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 ${(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)$ 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 $3.044\phantom{\rule{0.16em}{0ex}}51(7)$ 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 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 ${(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(8)$ 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 $3.044\phantom{\rule{0.16em}{0ex}}51(7)$ 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.

arXiv:1705.05931Physical review. B./Physical review. BSep 12, 201717 citations

Dynamical relaxation and universal short-time behavior in finite systems, The renormalization-group approach

Uwe Ritschel, Hans Werner Diehl, Fachbereich Physik +1 more

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.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.

Jan 1, 199611 citations

Relaxation theory of a strongly coupled system

Mio Murao, Fumiaki Shibata

Physica A Statistical Mechanics and its ApplicationsJun 1, 199510 citations

{(h/J)}_{c}=1.000\phantom{\rule{0.16em}{0ex}}03(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

3.044\phantom{\rule{0.16em}{0ex}}51(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.