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 $\theta$, 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 ${(h/J)}_c=1.00003\left(8\right)$ in the thermodynamic limit, and we estimate the critical initial slip exponent $\theta =0.3734\left(2\right)$ and the static exponent $\beta /\nu =0.1251\left(2\right)$ 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 $3.04451\left(7\right)$ in the thermodynamic limit, while the critical exponents are $\theta =0.209\left(4\right)$ and $\beta /\nu =0.518\left(1\right)$ 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.

• Received 18 May 2017
• Revised 21 August 2017

DOI:https://doi.org/10.1103/PhysRevB.96.094304