The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.

Progress of Theoretical Physics SupplementJan 1, 19868 citations

On dynamic scaling in the Heisenberg ferromagnet

Sergei V. Maleev

Physics Letters ADec 1, 19710 citations

Finite size scaling approach to a metamagnetic model in two dimensions

Hans Jürgen Herrmann

Physics Letters AJan 1, 198425 citations

Exact results for a finite-sized spherical model of ferromagnetism at the borderline dimensionality 4

Surjit Singh, Raj K. Pathria

We report exact results on the zero-field susceptibility \ensuremath{\chi}(T;L), the correlation length \ensuremath{\xi}(T;L), and the ``singular'' part of the specific heat ${\mathrm{c}}^{(\mathrm{s})}$ (T;L) of a finite-sized spherical model of ferromagnetism subjected to periodic boundary conditions. We take the total dimensionality of the system d to be 4 and deal with the geometry ${\mathrm{L}}^{4\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\ensuremath{'}}$ (d\ensuremath{'}\ensuremath{\leqslant}2). In the region of first-order phase transition (T ${\mathrm{T}}_{\mathrm{c}}$ ), our results are formally the same as in other cases with d>2. The ``core'' region (T\ensuremath{\simeq} ${\mathrm{T}}_{\mathrm{c}}$ ), however, is characterized by the appearance of factors involving lnL, which appear only when d=4. The relationship between these results and the corresponding ones following from the hyperscaling regime as d\ensuremath{\rightarrow}4- or from the mean-field regime as d\ensuremath{\rightarrow}4+ is explored, and a formulation in terms of the finite-size scaling theory is presented.We report exact results on the zero-field susceptibility \ensuremath{\chi}(T;L), the correlation length \ensuremath{\xi}(T;L), and the ``singular'' part of the specific heat ${\mathrm{c}}^{(\mathrm{s})}$ (T;L) of a finite-sized spherical model of ferromagnetism subjected to periodic boundary conditions. We take the total dimensionality of the system d to be 4 and deal with the geometry ${\mathrm{L}}^{4\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\ensuremath{'}}$ (d\ensuremath{'}\ensuremath{\leqslant}2). In the region of first-order phase transition (T${\mathrm{T}}_{\mathrm{c}}$), our results are formally the same as in other cases with d>2. The ``core'' region (T\ensuremath{\simeq}${\mathrm{T}}_{\mathrm{c}}$), however, is characterized by the appearance of factors involving lnL, which appear only when d=4. The relationship between these results and the corresponding ones following from the hyperscaling regime as d\ensuremath{\rightarrow}4- or from the mean-field regime as d\ensuremath{\rightarrow}4+ is explored, and a formulation in terms of the finite-size scaling theory is presented.

Physical review. B, Condensed matterMay 1, 199211 citations

First-order transitions breaking O( n ) symmetry: Finite-size scaling

Michael Ellis Fisher, Vladimir Privman A

The finite-size rounding of a first-order transition is studied in systems representable as n-vector ferromagnets, so that O(n) symmetry (n\ensuremath{\ge}2) is broken at the bulk transition point. Both ``block,'' V= ${L}^{d}$ , and ``cylinder,'' ${L}^{d\mathrm{\ensuremath{-}}1}$\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}, geometries are considered for general dimensionality d. Explicit expressions are obtained for the scaling functions describing the rounded transitions and the crossover in shape. Spin-wave effects are shown to be of relative order 1/${L}^{d\mathrm{\ensuremath{-}}2}$, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of N\'eel's phenomenological theory of superparamagnetism. The analysis for cylinders involves the formulation of a ``degeneracy kernel'' to describe the asymptotic rounding of first-order transitions and establishes a general relation between the helicity modulus (or ``spin-wave stiffness'' or ``superfluid density'') and the transfer operator spectrum. The relationship to finite-size scaling in the critical region is examined with emphasis on the extra scaling combination, ${\mathrm{Vt}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$, that is needed for d> ${d}_{>}$ =4. All the results found can be checked in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} against exact results for spherical models (described elsewhere).The finite-size rounding of a first-order transition is studied in systems representable as n-vector ferromagnets, so that O(n) symmetry (n\ensuremath{\ge}2) is broken at the bulk transition point. Both ``block,'' V=${L}^{d}$, and ``cylinder,'' ${L}^{d\mathrm{\ensuremath{-}}1}$\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}, geometries are considered for general dimensionality d. Explicit expressions are obtained for the scaling functions describing the rounded transitions and the crossover in shape. Spin-wave effects are shown to be of relative order 1/${L}^{d\mathrm{\ensuremath{-}}2}$, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of N\'eel's phenomenological theory of superparamagnetism. The analysis for cylinders involves the formulation of a ``degeneracy kernel'' to describe the asymptotic rounding of first-order transitions and establishes a general relation between the helicity modulus (or ``spin-wave stiffness'' or ``superfluid density'') and the transfer operator spectrum. The relationship to finite-size scaling in the critical region is examined with emphasis on the extra scaling combination, ${\mathrm{Vt}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$, that is needed for d>${d}_{>}$=4. All the results found can be checked in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} against exact results for spherical models (described elsewhere).

Physical review. B, Condensed matterJul 1, 1985179 citations

We report exact results on the zero-field susceptibility \ensuremath{\chi}(T;L), the correlation length \ensuremath{\xi}(T;L), and the ``singular'' part of the specific heat

{\mathrm{c}}^{(\mathrm{s})}

(T;L) of a finite-sized spherical model of ferromagnetism subjected to periodic boundary conditions. We take the total dimensionality of the system d to be 4 and deal with the geometry ${\mathrm{L}}^{4\mathrm{\ensuremath{-}}\mathrm{d}\ensuremath{'}}$\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{\mathrm{d}\ensuremath{'}}$ (d\ensuremath{'}\ensuremath{\leqslant}2). In the region of first-order phase transition (T

{\mathrm{T}}_{\mathrm{c}}

), our results are formally the same as in other cases with d>2. The ``core'' region (T\ensuremath{\simeq}

{\mathrm{T}}_{\mathrm{c}}

), however, is characterized by the appearance of factors involving lnL, which appear only when d=4. The relationship between these results and the corresponding ones following from the hyperscaling regime as d\ensuremath{\rightarrow}4- or from the mean-field regime as d\ensuremath{\rightarrow}4+ is explored, and a formulation in terms of the finite-size scaling theory is presented.

{L}^{d}

, and ``cylinder,'' ${L}^{d\mathrm{\ensuremath{-}}1}$\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}, geometries are considered for general dimensionality d. Explicit expressions are obtained for the scaling functions describing the rounded transitions and the crossover in shape. Spin-wave effects are shown to be of relative order 1/${L}^{d\mathrm{\ensuremath{-}}2}$, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of N\'eel's phenomenological theory of superparamagnetism. The analysis for cylinders involves the formulation of a ``degeneracy kernel'' to describe the asymptotic rounding of first-order transitions and establishes a general relation between the helicity modulus (or ``spin-wave stiffness'' or ``superfluid density'') and the transfer operator spectrum. The relationship to finite-size scaling in the critical region is examined with emphasis on the extra scaling combination, ${\mathrm{Vt}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$, that is needed for d>

{d}_{>}

=4. All the results found can be checked in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} against exact results for spherical models (described elsewhere).