Performance evaluation of the discrete truncated Wigner approximation for quench dynamics of quantum spin systems with long-range interactions

The discrete truncated Wigner approximation (DTWA) is a powerful tool for analyzing dynamics of quantum spin systems. Since the DTWA includes the leading-order quantum corrections to a mean-field approximation, it is naturally expected that the DTWA becomes more accurate when the range of interactions of the system increases. However, quantitative corroboration of this expectation is still lacking mainly because it is generally difficult in a large system to evaluate a timescale on which the DTWA is quantitatively valid. In order to investigate how the validity timescale depends on the interaction range, we analyze dynamics of quantum spin models with a step function type interaction subjected to a sudden quench of a magnetic field by means of both DTWA and its extension including the second-order correction, which is derived from the Bogoliubov-Born-Green-Kirkwood-Yvon equation. We also develop a formulation for calculating the second-order Rényi entropy within the framework of the DTWA. By comparing the time evolution of the Rényi entropy computed by the DTWA with that by the extension including the correction, we find that both in the one- and two-dimensional systems the validity timescale increases algebraically with the range of the step function type interaction.

Figure 1

Comparison with the DTWA and tDMRG results. Mean two-site Rényi entropy in 1D. (a) Ising model for $h^x=0.5J$. (b) XY model for $h^x=2.0J$. (c) Heisenberg model for $h^x=10.0J$. The black solid and dotted lines represent the tDMRG results. The open and closed symbols represent the first- and second-order BBGKY results, respectively. $r$ is the interaction range.

Figure 2

Time evolution after the sudden quench of the mean two-site Rényi entropy in 1D and 2D. (a), (b) Ising model for $h^x=0.5J$. (c), (d) XY model for $h^x=2.0J$. (e), (f) Heisenberg model for $h^x=10.0J$. The open and closed symbols represent the first- and second-order BBGKY results, respectively. $r$ denotes the interaction range.

Figure 3

Threshold time as a function of number of interacting sites for the Ising model. (a) One dimension. The black solid and dashed lines represent $N_r^{0.5}$ and $N_r^{0.75}$ for guide to eye. (b) Two dimensions. The black solid and dashed lines represent $N_r^{0.4}$ and $N_r^{0.6}$ for guide to eye.

Figure 4

Threshold time as a function of the number of interacting sites for the XY model. (a) One dimension. The black solid and dashed lines represent $N_r^{0.5}$ and $N_r^{0.3}$ for guide to eye. (b) Two dimensions. The black solid and dashed lines represent $N_r^{0.5}$ and $N_r^{0.3}$ for guide to eye.

Figure 5

Threshold time as a function of the number of interacting sites for the Heisenberg model. (a) One dimension. The black solid and dashed lines represent $N_r^{0.5}$ and $N_r^{0.8}$ for guide to eye. (b) Two dimensions. The black solid and dashed lines represent $N_r^{0.5}$ and $N_r^{1.0}$ for guide to eye.

Figure 6

Comparison with the threshold time for Ising model in 1D and 2D. (a) Ising model. (b) XY model. (c) Heisenberg model. Open and closed symbols represent 1D and 2D results, respectively.

Figure 7

One example for obtaining two candidates of the arrangement of site indices. Both candidates can be obtained by four swap operations from the present arrangement.