Rewriting the partition function of an Ising system as a multiple integral, dividing the integration space by a set of concentric spherical surfaces and sets of parallel planes into small domains, averaging separately the two factors constituting the integrand of the multiple integral over each of these domains, and replacing the two factors by their respective averages, logarithm of the partition function per spin is given by the extremum of a function with respect to a complex variable and a number of real variables. The specific heat curve of an Ising square lattice obtained by the present method is compared with those by the exact method, the spherical approximation and the mean field approximation.Rewriting the partition function of an Ising system as a multiple integral, dividing the integration space by a set of concentric spherical surfaces and sets of parallel planes into small domains, averaging separately the two factors constituting the integrand of the multiple integral over each of these domains, and replacing the two factors by their respective averages, logarithm of the partition function per spin is given by the extremum of a function with respect to a complex variable and a number of real variables. The specific heat curve of an Ising square lattice obtained by the present method is compared with those by the exact method, the spherical approximation and the mean field approximation.

お茶の水女子大學自然科學報告Dec 1, 19770 citations

Microscopic critical dynamics of one-dimensional spin systems

George Reiter

We present an exact solution for the spin-spin correlation function of the dynamical spherical model (the spherical-model equilibrium state together with the classical-Heisenberg-model equations of motion) in one dimension at $T=0$ . The physical picture of the system is clarified, as is the difference between the Heisenberg and dynamical spherical model arising from the presence of the constraint ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}=1$ in the former model. The effect of the higher-order correlations, induced by the constraint, on the dynamics is made explicit. The solution satisfies the dynamical-scaling hypothesis. The mode-coupling approximation, although it gives the correct dynamical exponents, yields a qualitatively incorrect spectral function. It cannot be improved by including higher-order terms in the expansion of the self energy in terms of the number of interactions (the extended-mode-coupling theory), as the higher-order terms yield physically unrealistic solutions. A qualitatively correct approximate solution has been found, that exhibits both the central peak and the spin wave sidebands of the exact solution. The solution cannot be improved by including higher-order terms either. There does not appear to be any way of constructing a series of approximations from the leading terms in a renormalized-perturbation-series expansion that will converge to the exact solution.We present an exact solution for the spin-spin correlation function of the dynamical spherical model (the spherical-model equilibrium state together with the classical-Heisenberg-model equations of motion) in one dimension at $T=0$. The physical picture of the system is clarified, as is the difference between the Heisenberg and dynamical spherical model arising from the presence of the constraint ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}=1$ in the former model. The effect of the higher-order correlations, induced by the constraint, on the dynamics is made explicit. The solution satisfies the dynamical-scaling hypothesis. The mode-coupling approximation, although it gives the correct dynamical exponents, yields a qualitatively incorrect spectral function. It cannot be improved by including higher-order terms in the expansion of the self energy in terms of the number of interactions (the extended-mode-coupling theory), as the higher-order terms yield physically unrealistic solutions. A qualitatively correct approximate solution has been found, that exhibits both the central peak and the spin wave sidebands of the exact solution. The solution cannot be improved by including higher-order terms either. There does not appear to be any way of constructing a series of approximations from the leading terms in a renormalized-perturbation-series expansion that will converge to the exact solution.

Physical review. B, Condensed matterFeb 1, 19795 citations

Constant coupling approximation for Ising spin systems

P.W. Kasteleijn

PhysicaJan 1, 195645 citations

New cluster approximation for Ising systems

Ümit Akıncı

Journal of Magnetism and Magnetic MaterialsDec 19, 20191 citation

T=0

. The physical picture of the system is clarified, as is the difference between the Heisenberg and dynamical spherical model arising from the presence of the constraint ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}=1$ in the former model. The effect of the higher-order correlations, induced by the constraint, on the dynamics is made explicit. The solution satisfies the dynamical-scaling hypothesis. The mode-coupling approximation, although it gives the correct dynamical exponents, yields a qualitatively incorrect spectral function. It cannot be improved by including higher-order terms in the expansion of the self energy in terms of the number of interactions (the extended-mode-coupling theory), as the higher-order terms yield physically unrealistic solutions. A qualitatively correct approximate solution has been found, that exhibits both the central peak and the spin wave sidebands of the exact solution. The solution cannot be improved by including higher-order terms either. There does not appear to be any way of constructing a series of approximations from the leading terms in a renormalized-perturbation-series expansion that will converge to the exact solution.