Dissipative quantum North-East-Center model: steady-state phase diagram, universality and nonergodic dynamics

We thank the Referee for this comment and for the opportunity of clarifying how our work differentiates from the one of Ref[45] (we assume the Referee was referring to this work, since Ref[42] discusses the bistability of the classical NEC).

The quest for open quantum systems that hosts genuine bistable behaviour is an active research area. In the context of open quantum lattice systems such behaviour has been found in several 2D models where, according to the initial conditions, the system dynamically approaches a different many-body steady state with distinct physical properties.

The main problem of these findings is that they unavoidably come with some mean-filed-like approximation of the full open many-body dynamics since this phenomenology can only emerge for 2D systems that, in general, do not allow for an analytical solution or are amenable for exact numerical simulations (as in D=1, thanks to tensor network techniques).

Since it is known that mean-field approximation can “artificially” lead to multiple stable solutions it is important to include and analyze systematically the effect of quantum fluctuations at increasing distances.

In Ref.[45] the authors use a variational method where the number of sites included in the ansatz is fixed (a single L-shaped plaquette made of three sites). On one hand this approach allows to compute analytically the variational norm that needs to be minimized in the steady state. On the other hand it does not allow to scale up the number of physical sites included in the treatment.

In this sense cluster mean-field tackles this issue directly and aims at a systematic way to include more and more sites to the ansatz in order to study the shift (or stationarity) of the phase boundaries. In this case it confirms the presence of a finite bistable region. This is not always the case, see e.g. Ref.[53].

Another important difference with respect to Ref. [45] is that we employ an non-translationally invariant version of our ansatz that allows us to study the real-time bubble dynamics in the quantum model. From these studies we deduced important information about the physical mechanism responsible for the bubble reabsorbtion due to quantum and thermal fluctuation.

For these reasons we believe that our study represents an important step forward in the quantitative study of bistable physics in open lattice systems and can be used as a reliable benchmark for future studies in kinetically constrained dissipative models.

We thank the Referee for their careful reading of our manuscript and for their comprehensive summary of our results. We appreciate the positive assessment of our work, and have updated our manuscript as suggested by including a more thorough analysis of the cluster mean field approximation. We agree with the Referee that these updates make our work stronger. We have now performed a more thorough analysis of the validity of the CMF approximation, adding the 4x4 cluster obtained through trajectories. As one can observe in the updated Fig.5, the phase boundaries obtained at $\ell=4$ are consistent with the ones for $\ell=2,3$, within error bars due to finite trajectories sampling (we attach a version of this figure to the reply for the Referee's convenience). As the nature of the transition at $h=0$ is hard to capture with the approximate CMF ansatz, and it is not crucial for the results shown in this work, we modified the text to de-emphasize it. In the revised version of the manuscript, we have further added a more insightful discussion of the consequences of our stability analysis. In particular, we now highlight that instability close to the critical point indicates that larger clusters are needed to quantitatively capture the critical point, as expected due to the build-up of long range correlations close to criticality. Additionally, we discuss that deep in the bistable and normal phases the steady state is stable to fluctuations at all wave vectors, suggesting that the results obtained through cluster mean field are reliable. Finally, we mention the fact that the largest eigenvalue $\mu_k = 0$ in the bistable phase is a consequence of bistability itself.

Regarding the Referee's last point on the stability wrt changes in the parameters: in the present work, we decided to fix some of the parameters, in order to simplify the analysis of our results. While drastic changes of the fixed parameters may qualitatively change our results (e.g. taking dramatically different values of gamma for the various dissipative processes, or setting one of the Omegas in Eqs.(11,12) to zero), we do believe that our conclusions are stable with respect to milder deviations from the current setting. A deeper investigation of the phase diagram beyond the fixed parameters chosen in this work represents an intriguing direction for future work.

Below, we address the requested changes point by point:

1) In the revised version of the manuscript we give a precise definition of the plaquette symmetry.

2) We thank the Referee for the request of clarification. While it is true that $\sigma_{j}^+ P_{plaq_j}^{\uparrow} = P_{j+e_x}^{\uparrow}P_{j+e_y}^{\uparrow}\sigma_j^+$ and similarly for the down majority projector, the operator projecting into the set of states with majority up or down in the plaquette is, by itself, more complicated. Their expression is needed for generality and for the definition of the $\bar{\nu}$ and $\bar{\mu}$ jump operators. As reported in Eq.(4) and (5) these are given by $P_{plaq_j}^{\uparrow} = ¼ (2 + \sigma_j^z + \sigma_{j+ex}^z + \sigma_{j+ey}^z - \sigma_j^z\sigma_{j+ex}^z \sigma_{j+ey}^z )$ and similarly for the down projector. As such, they include terms acting on all sites included in the plaquette. They define the unique combination of sigma^z acting on the plaquette and selecting up- and down-majority states in the plaquette. The symbol ∈j is actually the usual ∈, it was just too close to the j coming from the definition of the plaquette symbol. We have now increased the separation between the two to avoid any confusion.

3) We thank the Referee for spotting this imprecision. We have updated the figure with the correct indices.

4) We thank the Referee for the request of clarification. Eq.(22) is derived from the data, and from the fit of the phase boundary. We now highlight this in the main text.

5) The Referee is right, Fig. 8(b) doesn’t show results regarding the scaling with h. Accordingly, we have removed the reference to the bias from the caption. Notice that in Eq.(31) the linear dependence on h is inherited from the classical equation for the velocity in the NEC model.

6) We agree with the Referee that the description of the stability analysis may deserve a deeper and more clear derivation. We have now added an appendix where all the steps are derived in detail. The values attainable by $k_x$ and $k_y$ were mentioned after Eq.(28), we now further provide this information below Eq.(25).

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We thank the Referee for their careful reading of our manuscript and for evaluating our work as “solid” and “suitable for publication of SciPost Physics Core”. The Referee argues that the model “appears a bit artificial” and as such its study does not justify a highlighted article. While we agree that the model is certainly not standard, we would like to argue for its relevance as a paradigmatic model for the study of bistability in 2D open quantum many-body systems. In particular, the structure of the plaquette for the majority vote highlights the importance of chirality, i.e. lack of inversion symmetry, for the emergence of non-trivial phenomena such as bistability and the non-ergodic dynamics we observe.

Below, we reply to the comments of the Referee point by point.

1) The cluster mean-field (CMF) approach relies on the factorization of the system density matrix over clusters of variable size, typically square-shaped plaquettes made of \ell\times\ell sites in the case of 2D lattices. Such a method, originally proposed in Phys. Rev. X 6, 031011 (2016) for dissipative spin systems, does not have a “small parameter” but instead relies on the fact that quantum correlations within the cluster are treated exactly while correlations between nearest-neighbour clusters are treated at a mean-field level. The convergence of such technique is ensured by scaling the size of the cluster until possible (typically $\ell=4$), effectively including systematically the relevant correlation needed to describe the phases. Few remarks are needed. (a) this method has been widely used to analyze the phase diagram of D-dimensional (driven-)dissipative lattices. What is often found is that already a cluster of $\ell=2$ is able to capture the qualitative structure of the steady-state phase diagram. Larger clusters are needed for a quantitative location of the boundaries. (b) It is usually very difficult to benchmark against other approaches since 2D open lattices are intrinsically difficult to simulate. However, in this and other cases alternative variational methods have been devised (see Refs.[44,45] ) which confirms the structure of the phase diagram. In the revised version of the manuscript we provide a further check of the CMF ansatz by studying a cluster of size $\ell=4$, showing the stability of the phase boundaries.

2) Eq. (22) is derived from phenomenological observations of the phase boundaries obtained numerically. In the revised version of the manuscript we now state clearly the origin of this expression.

3) The referee is right in their observation. The CMF approach always gives mean-filed-like exponents in the proximity of a criticality. However, the progressive inclusion of quantum correlations among sites within the cluster allows for the proper location of the phase boundary as the size of the cluster is increased. Also, the dissipative dynamics is such that correlations decrease very fast with the distance and are thus short-range in nature.

4) In the revised version of the manuscript we changed Fig. 5 to a 2x2 pattern and changed the colors to make them more visible.

5) We thank the Referee for this comment. We have changed the caption of Fig. 6 accordingly.

6) We thank the Referee for their question. The stability analysis studies the fate of small fluctuations on top of the steady-state, calculated with a given cluster size. Instabilities indicate that close to the critical point the system is very susceptible to inhomogeneities with a given wavevector (note that the allowed periodicities must be larger than the cluster size since smaller ones are exactly accounted for in the ansatz). This is not an inconsistency (see again Phys. Rev. X 6, 031011 (2016), Fig. 13) but tells that the exact location of the boundary can be sensitive to the size of the cluster used to determine the steady state. Ideally, the next step in this analysis would be to use a cluster with a size able to encode the periodicities at which the system shows instabilities and see if such perturbations drive the emergence of a new steady state. As a matter of fact such sizes are not approachable numerically with any technique.

7) While typically the stability analysis would result in negative values of $\mu_k$, in our case the presence of bistability qualitatively changes this expectation. In Eq.(25) we write the state as fluctuations on top of the steady state. However, in the bistable phase there are two steady states, and one can consider a state $\rho^{(n)} = \rho^{(1)}_{ss} + \delta\rho^{(n)}$ = $\rho^{(1)}_{ss}$ +$ \rho^{(2)}_{ss}$, where $\rho^{(1)}_{ss}$ and $\rho^{(2)}_{ss}$ are the two steady states. As such, the state $\rho^{(n)}$ will be stationary itself. It then follows that the super-operator describing the dynamics of fluctuations on top of one steady state admits zero eigenvalues, corresponding to the situation described above. We have added a brief discussion of this issue at the end of Sec.5.

8) We agree that it is trivial that in the linear regime the effect of the two perturbations is additive. However, what is not trivial is the fact that such a linear regime exists (both for thermal and quantum fluctuations). A priori, the first corrections to the radial velocity could also be quadratic.

9) We thank the Referee for their comment. In panel (a) of Fig.8 the curve for $\Omega=0.2$ shows this behavior, with an initial decrease as the size of the island is increased. To increase clarity, we now refer the reader to panel (a) in the text when discussing this matter.

10) This feature is not captured by the stability analysis that, deep enough both in the normal and bistable phase, signals stability. However, this consideration is corroborated by Fig.7: in the bistable phase (left panels) the minority islands disappear in favor of the majority background from the top right corner and following the diagonal to the bottom left corner. Such mechanism is quantitatively validated by Eq.(31) and the subsequent analysis. In the normal phase (Fig. 7, right panels) the minority island melts with the background. The boundary of the island is stable (in position) and the entirety of the structure progressively disappears. We stress that this behavior is present in the classical NEC model (see Ref.[42] and Eq.(30)). Here we show its persistence in the presence of quantum fluctuations determining quantitatively the reabsorptions coefficients as in Eq.(32).

11) We agree with the Referee that our model shows no indication of fragmentation. Our comment in the discussion was meant more as a possible outlook for future work “Establishing such a connection between bistability and fragmentation could therefore be a very promising direction”. The fact that we do not observe in the present stage signatures of fragmentation doesn’t imply that there are none, and we stand by our statement that investigating this direction would be extremely interesting. In the revised version of the manuscript, we stressed further that the connection with fragmentation is a speculative one, originating from similar behaviors observed in closed systems.