Gutzwiller Monte Carlo approach for a critical dissipative spin model

We use the Gutzwiller Monte Carlo approach to simulate the dissipative $XYZ$ model in the vicinity of a dissipative phase transition. This approach captures classical spatial correlations together with the full on-site quantum behavior while neglecting nonlocal quantum effects. By considering finite two-dimensional lattices of various sizes, we identify a ferromagnetic and two paramagnetic phases, in agreement with earlier studies. The greatly reduced numerical complexity of the Gutzwiller Monte Carlo approach facilitates efficient simulation of relatively large lattice sizes. The inclusion of the spatial correlations allows to capture parts of the phase diagram that are completely missed by the widely applied Gutzwiller decoupling of the density matrix.

Figure 1

The magnetization in the $x$ direction (5) as a function of time on a $6\times 6$ lattice during individual trajectories for four representative values of $J_y$: (a) $\gamma$, (b) $1.2\gamma$, (c) $1.8\gamma$, and (d) $2.5\gamma$. Initially all spins are aligned parallel with the $x$ axis in the positive direction [except for the second panel with $J_y=1.2$ where results are shown with initial spins both in the positive (+) and in the negative ($-$) $x$ directions]. The different phases are clearly visible: A paramagnetic phase with $M_x=0$ for $J_y=\gamma$, a ferromagnetic phase with $M_x\ne 0$ for $J_y=1.2\gamma$, and another paramagnetic phase for $J_y=2.5\gamma$ with $M_x=0$ on average. For $J_y=1.8\gamma$ the system exhibits an intermediate regime where the system switches between the different phases on a relatively large time scale (note the different scales for the time).

Figure 2

The spin structure factor $S_{\mathrm{SS}}^{xx}\left(0\right)$ (6) for finite two-dimensional lattices as a function of the coupling $J_y$ (in units of $\gamma$). The full line is the single-site MF result (7) which predicts a nonzero value for $J_y\gtrsim 1.04\gamma$, corresponding to a ferromagnetic phase. The markers are the results from the Gutzwiller Monte Carlo approach obtained by averaging a single trajectory over a total time of $10000/\gamma$ for various system sizes (as indicated in the figure) and $40000/\gamma$ in the intermediate regime for the two largest lattices. This reveals a transition to a paramagnetic phase with $S_{\mathrm{SS}}^{xx}\left(0\right)=0$ for large $J_y$ which as the system size is increased becomes sharper with a shrinking ferromagnetic region.

Figure 3

The correlation function $\langle {\widehat{\sigma }}_{\mathbf{i}}^{\left(x\right)}{\widehat{\sigma }}_{\mathbf{j}}^{\left(x\right)}\rangle$ as a function of the distance $r=|\mathbf{i}-\mathbf{j}|$ for a $12\times 12$ lattice and for $J_y=1.2\gamma$ (circles), $1.7\gamma$ (squares), and $2\gamma$ (triangles). The results are obtained with the Gutzwiller Monte Carlo approach by averaging a single trajectory over a total time of $20000/\gamma$. For $J_y=1.2\gamma$ the correlation function converges to a finite value as expected in the ferromagnetic phase. For $J_y=1.7\gamma$ the correlation function decays and becomes negative at relatively long distances, corresponding to a long-scale antiferromagnetic ordering. For longer distances the correlation function is expected to eventually decay to zero as observed for $J_y=2\gamma$. The smaller symbols are the result from the cluster mean-field approach for $J_y=1.2\gamma$ and $1.7\gamma$ [28]. The inset shows the correlation function as a function of the distance in the $x$ and $y$ directions for $J_y=1.7\gamma$.