The Ising model provides a fundamental setting for investigating the emergence of phase transitions from simple interacting degrees of freedom. The current characterization study serves to investigate central requirements for phase transitions in terms of connectivity, i.e., the degree of coupled interactions between interaction sites. The impact of increasing connectivity between 1D-Ising chains mapped onto 2D-lattices with free boundary conditions were studied systematically, using exact free energy calculations. Starting from a reference system of non-interacting 1D-chains, interaction bonds between chains are introduced successively until the fully connected N×N-lattice is obtained. Two distinct construction schemes are analyzed, which differ in the connectivity of the intermediate partially coupled systems. The resulting free energies of the graphs along these paths are evaluated and compared with respect to their convergence behavior as a function of system size. We find that, despite topological differences between the schemes, strikingly, they converge to the same limiting straight line for increasing N when analyzed in terms of residual free energy differences. These findings provide insight into the relationship between interaction structure and thermodynamic behavior and suggest that appropriately chosen construction paths may serve as a basis for efficient extrapolation strategies toward the thermodynamic limit.
The Ising model provides a fundamental setting for investigating the emergence of phase transitions from simple interacting degrees of freedom. The current characterization study serves to investigate central requirements for phase transitions in terms of connectivity, i.e., the degree of coupled interactions between interaction sites. The impact of increasing connectivity between 1D-Ising chains mapped onto 2D-lattices with free boundary conditions were studied systematically, using exact free energy calculations. Starting from a reference system of non-interacting 1D-chains, interaction bonds between chains are introduced successively until the fully connected N×N-lattice is obtained. Two distinct construction schemes are analyzed, which differ in the connectivity of the intermediate partially coupled systems. The resulting free energies of the graphs along these paths are evaluated and compared with respect to their convergence behavior as a function of system size. We find that, despite topological differences between the schemes, strikingly, they converge to the same limiting straight line for increasing N when analyzed in terms of residual free energy differences. These findings provide insight into the relationship between interaction structure and thermodynamic behavior and suggest that appropriately chosen construction paths may serve as a basis for efficient extrapolation strategies toward the thermodynamic limit.
The fact that the Ising model in higher dimensions than 1D features a phase transition at the critical temperature Tc despite its apparent simplicity is one of the main reasons why it has lost none of its fascination and remains a central benchmark in modeling physical systems. Building on our previous work, where an approximative analytic free-energy expression for finite 2D-Ising lattices was introduced, we investigate different extrapolation strategies for estimating Tc of the infinite system from exactly solvable small lattices. Finite square lattices of linear dimension N with free and periodic boundary conditions were analyzed, exploiting their exactly accessible density of states to compute the heat capacity profiles C(T). Different approaches were compared, including scaling models for the peak temperature Tmax(N) and an envelope construction across the set of C(T)-profiles. We find that both approaches converge to the same asymptotic value and compare favorably to the established Binder cumulant method. Remarkably, a model for Tmax with a single model parameter following an N/(N+1)-law provides robust convergence, with a physical analogy motivating this proportionality. Our findings highlight that surprisingly few, but highly accurate, finite-size results are sufficient to obtain a precise extrapolation.
The fact that the Ising model in higher dimensions than 1D features a phase transition at the critical temperature Tcdespite its apparent simplicity is one of the main reasons why it has lost none of its fascination and remains a central benchmark in modeling physical systems. Building on our previous work, where an approximative analytic free-energy expression for finite 2D-Ising lattices was introduced, we investigate different extrapolation strategies for estimating Tcof the infinite system from exactly solvable small lattices. Finite square lattices of linear dimension N with free and periodic boundary conditions were analyzed, exploiting their exactly accessible density of states to compute the heat capacity profiles C(T). Different approaches were compared, including scaling models for the peak temperature Tmax(N)and an envelope construction across the set of C(T)-profiles. We find that both approaches converge to the same asymptotic value and compare favorably to the established Binder cumulant method. Remarkably, a model for Tmaxwith a single model parameter following an N/(N+1)-law provides robust convergence, with a physical analogy motivating this proportionality. Our findings highlight that surprisingly few, but highly accurate, finite-size results are sufficient to obtain a precise extrapolation.
For generalized 2D Ising model in an external magnetic field with the interaction of nearest neighbors, next nearest neighbors, all kinds of triple interactions and the quadruple interaction the formulas for finding free energy per lattice site in the thermodynamic limit were derived on a certain set of exact disordered solutions depending on seven parameters. Lattice models are considered with boundary conditions with a shift (similar to helical ones), and with cyclic closure of the set of all points in natural ordering. The elementary transfer matrix with nonnegative matrix elements are constructed. On the set of disorder solutions the largest eigenvalue of the transfer matrix is constant for every size of considering planar lattice, and, in particular, in the thermodynamic limit. Free energy per lattice site in the thermodynamic limit is expressed through the natural logarithm of the largest eigenvalue of transfer matrix. This largest eigenvalue can be found for a special form of eigenvector with positive components. The numerical example show the existence of nontrivial solutions of the resulting systems of equations. The system of equations and the value of free energy in the thermodynamic limit will remain the same for 2D generalized Ising models with Hamiltonians, in which the values of two (out of four) neighboring maximal spins in the natural ordering are replaced by the values of the spins at any other two lattice points adjacent in the natural ordering, this significantly expands the set of models having disordered exact solutions. The high degree of symmetry and inductive construction of the components of the eigenvectors, which disappear when going beyond the framework of the obtained set of exact solutions, is an occasion to search for phase transitions in the vicinity of this set of disordered solutions.
We perform a comprehensive numerical study of how variations in microscopic coupling strength affect critical behavior in the two-dimensional Ising model. Using extensive Monte Carlo simulations based on the Wolff cluster algorithm, we investigate the clean Ising model on a square lattice for interaction strengths in the range 0.80 ≤ J ≤ 1.20. While the exact solution at J = 1 has long established the model’s critical properties, a systematic numerical assessment of universality under continuous coupling rescaling has not been performed. Here, we address this gap by analyzing systems of linear size L = 20 − 500, allowing for high-precision finite-size scaling of thermodynamic observables. For each coupling, the critical temperature T_C(J) is determined via finite-size extrapolation of the susceptibility maxima, yielding results in quantitative agreement with Onsager’s exact prediction. At the extrapolated critical points, we extract the critical exponents ν, β, γ, and α through detailed scaling analyses. These exponents remain consistent with the exact Ising benchmarks across the full range of interaction strengths, confirming that the universal aspects of the phase transition are preserved. Our findings demonstrate that modifying the microscopic coupling strength rescales the critical temperature without altering the asymptotic critical behavior, thereby providing a clear and quantitative confirmation of the robustness of the two-dimensional Ising universality class.
We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy ΔF(T) and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, ΔF(T) at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, ΔF(T) decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .
For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent 2−2≈0.5858, comparable to the exponent 4/7≈0.5714 found from renormalization arguments.