Cluster Mean-Field Approach to the Steady-State Phase Diagram of Dissipative Spin Systems

We show that short-range correlations have a dramatic impact on the steady-state phase diagram of quantum driven-dissipative systems. This effect, never observed in equilibrium, follows from the fact that ordering in the steady state is of dynamical origin, and is established only at very long times, whereas in thermodynamic equilibrium it arises from the properties of the (free) energy. To this end, by combining the cluster methods extensively used in equilibrium phase transitions to quantum trajectories and tensor-network techniques, we extend them to nonequilibrium phase transitions in dissipative many-body systems. We analyze in detail a model of spin-$1/2$ on a lattice interacting through an $XYZ$ Hamiltonian, each of them coupled to an independent environment that induces incoherent spin flips. In the steady-state phase diagram derived from our cluster approach, the location of the phase boundaries and even its topology radically change, introducing reentrance of the paramagnetic phase as compared to the single-site mean field where correlations are neglected. Furthermore, a stability analysis of the cluster mean field indicates a susceptibility towards a possible incommensurate ordering, not present if short-range correlations are ignored.

Popular Summary

A phase transition is a transformation between two different states of matter that often involves a symmetry-breaking process (e.g., that between liquid water and ice). The theory of phase transitions in equilibrium systems is one of the triumphs of 20th century science, showing that apparently disparate physical systems undergo transitions in the same way; the details on small length scales do not matter. Phase transitions can also occur in an out-of-equilibrium context. Famous examples include systems of moving cars into traffic jams or individual flying birds exhibiting collective flocking. All of these situations are related to each other by the fact that the appearance of different steady-state ordering is of intimate dynamical origin and cannot be reduced to the equilibrium results. Here, we study a quantum dynamical phase transition.

We demonstrate that, contrary to what is observed in standard thermodynamics, short-range fluctuations drastically affect the steady-state phase diagram of driven-dissipative quantum systems. We focus on a magnetic system of spin-1/2 particles located on a two-dimensional square lattice. In our theoretical setup, spin-flip transitions may occur. Using extensive cluster mean-field calculations, we reveal the crucial importance of the interplay between dissipation and short-range fluctuations, which modify the phase-diagram topology. Our results are amenable to experimental verification in the near future using novel quantum-simulation platforms as trapped ions, highly excited “Rydberg” states of ultracold atoms, or arrays of coupled optical or microwave cavities. Our findings also have implications for the use of such physical systems in quantum computing.

We expect that our results will stimulate novel investigations of nonequilibrium critical phenomena, placing them in a completely different light compared with the paradigm for equilibrium systems.

Figure 1

A sketch of the phase diagram of the model defined by Eqs. (2) and (3), for $J_z=1$. The single-site mean field as worked out in Ref. [19] would predict the emergence of different phases: paramagnetic (PM), ferromagnetic (FM), antiferromagnetic (AFM), and spin-density-wave (SDW) phases (inset to the top panel). Here, we focus on the region highlighted by the red box, which displays a transition from PM to FM states (magnified in the top panel). A proper inclusion of short-range correlations (through the cluster mean field) shrinks the ferromagnetic region to a small “island,” thus suppressing the order at large couplings and hinting at a possible incommensurate (inc) ordering (bottom panel).

Figure 2

Sketch of the cluster mean-field approach in a dissipative system of interacting spin-$1/2$ particles. The figure refers to a $2\times 2$ cluster on a two-dimensional square lattice.

Figure 3

Quantum trajectories combined with the mean-field or cluster mean-field method. Colored boxes along a given line stand for the time-evolved state of the $k$th trajectory, which is stochastically chosen among the set of pure states $\{{|{\psi }_0(t)⟩}_k,{|{\psi }_j(t)⟩}_k\}$ according to Eq. (9). For each of those states, one finds the corresponding mean fields on each site $j$ inside the considered cluster, $⟨{\widehat{\mathbf{\sigma }}}_j(t){⟩}_k$. The mean field ${\mathbf{B}}_j^{\mathrm{eff}}(t)$ parametrizing the effective Hamiltonian in Eq. (13), to be used in the master equation for the cluster density matrix, is obtained by averaging over all the $N$ trajectories.

Figure 4

One-dimensional TEBD scheme for 1D systems with open boundaries, combined with the cluster mean-field method. Circles denote the sites of the lattice. The many-body state corresponding to the OBC cluster made up of $L$ black circles inside the orange box is written in a MPO representation and evolved in time with the TEBD scheme. The cluster is coupled to the rest of the system (gray circles) through the mean field at the edges. At regular small time intervals, the mean fields ${\mathbf{B}}_1^{\mathrm{eff}}(t)=\mathrm{Tr}[{\widehat{\mathbf{\sigma }}}_L\rho (t)]$ on the leftmost site and ${\mathbf{B}}_L^{\mathrm{eff}}(t)=\mathrm{Tr}[{\widehat{\mathbf{\sigma }}}_1\rho (t)]$ on the rightmost site are self-consistently evaluated and used to construct the Hamiltonian for the next TEBD iteration.

Figure 5

Ferromagnetic spin-structure factor along the $x$ direction, in a 1D setup, as a function of $J_y$. The various curves and symbols stand for different system sizes from $L=6$ to $L=16$, as indicated in the plot. The two arrows point at the positions of the two peaks ($J_y=0.4$ and $J_y=1.3$), for which we provide a finite-size scaling (Fig. 6) and an analysis of the two-point correlation functions (Fig. 7). Here, we have set $J_x=0.9$ and $J_z=1$, and we work in units of $\gamma$. Note that for $J_y=0.9$ and $J_y=1$, the spin structure factor is rigorously zero.

Figure 6

Scaling of $S_{\mathrm{SS}}^{xx}(0)$ as a function of the inverse system size $L$, for two values of $J_y$ in proximity of the peaks (see the arrows in Fig. 5). The symbols denote the numerical data, while the continuous lines are power-law fits performed for the data points to the left of the vertical dashed line. The black sets correspond to those of Fig. 5, obtained by simulating a small system with RK and QT approaches. The blue sets have been obtained with MPO simulations, where the CMF has been applied to the two edges. The other parameters are set as in Fig. 5.

Figure 7

Spatial decay of the correlation functions $⟨{\sigma }_j^x{\sigma }_{j+r}^x{⟩}_{\mathrm{SS}}$ with the distance $r$. Correlators have been chosen in a symmetric way with respect to the center of the chain. In the upper panel, $J_y=0.4$ (left peak in Fig. 5), while in the lower panel, $J_y=1.3$ (right peak in Fig. 5). The various data sets correspond to different system sizes: Results for $L=12$ have been obtained for systems with PBC by means of RK integration or the QT approach to the master equation; those for $L=24$ are, with MPO, used to simulate OBC and combined with the CMF; the thermodynamic limit $L\to \infty$ (diamonds-solid blue lines) has been addressed with a translationally invariant i-MPO method. The other parameter values are set as in Fig. 5.

Figure 8

Two-dimensional cluster mean-field phase diagram in the $J_x-J_y$ plane (with fixed $J_z=1$). The single-site MF ($1\times 1$) predicts a ferromagnetic steady state in the top left region with respect to the black curve. The extent of this region appears to be very fragile to a more accurate cluster mean-field treatment. At the $2\times 2$ level (red squares), the boundaries of the two phases are slightly deformed, while with a $3\times 3$ analysis, the FM phase shrinks down to a region of finite size (blue circles). The darkest color filling indicates the region that is PM in all simulations, while the lightest indicates that which is FM in all cases. The FM region shrinks with increasing cluster size, as indicated by the varying shades of color but, as discussed in the text, appears to converge with increasing cluster size. The five arrows denote the cuts along different values of $J_x$, which will be analyzed in detail in Figs. 9 and 10.

Figure 9

Cluster mean-field analysis of the ferromagnetic order parameter in two dimensions, for four different vertical cuts of Fig. 8, at constant $J_x$ (the corresponding values of $J_x$ are indicated in the various panels). The various data sets denote different sizes of the clusters, up to $\ell =3$.

Figure 10

Same analysis as in Fig. 9, but for $J_x=0.9$. In the inset, we show the rescaled data close to the left phase transition; the straight dashed line denotes a square-root behavior as in Eq. (22) and is plotted as a guide to the eye.

Figure 11

The two-point $xx$ (left panel) and $yy$ (right panel) correlations as a function of the distance $r$ in a two-dimensional square-lattice geometry. The calculations have been performed on a square lattice of size $\ell =4$, while the point $j$ has been chosen to be at one of the corners of the more-square cluster. The three sets of data refer to different values of $J_y$ according to the legend: two inside the PM phase ($J_y=1$ and $J_y=1.7$) and one inside the FM phase ($J_y=1.2$). We fixed $J_x=0.9$, $J_z=1$.

Figure 12

Magnification of the $2\times 2$ CMF analysis for $J_x=0.9$, close to $J_y=2$ (see the red data in Fig. 10). The two sets are calculated by sweeping both from upward and downward $J_y$ values, where the initial conditions for each point are based on the previous one, with a small offset. This evidences the presence of a first-order jump within the ordered phase for $J_y\approx 1.985$ (green arrow), followed by a second-order transition to the normal state, for $J_y^{c(\text{right})}\approx 2.04$.

Figure 13

Upper panel: Real part of the most unstable eigenvalue (negative is stable) as a function of $J_y$ and $k_x$ (for $k_y=0$). The parameters are set as in Fig. 10. Lower panel: High-resolution plot with the same range of $J_y$ as in Fig. 12, corresponding to upward trace.

Figure 14

Cluster mean-field analysis of the ferromagnetic order parameter for a system defined on a square lattice geometry in different dimensionalities. The upper panel refers to 2D ($z=4$), the middle one to 3D ($z=6$), and the lower one to 4D ($z=8$). Dashed lines denote the MF predictions, while symbols and continuous curves are the results of numerical simulations, with clusters of size $L=2^D$ ($D$ being the system dimensionality).

Figure 15

Magnetization along the $x$ (solid black line) and $y$ (dashed black line) directions and purity $\mathcal{P}=\mathrm{Tr}[{\rho }_{\mathrm{SS}}^2]$ (solid red line) as a function of $J_y$, for fixed $J_x=0.9$, $J_z=1$. The single-site MF ($1\times 1$, top panel) predicts a PM phase for $J_y<J_y^c$ and a FM phase for $J_y>J_y^c$. On the contrary, according to a CMF analysis ($2\times 1$, lower panel), the FM phase shrinks down to a finite region $J_y^c=J_y^{c(\mathrm{left})}<J_y<J_y^{c(\text{right})}$ allowing the reentrance of a PM phase for $J_y>J_y^{c(\text{right})}$.