results to the antiferromagnetic Ising model.

scholarly journals FINITE SIZE SCALING ON THE ISING COEXISTENCE LINE

1992 ◽  
Vol 03 (05) ◽  
pp. 1119-1124
Author(s):  
SOURENDU GUPTA ◽  
A. IRBÄCK
Keyword(s):  
Free Energy ◽  
Ising Model ◽  
Finite Size ◽  
Temperature Phase ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Coexistence Line ◽  
Low Temperature Phase ◽  
Good Agreement

We report tests of finite-size scaling ansatzes in the low temperature phase of the two-dimensional Ising model. For moments of the magnetisation density, we find good agreement with the new ansatz of Borgs and Kotecký, and clear evidence of violations of the double Gaussian ansatz. We note that certain consequences of the convexity of the free energy are not adequately treated in either of these approaches.

THE SIMULATION OF 2D SPIN-1 ISING MODEL WITH POSITIVE BIQUADRATIC INTERACTION ON A CELLULAR AUTOMATON

2003 ◽  
Vol 14 (10) ◽  
pp. 1305-1320 ◽  
Author(s):  
BÜLENT KUTLU
Keyword(s):  
Phase Transition ◽  
Ising Model ◽  
Cellular Automaton ◽  
Intermediate Phase ◽  
Finite Size ◽  
Square Lattice ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Creutz Cellular Automaton ◽  
Antiferromagnetic Spin

The two-dimensional antiferromagnetic spin-1 Ising model with positive biquadratic interaction is simulated on a cellular automaton which based on the Creutz cellular automaton for square lattice. Phase diagrams characterizing phase transition of the model are presented for a comparison with those obtained from other calculations. We confirm the existence of the intermediate phase observed in previous works for some values of J/K and D/K. The values of the static critical exponents (β, γ and ν) are estimated within the framework of the finite-size scaling theory for D/K<2J/K. Although the results are compatible with the universal Ising critical behavior in the region of D/K<2J/K-4, the model does not exhibit any universal behavior in the interval 2J/K-4<D/K<2J/K.

The Finite-Size Scaling Study of the Specific Heat and the Binder Parameter of the Two-Dimensional Ising Model for the Fractals Obtained by Using the Model of Diffusion-Limited Aggregation

2009 ◽  
Vol 64 (12) ◽  
pp. 849-854 ◽  
Author(s):  
Ziya Merdan ◽  
Mehmet Bayirli ◽  
Mustafa Kemal Ozturk
Keyword(s):  
Critical Temperature ◽  
Ising Model ◽  
Specific Heat ◽  
Finite Size ◽  
Two Dimensional ◽  
Finite Size Scaling ◽  
Size Scaling ◽  
Infinite Lattice ◽  
Straight Line ◽  
Binder Parameter

The two-dimensional Ising model with nearest-neighbour pair interactions is simulated on the Creutz cellular automaton by using the finite-size lattices with the linear dimensions L = 80, 120, 160, and 200. The temperature variations and the finite-size scaling plots of the specific heat and the Binder parameter verify the theoretically predicted expression near the infinite lattice critical temperature. The approximate values for the critical temperature of the infinite lattice Tc = 2.287(6), Tc = 2.269(3), and Tc =2.271(1) are obtained from the intersection points of specific heat curves, Binder parameter curves, and the straight line fit of specific heat maxima, respectively. These results are in agreement with the theoretical value (Tc =2.269) within the error limits. The values obtained for the critical exponent of the specific heat, α = 0.04(25) and α = 0.03(1), are in agreement with α = 0 predicted by the theory. The values for the Binder parameter by using the finite-size lattices with the linear dimension L = 80, 120, 160, and 200 at Tc = 2.269(3) are calculated as gL(Tc) = −1.833(5), gL(Tc) = −1.834(3), gL(Tc) = −1.832(2), and gL(Tc) = −1.833(2), respectively. The value of the infinite lattice for the Binder parameter, gL(Tc) = −1.834(11), is obtained from the straight line fit of gL(Tc) = −1.833(5), gL(Tc) = −1.834(3), gL(Tc) = −1.832(2), and gL(Tc) = −1.833(2) versus L = 80, 120, 160, and 200, respectively

Sign in / Sign up

Export Citation Format

Share Document