Finite-size scaling allows the extraction of a dimension from scaling ${\mathit{d}}_{\mathit{s}}$ , as well as critical exponents for systems with a second-order phase transition. The calculation of ${\mathit{d}}_{\mathit{s}}$ allows comparisons to be made with other expansions in dimension, in which ${\mathit{d}}_{\mathit{s}}$ plays the role of the physical dimension. This is demonstrated by a numerical transfer-matrix study of the pure ferromagnetic Ising model in two and three dimensions, as well as for quenched random bonds in the two-dimensional Ising ferromagnet.Finite-size scaling allows the extraction of a dimension from scaling ${\mathit{d}}_{\mathit{s}}$, as well as critical exponents for systems with a second-order phase transition. The calculation of ${\mathit{d}}_{\mathit{s}}$ allows comparisons to be made with other expansions in dimension, in which ${\mathit{d}}_{\mathit{s}}$ plays the role of the physical dimension. This is demonstrated by a numerical transfer-matrix study of the pure ferromagnetic Ising model in two and three dimensions, as well as for quenched random bonds in the two-dimensional Ising ferromagnet.

Physical Review LettersJan 11, 199323 citations

Bound states of the transfer matrix of the Ising model

Sh. S. Mamatov

Theoretical and Mathematical PhysicsFeb 1, 19840 citations

Transfer-matrix scaling for diluted Ising systems

S. L. A. de Queiroz, Robin B. Stinchcombe

A transfer-matrix scaling technique is developed for randomly diluted systems and applied to the site-diluted Ising model on a square lattice. For each connected configuration between adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of the two largest eigenvalues, allowing an economical incorporation of configurational averages. Predictions for the phase boundary at and near the percolation and pure Ising limits, and for the correlation exponent \ensuremath{\eta} at those limits, agree with exactly known results to within 1% error with largest strip widths of only L=5. The exponent \ensuremath{\eta} remains near the pure value (1/4) for all intermediate concentrations until it turns over to the percolation value at the threshold.A transfer-matrix scaling technique is developed for randomly diluted systems and applied to the site-diluted Ising model on a square lattice. For each connected configuration between adjacent columns, the contribution of the respective transfer matrix to the decay of correlations is considered only as far as the ratio of the two largest eigenvalues, allowing an economical incorporation of configurational averages. Predictions for the phase boundary at and near the percolation and pure Ising limits, and for the correlation exponent \ensuremath{\eta} at those limits, agree with exactly known results to within 1% error with largest strip widths of only L=5. The exponent \ensuremath{\eta} remains near the pure value (1/4) for all intermediate concentrations until it turns over to the percolation value at the threshold.

Physical review. B, Condensed matterSep 1, 199216 citations

Phase transition of the two-dimensional Ising model

Tomoyasu Tanaka

Cambridge University Press eBooksMar 14, 20020 citations

Finite-size scaling allows the extraction of a dimension from scaling

{\mathit{d}}_{\mathit{s}}

, as well as critical exponents for systems with a second-order phase transition. The calculation of

{\mathit{d}}_{\mathit{s}}

allows comparisons to be made with other expansions in dimension, in which

{\mathit{d}}_{\mathit{s}}

plays the role of the physical dimension. This is demonstrated by a numerical transfer-matrix study of the pure ferromagnetic Ising model in two and three dimensions, as well as for quenched random bonds in the two-dimensional Ising ferromagnet.