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

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 $d=4\mathrm{}$.