Phenomenological arguments on the critical slowing down is presented and “similarity law” is proposed on the indices of the critical slowing down. The similarity law is confirmed in linear spin chains near the critical field and in the kinetic Ising model near the critical temperature. It is exactly shown in the linear spin chains that the critical index of slowing down is different from that of the static susceptibility and that the dynamical susceptibility has a logarithmic singularity with respect to the frequency at the critical field and at zero temperature.Phenomenological arguments on the critical slowing down is presented and “similarity law” is proposed on the indices of the critical slowing down. The similarity law is confirmed in linear spin chains near the critical field and in the kinetic Ising model near the critical temperature. It is exactly shown in the linear spin chains that the critical index of slowing down is different from that of the static susceptibility and that the dynamical susceptibility has a logarithmic singularity with respect to the frequency at the critical field and at zero temperature.

Progress of Theoretical PhysicsApr 1, 197045 citations

Critical dynamics of the four-dimensional XY model

Lars Melwyn Jensen

Physica B Condensed MatterJul 1, 20005 citations

Inequalities for the exponents of critical slowing down in the two-dimensional one-spin-flip Ising model

T. Schneider

Exact lower bounds for the exponent of critical slowing down of the time-dependent order-parameter correlation and autocorrelation functions have been derived by Abe and Hatano and by Halperin, respectively. Moreover, Suzuki proved an exact inequality among the exponents characterizing the critical slowing down of the order parameter and energy fluctuations. In the present work we extend these results to include lower bounds for the exponents associated with the critical energy fluctuations and an inequality among the exponents characterizing the critical slowing down of the time-dependent autocorrelation functions. This is achieved by extending Mori's generalized Langevin-equation approach to the case of two variables, namely the order-parameter fluctuations and the energy fluctuations. The general results based on this approach hold for all Hermitian kinetic Ising models. Exact results are found revealing the failure of the extended conventional theory of slowing down. It is shown that this theory does not even hold in the mean-field approximation, which is expected to be exact at dimensionality $d=4$ .Exact lower bounds for the exponent of critical slowing down of the time-dependent order-parameter correlation and autocorrelation functions have been derived by Abe and Hatano and by Halperin, respectively. Moreover, Suzuki proved an exact inequality among the exponents characterizing the critical slowing down of the order parameter and energy fluctuations. In the present work we extend these results to include lower bounds for the exponents associated with the critical energy fluctuations and an inequality among the exponents characterizing the critical slowing down of the time-dependent autocorrelation functions. This is achieved by extending Mori's generalized Langevin-equation approach to the case of two variables, namely the order-parameter fluctuations and the energy fluctuations. The general results based on this approach hold for all Hermitian kinetic Ising models. Exact results are found revealing the failure of the extended conventional theory of slowing down. It is shown that this theory does not even hold in the mean-field approximation, which is expected to be exact at dimensionality $d=4$.

Physical review. B, Solid stateMay 1, 197418 citations

Critical properties of the XY model

Donald D. Betts

Physica B+CJan 1, 19779 citations

d=4