We study the imaginary-time relaxation critical dynamics of a quantum system with a vanishing initial correlation length and an arbitrary initial order parameter M0. We find that in quantum critical dynamics, the behavior of M0 under scale transformations deviates from a simple power law, which was proposed for very small M0 previously. A universal characteristic function is then suggested to describe the rescaled initial magnetization, similar to classical critical dynamics. This characteristic function is shown to be able to describe the quantum critical dynamics in both short- and long-time stages of the evolution. The one-dimensional transverse-field Ising model is employed to numerically determine the specific form of the characteristic function. We demonstrate that it is applicable as long as the system is in the vicinity of the quantum critical point. The universality of the characteristic function is confirmed by numerical simulations of models belonging to the same universality class.
A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing
l spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of
r, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.