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

This work studies the dissipative quantum North-East-Center (NEC) model, a two-dimensional kinetically constrained system with chiral majority-rule dissipation, which does not satisfy detailed balance. In the classical case this is bistable, even in the presence of bias and noise. Here, the authors study a quantum version of this model and establish the survival of a bistable phase against both coherent (Hamiltonian) and incoherent perturbations. They further analyze nonergodic dynamics through the fate of minority spin islands, showing universal reabsorption mechanisms with implications for error correction. To do so they use a cluster mean-field approach and its inhomogeneous extension.

The work is technically sound, and of clear interest to the SciPost Physics community, both for the result and for the technical approach used.

I am comfortable recommending that this paper should be published in one of the SciPost journals. I am less comfortable deciding whether it should be published in SciPost Physics or SciPost Physics Core. The reason is that there is previous work, namely Ref 42, which demonstrates that the quantum model also has a bistable phase, just like the classical one. The manuscript here has more results, but I find that this is the most interesting one; so if it already appears in previous work, then this somewhat dulls the impact of this work.

I would invite the authors to contrast their results with those of Ref 42, clarifying what advances they make and why this merits publication in SciPost Physics (rather than Core).

Ask for major revision

1-new results regarding the interplay of constraints, dissipation and coherent terms in quantum dynamics;
2-generalizes previous findings regarding quantum noise in the model;
3-the robustness results of the bistable phase could have implication for error correction.

1-the analysis of the numerical approximation (errors, reliability of conclusions) is not sufficiently detailed;
2-the complex notation makes it difficult to follow at some points

The authors investigate the interplay of constraints, coherence and dissipation in a constrained dissipative model on a 2D square lattice. The starting point is the North-East model, described by a Lindblad master equation with constrained jump operators that implement a North-East majority voting. This dissipative model is known to exhibit a bistable phase with two stationary states, which survives in the presence of classical noise.
Also for quantum fluctuations (namely adding a coherent Hamiltonian term to the dissipative model) it was shown in Ref. [45] that the phase could persist. In that previous study, the Hamiltonian had the same geometric structure as the dissipative constraints. In this study, the authors want to explore more systematically such robustness to quantum fluctuations for Hamiltonians with different structures. To this end, three types of Hamiltonians are analyzed, with or without constraints, which may also preserve or break the asymmetric structure of the dissipative terms.

The main conclusion is that the bistable phase can persist over a finite region of the parameter space, independently of the microscopic details of the coherent Hamiltonian, while the extension of the phase depends on the parameters of the model. Additionally, the dynamics is simulated for initial states with an island of “wrong” magnetization, and it is shown that in the bistable region, they are always absorbed by the majority background.

The results are original, and extend the previous understanding of the NEC dissipative model. The bistability of the model, moreover, makes it interesting as potential memory, and thus the robustness of the bistable phase and the dynamics of defects, such as islands of different magnetization, is a relevant question. More generally, the study contributes to exploring the rich dynamics of constrained models and the competition of classical and quantum noise. As such, the work merits publication, but since the conclusions are based on a numerical study, a more careful discussion of the potential numerical effects and the robustness of the convergence analysis would make the paper stronger.

The analysis is done numerically, using a non translationally invariant generalization of the cluster mean field approach, which approximates the state of the system as a tensor product of small clusters and thus discards quantum correlations beyond some short-range. Thus, I miss a more careful discussion of the convergence of the method, or the robustness of the conclusions. To check their validity, the paper compares plots for clusters of sizes 2x2 and 3x3 in a few particular cases. Even though the results in the plots look similar, this is far from demonstrating convergence. In particular, to conclude, as the text states, that at h=0 the phase transition is continuous, while at other points is first order, would require a more careful analysis (or at least discussion, if that’s not necessary) of how the CMF approximation, which cuts all correlations beyond a short range, would affect this behaviour.

Another analysis of convergence is done in Sec. 5 by introducing non-translationally-invariant perturbations on top of the steady state found by CMF, and evolving them in time to probe for instability. The conclusion is that there is instability around the critical point, but there is almost no discussion about how this affects the reliability of the previous conclusions, or whether this effect is physical or an artifact of the method.

Finally, the analysis considers some simplifications of the parameters (e.g. fixing a single $gamma$). How relevant are such steps for the conclusions?

i) There seems to be no precise definition of the term “plaquette chiral symmetry”;

ii) Some of the notation is confusing. In particular, the definition (1), including three vertices, seems to contradict the projectors (4,5), which look like the product over two vertices. Moreover, one of these two should have all positive signs as a product of (1+sigma_z) factors.
And the symbol $\in_j$ is not properly defined

iii) in Fig.1c, the up and down subindices of the last part of the figure seem to be reversed.

iv) How is equation (22) determined? Is it derived/fitted from the data?

v) The caption of Fig. 8(b) states that the absorption velocity increases monotonically with h, but all the data seem to be at constant h (bias). How is the trend determined?

vi) The description of the procedure to analyze the stability in Sec. 5 is not very clear. For instance, in (25) it is not explained which values k_x and k_y can take, or in (26) it is not mentioned that corner terms are discarded (these things are only mentioned much later), or how the trace squared term in the last line of (26) appears (and why not something like $tr(l_{j+e_q}^{\dagger l}_{j+e_q} \rho_{n+e_q})$ . It would be useful to show a more clear derivation, maybe in an appendix, with all necessary steps to arrive to this form.

Ask for minor revision

The authors investigate the dissipative quantum North-East-Center model
by a cluster mean-field approach. They determine steady states
and how the system approaches them. In particular, they find two distinct
phases which they examine for their stability. The combined
effects of thermal and quantum fluctuations are analyzed.

The model is motivated as an extension of a classical model to the
quantum realm. It is claimed that it has relevance for quantum error
correction, but the study is only concerned with phases so that I see some
relevance for conventional error correction, but not for QEC.
Although some reference to experiments is made there is no relevance of
the model for any experiment nor for a universal behavior. Thus, the model
appears a bit artificial.

This is decent work of which the presentation can be improved
in some points.

All in all, I recommend to improve the presentation along the lines
indicated above. Then, this solid piece of work is suitable for
publication of SciPost Physics Core because neither the model nor
the approximate approach stand out to justify a highlighted article.

1)
What is the small parameter of the cluster mean-field method?
Is there any independent check by alternative methods?
I find it difficult to trust this approach only by stating
that a larger cluster does not change the results visibly.

2)
Eq. (22) appears out of nowhere. Why should the critical field
behave like this?

3)
Since no fluctuations between the clusters are incorporated
I wonder whether one can expect any exponent to be different from
a mean-field value. Why are fluctuations so unimportant?
Is it because damping is dominating?

4)
In Fig. 5, the lines are not discernible and the colors partially
too light. Please arrange the panels in a 2x2 pattern to increase
their size.

5)
Please denote all parameters used in Fig. 6 so that the reader
does not need to look around.

6)
Isn't it an inconsistency that the mu_k values become positive
before the dashed vertical lines (please explain them in the caption)
are reached in Fig. 6?

7)
If the system converges to some steady-state this should appear in
the stability analysis as negative real values of mu_k.
Why does this not happen?

8)
It is mathematically trivial that in the linear regime the effects of
h and of T are additive, see multivariate Taylor expansions, in Eq. (31).

9)
At the end of Sect. 6, the statement "tau gains an inverse proportionality
to the size of the island." appears. Where can this be seen in the
depicted data?

10)
In the Conclusion one finds "... a bulk mechanism is responsible for the
relaxation to the (unique) steady-state." at the end of the first
paragraph on page 16.
Why does this not appear in the stability analysis?
Or is the claim not warranted?

11)
From the shown data and the discussion I do not see evidence for
Hilbert space "fragmentation". This appears to me to be overselling
the relevance of the results.

Accept in alternative Journal (see Report)