Real-time dynamics of open quantum spin systems driven by dissipative processes

We study the real-time evolution of large open quantum spin systems in two spatial dimensions, whose dynamics is entirely driven by a dissipative coupling to the environment. We consider different dissipative processes and investigate the real-time evolution from an ordered phase of the Heisenberg or XY model towards a disordered phase at late times, disregarding unitary Hamiltonian dynamics. The corresponding Kossakowski-Lindblad equation is solved via an efficient cluster algorithm. We find that the symmetry of the dissipative process determines the time scales, which govern the approach towards a new equilibrium phase at late times. Most notably, we find a slow equilibration if the dissipative process conserves any of the magnetization Fourier modes. In these cases, the dynamics can be interpreted as a diffusion process of the conserved quantity.

Figure 1

Visualization of the discretization of the physical system. The oval-shaped objects denote the nearest-neighbor spin pairs, which may interact at discretization step $1-4$, respectively.

Figure 2

(Heisenberg antiferromagnet initial state) Time evolution of certain Fourier modes $\langle |S\left(p\right){|}^2\rangle \left(t\right)$ for the measurement process $O^{\left(1\right)}$. (Left) Linear plot for a short-time interval, cf. also Fig. 1 in Ref. [56]. (Right) Log-log plot for a long time interval. The error bars are of the order of the symbol sizes and the lines are included to guide the eye. We initialize an antiferromagnetic Heisenberg model at low temperatures such that the $(\pi ,\pi )$ mode is large whereas the $(0,0)$ mode vanishes. The remaining parameters are $4N_{\tau }=512,L=16a,\beta J=5L/2a=40$, and $\epsilon \gamma =0.05$. In order to determine the time evolution, we performed ${10}^6$ Monte Carlo measurements. The horizontal line corresponds to the analytically derived final equilibrium value (61b). We emphasize that the $(0,0)$ mode is exactly conserved during the time evolution.

Figure 3

(Heisenberg antiferromagnet initial state) Time evolution of certain Fourier modes $\langle |S\left(p\right){|}^2\rangle \left(t\right)$ for the measurement process $O^{\left(2\right)}$ on a linear plot. The error bars are again of the order of the symbol sizes and the lines are included to guide the eye. The parameters are as in Fig. 2 and the horizontal line corresponds to the analytically derived final equilibrium value (63). We emphasize that all Fourier modes equilibrate very rapidly.

Figure 4

(Heisenberg antiferromagnet initial state) Time evolution of certain Fourier modes $\langle |S\left(p\right){|}^2\rangle \left(t\right)$ for the measurement process $O^{\left(3\right)}$. (Left) Linear plot for a short-time interval. (Right) Log-log plot for a long time interval. The error bars are again of the order of the symbol sizes and the lines are included to guide the eye. The parameters are as in Fig. 2 and the horizontal line corresponds to the analytically derived final equilibrium value (71b). We emphasize that the $(\pi ,\pi )$ mode is exactly conserved during the time evolution.

Figure 5

(Heisenberg antiferromagnet initial state) Time evolution of certain Fourier modes $\langle |S\left(p\right){|}^2\rangle \left(t\right)$ for the measurement processes $O^{\left(1\right)}$ (left) and $O^{\left(3\right)}$ (right). The error bars are of the order of the symbol sizes. The solid lines correspond to a simultaneous fit of the twelve slowest Fourier modes upon taking into account the analytically predicted attractor behavior. The parameters are $4N_{\tau }=512,L=16a,\beta J=5L/2a=40,\epsilon \gamma =0.05$ and we performed $4\times {10}^6$ Monte Carlo measurements.

Figure 6

(Heisenberg antiferromagnet initial state) Momentum dependence of the attraction rate $1/T\left(p\right)$ for the measurement processes $O^{\left(1\right)}$ and $O^{\left(2\right)}$ (left) as well as $O^{\left(3\right)}$ and $O^{\left(2\right)}$ (right). The data points for the measurement processes $O^{\left(1\right)}$ and $O^{\left(3\right)}$, respectively, are calculated by a simultaneous fit of the twelve slowest Fourier modes upon taking into account the analytically predicted attractor behavior. On the other hand, we performed independent exponential fits for the measurement process $O^{\left(2\right)}$. The parameters are as in Fig. 5.

Figure 7

(Heisenberg antiferromagnet initial state) Time-dependent Binder ratio $B_4\left(t\right)$ for the measurement process $O^{\left(2\right)}$ for different values of $L/a$ along with variable $\beta J=3L/4a$. The error bars are of the order of the symbol sizes and the lines are included to guide the eye. We emphasize that the results for the measurement process $O^{\left(1\right)}$ are essentially identical, cf. also Fig. 2 in Ref. [56].

Figure 8

(Heisenberg antiferromagnet initial state) (Left) Time-dependent staggered magnetization density ${\mathcal{M}}_s\left(t\right)/{\mathcal{M}}_s\left(0\right)$ for the measurement process $O^{\left(2\right)}$ on a logarithmic plot along with an exponential fit of the data. (Right) Time-dependent length scale $\xi \left(t\right)/\xi \left(0\right)$ for the measurement process $O^{\left(2\right)}$ on a linear plot with the lines included to guide the eye. In both cases, the error bars are of the order of the symbol sizes. We again emphasize that the results for the measurement process $O^{\left(1\right)}$ are essentially identical, cf. also Figs. 2 and 2 in Ref. [56].

Figure 9

(Heisenberg ferromagnet initial state) Time evolution of certain Fourier modes $\langle |S\left(p\right){|}^2\rangle \left(t\right)$ for a short time interval for the measurement processes $O^{\left(2\right)}$ (left) and $O^{\left(3\right)}$ (right). The error bars are again of the order of the symbol sizes and the lines are included to guide the eye. We initialize a ferromagnetic Heisenberg model at low temperatures such that the $(0,0)$ mode is very large whereas the $(\pi ,\pi )$ mode is small. The remaining parameters are $4N_{\tau }=512,L=16a,\beta J=5L/2a=40$, and $\epsilon \gamma =0.05$. In order to determine the time evolution, we performed ${10}^6$ Monte Carlo measurements. The horizontal lines corresponds to the analytically derived final equilibrium values (B4) and (B6b), respectively.

Figure 10

(XY model initial state) Time evolution of certain Fourier modes $\langle |S\left(p\right){|}^2\rangle \left(t\right)$ for a long time interval for the measurement processes $O^{\left(1\right)}$ (left) and $O^{\left(3\right)}$ (right). The error bars are of the order of the line width. We initialize the XY model at low temperatures such that the $(0,0)$ mode is large, whereas the $(\pi ,\pi )$ mode is small. The remaining parameters are $4N_{\tau }=512,L=16a,\beta J=5L/2a=40$, and $\epsilon \gamma =0.05$. In order to determine the time evolution, we performed ${10}^6$ Monte Carlo measurements. The horizontal lines corresponds to the final equilibrium values (C4b) and (C5b), respectively.

Figure 11

(XY-model initial state) (Left) Time-dependent Binder ratio $B_4\left(t\right)$ for the measurement process $O^{\left(3\right)}$ for different values of $L/a$ along with variable $\beta J=3L/4a$. The lines are included to guide the eye. (Right) Time-dependent magnetization density $\mathcal{M}\left(t\right)/\mathcal{M}\left(0\right)$ for the measurement process $O^{\left(3\right)}$ on a logarithmic plot along with an exponential fit of the data. In both cases, the error bars are of the order of the symbol sizes. We emphasize that the results for the measurement process $O^{\left(2\right)}$ are essentially identical.