Renormalization-group methods provide a viable approach for investigating the emergent collective behavior of classical and quantum statistical systems in both equilibrium and nonequilibrium conditions. Within this approach we investigate here the dynamics of an isolated quantum system represented by a scalar ${\ensuremath{\phi}}^{4}$ theory after a global quench of the potential close to a dynamical critical point. We demonstrate that, within a prethermal regime, the time dependence of the relevant correlations is characterized by a short-time universal exponent, which we calculate at the lowest order in a dimensional expansion.
The influence of nonequilibrium initial values of the order parameter on its evolution at a critical point is described using a renormalization group approach of the field theory. The dynamic critical exponent θ of the short time evolution of a system with an n -component order parameter is calculated within a dynamical dissipative model using the method of Σ-expansion in a three-loop approximation. Numerical values of θ for three-dimensional systems are determined using the Padé-Borel method for the summation of asymptotic series.
The influence of nonequilibrium initial states on the evolution of anisotropic systems with quenched uncorrelated structural defects at the critical point is studied. The field-theoretical description of the nonequilibrium critical behavior of 3D systems is obtained for the first time, and the dynamic critical exponent of the short-time evolution in the two-loop approximation without the use of ɛ expansion is calculated. The values of dynamic critical exponents calculated using the series resummation methods are compared with the results of computer simulation of nonequilibrium critical behavior of the 3D disordered Ising model in the short-time regime. It is demonstrated that the values of the critical exponents calculated in this paper are in better agreement with the results of computer simulation than the results of application of ɛ expansion.
We investigate the short-time quantum critical dynamics in the imaginary-time relaxation processes of finite-size systems. Universal scaling behaviors exist in the imaginary-time evolution. In particular, the system undergoes a critical initial slip stage characterized by an exponent $\ensuremath{\theta}$, in which an initial power-law increase emerges in the imaginary-time correlation function when the initial state has a zero order parameter and a vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo (QMC) simulations. In the one-dimensional case, we locate the quantum critical point at
(h/J)c=1.00003(8) in the thermodynamic limit, and we estimate the critical initial slip exponent $\ensuremath{\theta}=0.3734(2)$ and the static exponent $\ensuremath{\beta}/\ensuremath{\nu}=0.1251(2)$ by analyzing data on chains of length
L=32−−256 and 48--256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by
3.04451(7) in the thermodynamic limit, while the critical exponents are $\ensuremath{\theta}=0.209(4)$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.518(1)$ estimated by data on systems of size
L=24--64 and 32--64, respectively. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts due to the essential differences between classical and quantum dynamics. The short-time critical dynamics and the imaginary-time relaxation QMC approach can be readily adapted to various models.
We study the short-time dynamics of systems that develop quasi-long-range order after a quench to the Kosterlitz-Thouless phase. With the working hypothesis that the universal short-time behavior, previously found in Ising-like systems, also occurs in the Kosterlitz-Thouless phase, we explore the scaling behavior of thermodynamic variables during the relaxational process following the quench. As a concrete example, we investigate the two-dimensional six-state clock model by Monte Carlo simulation. The exponents governing the magnetization, the second moment, and the autocorrelation function are calculated. From them, by means of scaling relations, estimates for the equilibrium exponents
z and $\ensuremath{\eta}$ are derived. In particular, our estimates for the temperature-dependent anomalous dimension $\ensuremath{\eta}$ that governs the static correlation function are consistent with existing analytical and numerical results, and, thus, confirm our working hypothesis.
We study how the finite-sized n-component model A with periodic boundary conditions relaxes near its bulk critical point from an initial nonequilibrium state with short-range correlations. Particular attention is paid to the universal long-time traces that the initial condition leaves. An approach based on renormalization-group improved perturbation theory in 4−ǫ space dimensions and a nonperturbative treatment of the q = 0 mode of the fluctuating orderparameter field is developed. This leads to a renormalized effective stochastic equation for this mode in the background of the other, q = 0 modes; we explicitly derive it to one-loop order, show that it takes the expected finitesize scaling form at the fixed point, and solve it numerically. Our results confirm for general n that the amplitude of the magnetization density m(t) in the linear relaxation-time regime depends on the initial magnetization in the universal fashion originally found in our large-n analysis [J. Stat. Phys. 73 (1993) 1]. The anomalous short-time power-law increase of m(t) also is recovered. For n = 1, our results are in fair agreement with recent Monte Carlo simulations by Li, Ritschel, and Zheng [J. Phys. A 27 (1994) L837] for the three-dimensional Ising model.