We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization $M(H,T)$ . The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of $M(H,T)$ converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for $H=O$ , which reproduces exactly the critical-point behavior of the full Ising model.We present the results of numerical calculations giving accurate estimates of the magnetization of the two-dimensional Ising model on a square lattice. Moreover, we argue that these results are strict lower bounds to the correct magnetization $M(H,T)$. The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of $M(H,T)$ converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for $H=O$, which reproduces exactly the critical-point behavior of the full Ising model.

Physical review. B, Solid stateOct 1, 197011 citations

Monte Carlo studies of critical phenomena in spin lattice systems

Akai Kurbanovich Murtazaev

Uspekhi Fizicheskih NaukJan 1, 20069 citations

Nonuniversal and surface critical behavior near a defect plane

Ferenc Iglói

The existence of nonuniversal, as well as surface (universal) critical behavior near a defect plane is demonstrated on a one-dimensional quantum lattice gas model, belonging to the two-dimensional Ising universality class. According to numerical results the weak defect region is attracted by a line of fixed points starting at the bulk critical point, while the strong defect region is controlled by a separate fixed point describing the surface transition of the model. This new kind of phase diagram is argued to be general for two-dimensional lattice gas models.The existence of nonuniversal, as well as surface (universal) critical behavior near a defect plane is demonstrated on a one-dimensional quantum lattice gas model, belonging to the two-dimensional Ising universality class. According to numerical results the weak defect region is attracted by a line of fixed points starting at the bulk critical point, while the strong defect region is controlled by a separate fixed point describing the surface transition of the model. This new kind of phase diagram is argued to be general for two-dimensional lattice gas models.

Physical review. B, Condensed matterSep 1, 19893 citations

Critical properties of a one-dimensional interacting Fermi system in a magnetic field

V.I. Ponomarev

Physics Letters AApr 1, 19841 citation

Critical Behavior in Random Magnetic Systems

Alfred Nash

Jan 1, 19910 citations

M(H,T)

. The estimates are obtained by dividing the infinite lattice into finite strips of width between two and nine spins and infinite length. The largest eigenvalue and the corresponding eigenvector of the transfer matrix are then obtained by an iterative process. The estimates of

M(H,T)

converge to the correct answer for the infinite lattice everywhere except for a small region in the $T\ensuremath{-}H$ plane. We also compute isotherms and critical isobar for the corresponding lattice gas. Finally, we propose a new approximation to the transfer matrix, exactly solvable in two dimensions for

H=O

, which reproduces exactly the critical-point behavior of the full Ising model.